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Geometry of the eigencurve at critical Eisenstein series of weight 2

2015-01-01
Dipramit Majumdar
In this paper we show that the critical Eisenstein series of weight 2, E 2 crit p , defines a smooth point in the eigencurve ℂ(l), where l is a … In this paper we show that the critical Eisenstein series of weight 2, E 2 crit p , defines a smooth point in the eigencurve ℂ(l), where l is a prime different from p. We also show that E 2 crit p ,ord l defines a smooth point in the full eigencurve ℂ full (l) and E 2 crit p ,ord l 1 ,ord l 2 defines a non-smooth point in the full eigencurve ℂ full (l 1 l 2 ). Further, we show that ℂ(l) is étale over the weight space at the point defined by E 2 crit p . As a consequence, we show that level lowering conjecture of Paulin fails to hold at E 2 crit p ,ord l .
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Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

2017-01-01
Somnath Jha , Dipramit Majumdar
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension … We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.
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Cyclic cubic extensions of ℚ

2022-04-25
Dipramit Majumdar , B. Sury
We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The … We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The proof, although involved, is elementary and one can parametrize all these polynomials explicitly. As an accidental by-product of the results, we prove that infinitely many primes congruent to [Formula: see text] or [Formula: see text] mod [Formula: see text] are sums of two rational cubes - thereby, giving the first unconditional result on a classical open problem.
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Geometry of the eigencurve at critical Eisenstein series of weight 2

2013-07-18
Dipramit Majumdar
In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} … In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.

Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

2015-04-12
Somnath Jha , Dipramit Majumdar
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension … We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.

l-Class groups of cyclic extensions of prime degree l

2014-01-01
Manisha Kulkarni , Dipramit Majumdar , B. Sury
Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow … Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we obtain bounds for the F_ rank of the l-Sylow subgroup of K using genus theory. We obtain some results valid for general l. Following that, we obtain more complete results for l=5 and F =Q(ζ_5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (the latter is under GRH). We obtain explicit results in several cases. Using these results, and duality theory, we deduce results on the 5-class numbers of fields of the form Q(n^1/5).

Fruit Diophantine Equation

2021-01-01
Dipramit Majumdar , B. Sury
We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the … We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral point.
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107.15 Fruit diophantine equation

2023-07-01
Dipramit Majumdar , B. Sury
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Ordinary families of automorphic forms on definite unitary groups

2016-06-21
Baskar Balasubramanyam , Dipramit Majumdar

Endoscopic transfer between eigenvarieties for definite unitary groups

2014-01-01
Dipramit Majumdar
In this paper, we extend the endoscopic transfer of definite unitary group U(n), which sends a pair of automorphic forms of U(m),U(n) to an automorphic form of U(m+n), to finite … In this paper, we extend the endoscopic transfer of definite unitary group U(n), which sends a pair of automorphic forms of U(m),U(n) to an automorphic form of U(m+n), to finite slope p-adic automorphic forms for definite unitary groups by constructing a rigid analytic map between eigenvarieties E_m \times E_n \to E_{m+n}, which at classical points interpolates endoscopic transfer map.

p-adic Asai transfer

2017-02-03
Baskar Balasubramanyam , Dipramit Majumdar
Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In … Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $GL_{2}/K$ to the universal eigenvariety of $GL_{4}/Q$, which at nice classical points interpolate this Asai transfer.

$p^r$-Selmer companion modular forms

2018-06-13
Somnath Jha , Dipramit Majumdar , Sudhanshu Shekhar
The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given … The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

A generalization of Mazur's theorem (Ogg's conjecture) for number fields

2018-01-01
Debargha Banerjee , Narasimha Kumar , Dipramit Majumdar
In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is … In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the congruence subgroups of the form $\Ga_0(N)$ for any $N \in \N$. This in turn help us to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups.

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:msup></mml:math>-Selmer companion modular forms

2021-04-15
Somnath Jha , Dipramit Majumdar , Sudhanshu Shekhar
The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given … The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves E 1 and E 2 over a number field K, Mazur–Rubin have defined them to be n-Selmer companion if for every quadratic character χ of K, the n-Selmer groups of E 1 χ and E 2 χ over K are isomorphic. Given a prime p, they have given sufficient conditions for two elliptic curves to be p r -Selmer companion in terms of mod-p r congruences between the curves. We discuss an analogue of this for Bloch–Kato p r -Selmer groups of modular forms. We compare the Bloch–Kato Selmer group of a modular form respectively with the Greenberg Selmer group when the modular form is p-ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary at p. We also indicate the relation between our results and the well-known congruence results for the special values of the corresponding L-functions due to Vatsal.
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Cyclic Cubic Extensions of Q.

2021-07-19
Dipramit Majumdar , B. Sury
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of … In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1.

Towards the Ribet's conjecture for non-rational Eisenstein maximal ideals

2021-11-15
Debargha Banerjee , Narasimha Kumar , Dipramit Majumdar
According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form X_0(N) for a prime number N. There … According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form X_0(N) for a prime number N. There is a recent interest to generalize the conjecture for arbitrary N by Ribet, Ohta and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are ``cuspidal. Yoo proved the conjecture (under certain hypothesis) provided that those ideals are rational. In this article, we show that under certain hypothesis, Ribet's conjecture is true for non-rational Eisenstein maximal ideals.

On characteristic ideal of Selmer group associated to Artin representations

2022-01-01
Dipramit Majumdar , Subhasis Panda
Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a … Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a totally complex field. We establish an algebraic function of the characteristic ideal of the Selmer group associated to Artin representation over the cyclotomic $\Z_p$- extension of the rational numbers under certain mild hypotheses and construct several examples to illustrate our result. We also prove that in this situation $\mu$-invariant of the dual Selmer group is independent of the choice of the lattice.
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$3$-Selmer group, ideal class groups and cube sum problem

2022-01-01
Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar
Given an elliptic curve $E$ over a number field $F$ and an isogeny $\varphi$ of $E$ defined over $F$, the study of the $\varphi$-Selmer group has a rich history going … Given an elliptic curve $E$ over a number field $F$ and an isogeny $\varphi$ of $E$ defined over $F$, the study of the $\varphi$-Selmer group has a rich history going back to the works of Cassels and the recent works of Bhargava et al. and Chao Li. Let $E/\mathbb Q$ be an elliptic curve with a rational $3$-isogeny. In this article, we give an upper bound and a lower bound of the rank of the Selmer group of $E$ over $\mathbb Q(\zeta_3)$ induced by the $3$-isogeny in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(\zeta_3)$. Using our bounds on the Selmer groups, we prove some cases of Sylvester's conjecture on the rational cube sum problem and also exhibit infinitely many elliptic curves of arbitrary large $3$-Selmer rank over $\mathbb Q(\zeta_3)$. Our method also produces infinitely many imaginary quadratic fields and biquadratic fields with non-trivial $3$-class groups.
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Ribet's conjecture for Eisenstein maximal ideals

2021-01-01
Debargha Banerjee , Narasimha Kumar , Dipramit Majumdar
According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. … According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. There is a recent interest to generalize the conjecture for arbitrary $N$ by Ribet, Ohta and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are "cuspidal". Hwajong Yoo proved the conjecture ( under certain hypothesis) provided that those ideals are {\it rational}. In this article, we show that ( under certain hypothesis), Ribet's conjecture is true for {\it non-rational} Eisenstein maximal ideals.

Geometry of the eigencurve at critical Eisenstein series of weight 2

2013-01-01
Dipramit Majumdar
In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} … In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.
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p-adic Asai transfer

2017-01-01
Baskar Balasubramanyam , Dipramit Majumdar
Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In … Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $GL_{2}/K$ to the universal eigenvariety of $GL_{4}/Q$, which at nice classical points interpolate this Asai transfer.

$p^r$-Selmer companion modular forms

2018-01-01
Somnath Jha , Dipramit Majumdar , Sudhanshu Shekhar
The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given … The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.
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Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

2015-01-01
Somnath Jha , Dipramit Majumdar
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension … We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.
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Cube sum problem for integers having exactly two distinct prime factors

2022-01-01
Dipramit Majumdar , Pratiksha Shingavekar
Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several … Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satg\'e, Selmer etc. and up to the recent works of Alp\"oge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.

Binary Cubic Forms and Rational Cube Sum Problem

2023-01-01
Somnath Jha , Dipramit Majumdar , B. Sury
The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Selmer, Satg{\'e}, … The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Selmer, Satg{\'e}, {Lieman} etc. and up to the recent work of Alp{\"o}ge-Bhargava-Shnidman. In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes.
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Cube sum problem for integers having exactly two distinct prime factors

2023-11-30
Dipramit Majumdar , Pratiksha Shingavekar

Infinite families of congruences modulo $2$ for $(\ell, k)$-regular partitions

2024-02-15
T Kathiravan , Usha K. Sangale , Dipramit Majumdar
Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and … Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and modulo $12$ for $(4, 9)$-regular partition has been studied. In this paper, we use theta function identities and Newman results to prove some infinite families of congruences modulo $2$ for $(2, 7)$, $(5, 8)$, $(4, 11)$-regular partition and modulo $4$ for $(4, 5)$-regular partition.
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Hilbert's 10th Problem via Mordell curves

2024-12-05
Somnath Jha , Debanjana Kundu , Dipramit Majumdar
We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free … We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(\zeta_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.
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Relative $p$-class groups and $p$-Selmer groups

2024-12-17
Debajyoti De , Dipramit Majumdar , Sudipa Mondal
Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction … Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group.
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$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

2025-02-03
Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar
We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on … We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.
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Hilbert’s 10<sup>th</sup> Problem via Mordell curves

2025-02-26
Somnath Jha , Debanjana Kundu , Dipramit Majumdar

3-Selmer groups, ideal class groups and the cube sum problem

2025-05-01
Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar

3-Selmer groups, ideal class groups and the cube sum problem

2025-05-01
Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar

Hilbert’s 10<sup>th</sup> Problem via Mordell curves

2025-02-26
Somnath Jha , Debanjana Kundu , Dipramit Majumdar

$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

2025-02-03
Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar
We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on … We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.
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Relative $p$-class groups and $p$-Selmer groups

2024-12-17
Debajyoti De , Dipramit Majumdar , Sudipa Mondal
Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction … Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group.
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Hilbert's 10th Problem via Mordell curves

2024-12-05
Somnath Jha , Debanjana Kundu , Dipramit Majumdar
We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free … We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(\zeta_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.
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Infinite families of congruences modulo $2$ for $(\ell, k)$-regular partitions

2024-02-15
T Kathiravan , Usha K. Sangale , Dipramit Majumdar
Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and … Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and modulo $12$ for $(4, 9)$-regular partition has been studied. In this paper, we use theta function identities and Newman results to prove some infinite families of congruences modulo $2$ for $(2, 7)$, $(5, 8)$, $(4, 11)$-regular partition and modulo $4$ for $(4, 5)$-regular partition.
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Cube sum problem for integers having exactly two distinct prime factors

2023-11-30
Dipramit Majumdar , Pratiksha Shingavekar

107.15 Fruit diophantine equation

2023-07-01
Dipramit Majumdar , B. Sury
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Binary Cubic Forms and Rational Cube Sum Problem

2023-01-01
Somnath Jha , Dipramit Majumdar , B. Sury
The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Selmer, Satg{\'e}, … The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Selmer, Satg{\'e}, {Lieman} etc. and up to the recent work of Alp{\"o}ge-Bhargava-Shnidman. In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes.
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Cyclic cubic extensions of ℚ

2022-04-25
Dipramit Majumdar , B. Sury
We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The … We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The proof, although involved, is elementary and one can parametrize all these polynomials explicitly. As an accidental by-product of the results, we prove that infinitely many primes congruent to [Formula: see text] or [Formula: see text] mod [Formula: see text] are sums of two rational cubes - thereby, giving the first unconditional result on a classical open problem.
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On characteristic ideal of Selmer group associated to Artin representations

2022-01-01
Dipramit Majumdar , Subhasis Panda
Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a … Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a totally complex field. We establish an algebraic function of the characteristic ideal of the Selmer group associated to Artin representation over the cyclotomic $\Z_p$- extension of the rational numbers under certain mild hypotheses and construct several examples to illustrate our result. We also prove that in this situation $\mu$-invariant of the dual Selmer group is independent of the choice of the lattice.
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$3$-Selmer group, ideal class groups and cube sum problem

2022-01-01
Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar
Given an elliptic curve $E$ over a number field $F$ and an isogeny $\varphi$ of $E$ defined over $F$, the study of the $\varphi$-Selmer group has a rich history going … Given an elliptic curve $E$ over a number field $F$ and an isogeny $\varphi$ of $E$ defined over $F$, the study of the $\varphi$-Selmer group has a rich history going back to the works of Cassels and the recent works of Bhargava et al. and Chao Li. Let $E/\mathbb Q$ be an elliptic curve with a rational $3$-isogeny. In this article, we give an upper bound and a lower bound of the rank of the Selmer group of $E$ over $\mathbb Q(\zeta_3)$ induced by the $3$-isogeny in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(\zeta_3)$. Using our bounds on the Selmer groups, we prove some cases of Sylvester's conjecture on the rational cube sum problem and also exhibit infinitely many elliptic curves of arbitrary large $3$-Selmer rank over $\mathbb Q(\zeta_3)$. Our method also produces infinitely many imaginary quadratic fields and biquadratic fields with non-trivial $3$-class groups.
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Cube sum problem for integers having exactly two distinct prime factors

2022-01-01
Dipramit Majumdar , Pratiksha Shingavekar
Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several … Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satg\'e, Selmer etc. and up to the recent works of Alp\"oge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.

Towards the Ribet's conjecture for non-rational Eisenstein maximal ideals

2021-11-15
Debargha Banerjee , Narasimha Kumar , Dipramit Majumdar
According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form X_0(N) for a prime number N. There … According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form X_0(N) for a prime number N. There is a recent interest to generalize the conjecture for arbitrary N by Ribet, Ohta and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are ``cuspidal. Yoo proved the conjecture (under certain hypothesis) provided that those ideals are rational. In this article, we show that under certain hypothesis, Ribet's conjecture is true for non-rational Eisenstein maximal ideals.

Cyclic Cubic Extensions of Q.

2021-07-19
Dipramit Majumdar , B. Sury
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of … In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1.

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:msup></mml:math>-Selmer companion modular forms

2021-04-15
Somnath Jha , Dipramit Majumdar , Sudhanshu Shekhar
The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given … The study of n-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves E 1 and E 2 over a number field K, Mazur–Rubin have defined them to be n-Selmer companion if for every quadratic character χ of K, the n-Selmer groups of E 1 χ and E 2 χ over K are isomorphic. Given a prime p, they have given sufficient conditions for two elliptic curves to be p r -Selmer companion in terms of mod-p r congruences between the curves. We discuss an analogue of this for Bloch–Kato p r -Selmer groups of modular forms. We compare the Bloch–Kato Selmer group of a modular form respectively with the Greenberg Selmer group when the modular form is p-ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary at p. We also indicate the relation between our results and the well-known congruence results for the special values of the corresponding L-functions due to Vatsal.
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Fruit Diophantine Equation

2021-01-01
Dipramit Majumdar , B. Sury
We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the … We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral point.
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Ribet's conjecture for Eisenstein maximal ideals

2021-01-01
Debargha Banerjee , Narasimha Kumar , Dipramit Majumdar
According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. … According to Ogg's conjecture (Mazur's Theorem), cuspidal subgroup coincides with rational torsion points of the Jacobian variety of modular curves of the form $X_0(N)$ for a {\it prime} number $N$. There is a recent interest to generalize the conjecture for arbitrary $N$ by Ribet, Ohta and Yoo. In this direction, Ribet conjectured that all the Eisenstein maximal ideals are "cuspidal". Hwajong Yoo proved the conjecture ( under certain hypothesis) provided that those ideals are {\it rational}. In this article, we show that ( under certain hypothesis), Ribet's conjecture is true for {\it non-rational} Eisenstein maximal ideals.

$p^r$-Selmer companion modular forms

2018-06-13
Somnath Jha , Dipramit Majumdar , Sudhanshu Shekhar
The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given … The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

A generalization of Mazur's theorem (Ogg's conjecture) for number fields

2018-01-01
Debargha Banerjee , Narasimha Kumar , Dipramit Majumdar
In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is … In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the congruence subgroups of the form $\Ga_0(N)$ for any $N \in \N$. This in turn help us to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups.

$p^r$-Selmer companion modular forms

2018-01-01
Somnath Jha , Dipramit Majumdar , Sudhanshu Shekhar
The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given … The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $\chi$ of $K$, the $n$-Selmer groups of $E_1^\chi $ and $E_2^\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.
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p-adic Asai transfer

2017-02-03
Baskar Balasubramanyam , Dipramit Majumdar
Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In … Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $GL_{2}/K$ to the universal eigenvariety of $GL_{4}/Q$, which at nice classical points interpolate this Asai transfer.

Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

2017-01-01
Somnath Jha , Dipramit Majumdar
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension … We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.
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p-adic Asai transfer

2017-01-01
Baskar Balasubramanyam , Dipramit Majumdar
Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In … Let $K/Q$ be a real quadratic field. Given an automorphic representation $\pi$ for $GL_{2}/K$, let $As^{\pm}(\pi)$ denote the plus/minus Asai transfer of $\pi$ to an automorphic representation for $GL_{4}/Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $GL_{2}/K$ to the universal eigenvariety of $GL_{4}/Q$, which at nice classical points interpolate this Asai transfer.

Ordinary families of automorphic forms on definite unitary groups

2016-06-21
Baskar Balasubramanyam , Dipramit Majumdar

Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

2015-04-12
Somnath Jha , Dipramit Majumdar
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension … We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.

Geometry of the eigencurve at critical Eisenstein series of weight 2

2015-01-01
Dipramit Majumdar
In this paper we show that the critical Eisenstein series of weight 2, E 2 crit p , defines a smooth point in the eigencurve ℂ(l), where l is a … In this paper we show that the critical Eisenstein series of weight 2, E 2 crit p , defines a smooth point in the eigencurve ℂ(l), where l is a prime different from p. We also show that E 2 crit p ,ord l defines a smooth point in the full eigencurve ℂ full (l) and E 2 crit p ,ord l 1 ,ord l 2 defines a non-smooth point in the full eigencurve ℂ full (l 1 l 2 ). Further, we show that ℂ(l) is étale over the weight space at the point defined by E 2 crit p . As a consequence, we show that level lowering conjecture of Paulin fails to hold at E 2 crit p ,ord l .
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Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

2015-01-01
Somnath Jha , Dipramit Majumdar
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension … We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.
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Endoscopic transfer between eigenvarieties for definite unitary groups

2014-01-01
Dipramit Majumdar
In this paper, we extend the endoscopic transfer of definite unitary group U(n), which sends a pair of automorphic forms of U(m),U(n) to an automorphic form of U(m+n), to finite … In this paper, we extend the endoscopic transfer of definite unitary group U(n), which sends a pair of automorphic forms of U(m),U(n) to an automorphic form of U(m+n), to finite slope p-adic automorphic forms for definite unitary groups by constructing a rigid analytic map between eigenvarieties E_m \times E_n \to E_{m+n}, which at classical points interpolates endoscopic transfer map.

l-Class groups of cyclic extensions of prime degree l

2014-01-01
Manisha Kulkarni , Dipramit Majumdar , B. Sury
Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow … Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we obtain bounds for the F_ rank of the l-Sylow subgroup of K using genus theory. We obtain some results valid for general l. Following that, we obtain more complete results for l=5 and F =Q(ζ_5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (the latter is under GRH). We obtain explicit results in several cases. Using these results, and duality theory, we deduce results on the 5-class numbers of fields of the form Q(n^1/5).

Geometry of the eigencurve at critical Eisenstein series of weight 2

2013-07-18
Dipramit Majumdar
In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} … In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.

Geometry of the eigencurve at critical Eisenstein series of weight 2

2013-01-01
Dipramit Majumdar
In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} … In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.
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Coauthor Papers Together
Somnath Jha 6
Somnath Jha 6
B. Sury 5
Pratiksha Shingavekar 5
Baskar Balasubramanyam 3
Narasimha Kumar 3
Debargha Banerjee 3
Sudhanshu Shekhar 2
Debanjana Kundu 2
Sudipa Mondal 1
Debajyoti De 1
Usha K. Sangale 1
Subhasis Panda 1
T Kathiravan 1
Sudhanshu Shekhar 1
Manisha Kulkarni 1
B. Sury 1

Families of Galois representations and Selmer groups

2018-11-06
Joël Bellaïche , Gaëtan Chenevier
This is the final version of a book about p-adic families of Galois representations, Selmer groups, eigenvarieties and Arthur's conjectures This is the final version of a book about p-adic families of Galois representations, Selmer groups, eigenvarieties and Arthur's conjectures
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Critical p-adic L-functions

2011-11-02
Joël Bellaïche
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Un analogue du calcul de Heegner

1987-06-01
Philippe Satg�

Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties

2013-07-01
Masami Ohta
Let $N \geq 5$ be a prime number. Conrad, Edixhoven and Stein have conjectured that the rational torsion subgroup of the modular Jacobian variety $J_1(N)$ coincides with the 0-cuspidal class … Let $N \geq 5$ be a prime number. Conrad, Edixhoven and Stein have conjectured that the rational torsion subgroup of the modular Jacobian variety $J_1(N)$ coincides with the 0-cuspidal class group. We prove this conjecture up to 2-torsion. To do this, we study certain ideals of the Hecke algebras, called the Eisenstein ideals, related to modular forms of weight 2 with respect to $\varGamma_1(N)$ that vanish at the 0-cusps.

On the $8$ case of the Sylvester conjecture

2021-12-20
Hongbo Yin
Let $p\equiv 8\mod 9$ be a prime. In this paper we give a sufficient condition such that at least one of $p$ and $p^2$ is the sum of two rational … Let $p\equiv 8\mod 9$ be a prime. In this paper we give a sufficient condition such that at least one of $p$ and $p^2$ is the sum of two rational cubes. This is the first general result on the $8$ case of the so-called Sylvester conjecture.
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PARABOLIC POINTS AND ZETA-FUNCTIONS OF MODULAR CURVES

1972-02-28
Ju I Manin
In this paper we obtain explicit formulas for the values at the center of the critical strip of Dirichlet series connected with weight 2 parabolic forms of the group Γ0(N). … In this paper we obtain explicit formulas for the values at the center of the critical strip of Dirichlet series connected with weight 2 parabolic forms of the group Γ0(N). In particular, these formulas allow us to verify the Birch-Swinnerton-Dyer conjecture on the order of a zero for uniformizable elliptic curves over certain Γ-extensions. We also give applications to noncommutative reciprocity laws.

The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

2019-08-12
Manjul Bhargava , Noam D. Elkies , Ari Shnidman
The elliptic curve E k : y 2 = x 3 + k admits a natural 3-isogeny ϕ k : E k → E − 27 k . We compute … The elliptic curve E k : y 2 = x 3 + k admits a natural 3-isogeny ϕ k : E k → E − 27 k . We compute the average size of the ϕ k -Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of E k and E − 27 k . As a consequence, we prove that the average rank of the curves E k , k ∈ Z , is less than 1.21 and over 23 % (respectively, 41 % ) of the curves in this family have rank 0 (respectively, 3-Selmer rank 1).
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Motives for modular forms

1990-12-01
A. J. Scholl

Cyclic cubic extensions of ℚ

2022-04-25
Dipramit Majumdar , B. Sury
We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The … We determine the irreducible trinomials [Formula: see text] for integers [Formula: see text] which generate precisely all possible Galois extensions of degree [Formula: see text] over [Formula: see text]. The proof, although involved, is elementary and one can parametrize all these polynomials explicitly. As an accidental by-product of the results, we prove that infinitely many primes congruent to [Formula: see text] or [Formula: see text] mod [Formula: see text] are sums of two rational cubes - thereby, giving the first unconditional result on a classical open problem.
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A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial

2020-01-01
Yukako Kezuka , Yongxiong Li
We study elliptic curves of the form x^3+y^3=2p and x^3+y^3=2p^2 where p is any odd prime satisfying p\equiv 2 \mod 9 or p\equiv 5 \mod 9 . We first show … We study elliptic curves of the form x^3+y^3=2p and x^3+y^3=2p^2 where p is any odd prime satisfying p\equiv 2 \mod 9 or p\equiv 5 \mod 9 . We first show that the 3 -part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 2 -Selmer group to the 2 -rank of the ideal class group of \mathbb{Q}(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 2 -part of the Tate-Shafarevich group.
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Variation of Iwasawa invariants in Hida families

2005-12-22
Matthew Emerton , Robert Pollack , Tom Weston
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Diophantine equations and modular forms

1975-01-01
A. P. Ogg
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Cube sum problem and an explicit Gross-Zagier formula

2017-01-01
Li Cai , Jie Shu , Ye Tian
A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer … A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.
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L-Functions and Tamagawa Numbers of Motives

2007-01-01
Spencer Bloch , Kazuya Katô
The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. … The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture on the values at integer points of L-functions associated to motives. Conjectures due to Deligne and Beilinson express these values "modulo Q* multiples" in terms of archimedean period or regulator integrals. Our aim is to remove the Q* ambiguity by defining what are in fact Tamagawa numbers for motives. The essential technical tool for this is the Fontaine-Messing theory of p-adic cohomology. As evidence for our Tamagawa number conjecture, we show that it is compatible with isogeny, and we include strong results due to one of us (Kato) for the Riemann zeta function and for elliptic curves with complex multiplication.

Nonvanishing of L-Series Associated to Cubic Twists of Elliptic Curves

1994-07-01
Daniel Lieman
has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly … has a rational solution. An extensive compilation of the older history of the problem is given in Dickson [6]. Within the past century, researchers have tried to exploit (either explicitly or unknowingly) the fact that the curve (0.1) is in fact an elliptic curve. During the nineteenth century, Lucas, and later Sylvester, used a descent argument to prove that (0.1) had no solution for infinitely many D in certain congruence classes mod 9 and 18 (see [6], Ch. XXI). Zagier and Kramarz [19] have produced a great deal of numerical evidence about the L-series of the curves; based on these computations, they have argued heuristically that for

Quadratic Twists of Modular Forms and Elliptic Curves

2023-03-13
Ken Ono , Matthew A. Papanikolas
Here we summarize the results presented in the first author's lecture at the Millennial Conference on Number Theory. These results appear in [16] in full detail. In addition, we present … Here we summarize the results presented in the first author's lecture at the Millennial Conference on Number Theory. These results appear in [16] in full detail. In addition, we present a new result regarding the growth of Tate-Shafarevich groups of certain elliptic curves over elementary abelian simple 2-extensions.

On the conductors of mod/Galois representations coming from modular forms

1989-02-01
Ron Livné

On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions

2016-09-29
Antonio Lei , David Loeffler , Sarah Livia Zerbes
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In … We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
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Iwasawa theory for modular forms at supersingular primes

2011-02-07
Antonio Lei
Abstract We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus … Abstract We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p -adic L -functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.
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Cube sums of the forms 3p and $$3p^2$$ II

2022-02-11
Jie Shu , Hongbo Yin

On Certain Ternary Cubic-Form Equations

1879-12-01
J.J. Sylvester

Modular Forms, a Computational Approach

2007-02-13
William Stein
Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing … Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.
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Galois representations into GL2 (Z p [[X]]) attached to ordinary cusp forms

1986-10-01
Haruzo Hida

Arithmetic on Modular Curves

1982-01-01
Glenn Stevens

On ordinary ?-adic representations associated to modular forms

1988-10-01
Andrew Wiles

An analogue of Kida's formula for the Selmer groups of elliptic curves

1999-07-01
Yoshitaka Hachimori , Kazuo Matsuno

How to do a 𝑝-descent on an elliptic curve

2003-10-27
Edward F. Schaefer , Michael Stoll
In this paper, we describe an algorithm that reduces the computation of the (full) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer group of an elliptic … In this paper, we describe an algorithm that reduces the computation of the (full) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer group of an elliptic curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a number field to standard number field computations such as determining the (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-torsion of) the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-class group and a basis of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-units modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th powers for a suitable set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of primes. In particular, we give a result reducing this set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of ‘bad primes’ to a very small set, which in many cases only contains the primes above <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As of today, this provides a feasible algorithm for performing a full <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-descent on an elliptic curve over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-bracket p right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E[p]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is favorable, simplifications are possible and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-descents for larger <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are accessible even today. To demonstrate how the method works, several worked examples are included.

Control Theorem for Bloch–Kato's Selmer Groups of p-Adic Representations

2000-05-01
Tadashi Ochiai

Selmer companion curves

2014-09-04
Barry Mazur , Karl Rubin
We say that two elliptic curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation … We say that two elliptic curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">E_1, E_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer companions</italic> for a positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if for every quadratic character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi>χ</mml:mi> <mml:annotation encoding="application/x-tex">\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is an isomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S e l Subscript n Baseline left-parenthesis upper E 1 Superscript chi Baseline slash upper K right-parenthesis approximately-equals upper S e l Subscript n Baseline left-parenthesis upper E 2 Superscript chi Baseline slash upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Sel</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> <mml:mi>χ</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≅</mml:mo> <mml:msub> <mml:mi>Sel</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> <mml:mi>χ</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sel}_n(E_1^\chi /K) \cong \operatorname {Sel}_n(E_2^\chi /K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer groups of the quadratic twists <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 Superscript chi"> <mml:semantics> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> <mml:mi>χ</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">E_1^\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 2 Superscript chi"> <mml:semantics> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> <mml:mi>χ</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">E_2^\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give sufficient conditions for two elliptic curves to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Selmer companions, and give a number of examples of non-isogenous pairs of companions.
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Local to global compatibility on the eigencurve

2011-02-28
Alexander G. M. Paulin
We generalize Coleman's construction of Hecke operators in families to define an action of GL 2 ( A f p ) on the space of overconvergent p-adic modular forms, extending … We generalize Coleman's construction of Hecke operators in families to define an action of GL 2 ( A f p ) on the space of overconvergent p-adic modular forms, extending the classical construction. Using this, we construct a family of ‘p-adic automorphic representations’ of GL 2 ( A f p ) on the Coleman–Mazur eigencurve. We compare this with the natural family of Galois representations carried by the eigencurve using the local Langlands correspondence and deduce local to global compatibility away from a discrete subset, extending the results of Carayol.
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Rational torsion subgroups of modular Jacobian varieties

2018-03-21
Yuan Ren

Canonical periods and congruence formulae

1999-06-01
Vinayak Vatsal
The purpose of this article is to show how congruences between the Fourier coefficients of Hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values … The purpose of this article is to show how congruences between the Fourier coefficients of Hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated L-functions. This study was initiated by B. Mazur in his fundamental work on the Eisenstein ideal (see [Maz77] and [Maz79]) where it was made clear that congruences for analytic L-values were closely related to the integral structure of certain Hecke rings and cohomology groups. The results of [Maz79] also showed that congruences were useful in the study of nonvanishing of L-functions. This idea was then further developed by Stevens [Ste82] and Rubin-Wiles [RW82]. The work of Rubin and Wiles, in particular, used congruences to study the behavior of elliptic curves in towers of cyclotomic fields. A key ingredient here was a theorem of Washington, which states, roughly, that almost L-values in certain families are nonzero modulo p. This theme has recently been taken up again, in the work of Ono-Skinner [OSa], [OSb], James [Jam], and Kohnen [Koh97]. While the earlier history was primarily concerned with cyclotomic twists, the current emphasis is on families of twists by quadratic characters. Here one wants quantitative estimates for the number of quadratic twists of a given modular form, which have nonvanishing L-function at s = 1. We continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three. This generalizes a beautiful example due to Kevin James, and provides new evidence for a conjecture of Goldfeld [Gol79]. It should, however, be pointed out that even the study of quadratic twists may be traced back to Mazur: the reader is urged to look at pages 212–213 of [Maz79], and especially at the footnote at the bottom of page 213. The theorems of Davenport-Heilbronn [DH71] and Washington [Was78], which are crucial in this paper, are both mentioned in Mazur’s article. We want to begin by discussing the congruences that lie at the heart of this article. Thus let f = ∑ anq n be an elliptic modular cuspform of level M and weight k ≥ 2. Assume that f is a simultaneous eigenform for all the Hecke operators and that a1(f) = 1. The L-function associated to f is defined by the Dirichlet series L(s, f) = ∑ ann −s, which converges for the real part of s sufficiently large, and has analytic continuation to s ∈ C. A fundamental theorem of Shimura [Shi76] states that L(s, f) enjoys the following algebraicity property:

Overconvergent modular forms and the Fontaine-Mazur conjecture

2003-08-01
Mark Kisin

Visualizing Elements in the Shafarevich—Tate Group

2000-01-01
J. E. Cremona , Barry Mazur
Abstract We review a number of ways of "visualizing" the elements of the Shafarevich–Tate group of an elliptic curve Eover a number field K. We are specifically interested in caseswhere … Abstract We review a number of ways of "visualizing" the elements of the Shafarevich–Tate group of an elliptic curve Eover a number field K. We are specifically interested in caseswhere the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of X0(N), where N is the conductor of the elliptic curve). For a given such E with nontrivial Shafarevich–Tate group, we pose the question: Are all the curves of genus one representing elements of the Shafarevich–Tate group of E isomorphic (over the rationals) to curves contained in a (single) abelian surface A, itself defined over the rationals, containing E as a subelliptic curve, and contained in turn in the new part of the jacobian of a modular curve X0(N)? At first view, one might imagine that there are few E with nontrivial Shafarevich–Tate group for which the answer is yes. Indeed we have a small number of examples where the answer is no, and it is very likely that the answer will be no if the order of the Shafarevich–Tategroup is large enough. Nonetheless, among all (modular) elliptic curves Eas above, with conductors up to 5500 and with no rational point of order 2, we have found the answer to the question to be yes in the vast majority of cases. We are puzzled by this and wonder whether there is some conceptual reason for it. We present a substantial amount of data relating to the curves investigated.
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The weight in Serre's conjectures on modular forms

1992-12-01
Bas Edixhoven

Advanced Topics in the Arithmetic of Elliptic Curves

1994-01-01
Joseph H. Silverman

On certain Dirichlet series associated with Hilbert modular forms and Rankin's method

1977-02-01
Tetsuya Asai

On higher p-adic regulators

1981-01-01
Christophe Soulé

3-Selmer groups for curves y 2 = x 3 + a

2008-06-01
Andrea Bandini

Automorphic representations of unitary groups in three variables

1990-01-01
Jonathan David Rogawski
The purpose of this book is to develop the stable trace formula for unitary groups in three variables. The stable trace formula is then applied to obtain a classification of … The purpose of this book is to develop the stable trace formula for unitary groups in three variables. The stable trace formula is then applied to obtain a classification of automorphic representations. This work represents the first case in which the stable trace formula has been worked out beyond the case of SL (2) and related groups. Many phenomena which will appear in the general case present themselves already for these unitary groups.

Control Theorem for Greenberg's Selmer Groups of Galois Deformations

2001-05-01
Tadashi Ochiai

Groupes de Selmer et corps cubiques

1986-07-01
P Satgé

Introduction to Cyclotomic Fields

1982-01-01
Lawrence C. Washington

Galois representations associated to Siegel modular forms of low weight

1991-07-01
Richard Taylor

The Arithmetic of Elliptic Curves

1986-01-01
Joseph H. Silverman
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New parity results for broken 11-diamond partitions

2014-03-06
Olivia X. M. Yao
The notation of broken k-diamond partitions was introduced in 2007 by Andrews and Paule. For a fixed positive integer k, let Δk(n) denote the number of broken k-diamond partitions of … The notation of broken k-diamond partitions was introduced in 2007 by Andrews and Paule. For a fixed positive integer k, let Δk(n) denote the number of broken k-diamond partitions of n. Recently, Radu and Sellers established numerous congruence properties for (2k+1)-cores by using the theory of modular forms, where k=2,3,5,6,8,9,11. Employing their congruences for (2k+1)-cores, Radu and Sellers obtained a number of nice parity results for Δk(n). In particular, they proved that for n⩾0, Δ11(46n+r)≡0(mod2), where r∈{11,15,21,23,29,31,35,39,41,43,45}. In this paper, we derive several new infinite families of congruences modulo 2 for Δ11(n) by using an identity given by Chan and Toh, and the p-dissection of Ramanujan's theta function f1 due to Cui and Gu. For example, we prove that for n⩾0 and k,α⩾1, Δ11(23α−2×23kn+23α−2s×23k−1+1)≡0(mod2), where s∈{5,7,10,11,14,15,17,19,20,21,22}. This generalizes the parity results for Δ11(n) discovered by Radu and Sellers.

Automorphic symbols, p -adic L -functions and ordinary cohomology of Hilbert modular varieties

2013-07-28
Mladen Dimitrov
We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general $p$-adic $L$-functions to nearly ordinary Hilbert automorphic forms. Then we establish … We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general $p$-adic $L$-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for the $p$-adically completed cohomology of a Hilbert modular variety localized at a suitable nearly ordinary maximal ideal of the Hecke algebra. We also show its freeness over the corresponding Hecke algebra which turns out to be a universal deformation ring. In the last part of the paper we combine the above results to construct $p$-adic $L$-functions for Hida families of Hilbert automorphic forms in universal deformation rings of Galois representations.

Geometric level raising and lowering on the eigencurve

2011-06-08
Alexander G. M. Paulin

The Iwasawa Main Conjectures for GL2

2013-05-22
Christopher Skinner , Eric Urban
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Selmer groups of elliptic curves that can be arbitrarily large

2003-03-01
Remke Kloosterman , Edward F. Schaefer