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.
In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations …
In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of type<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of types<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of type<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>.
We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric $SU(5)$ in 6d, where …
We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified on a $T_2/\mathbb{Z}_2$ orbifold. We have shown that, if there is a finite modular symmetry, then it can only be $A_4$ with an (infinite) discrete choice of moduli, where we focus on $\tau = \omega=e^{i2\pi/3}$, the unique solution with $|\tau|=1$. The fields on the branes respect a generalised CP and flavour symmetry $A_4\ltimes \mathbb{Z}_2$ which is isomorphic to $S_4$ which leads to an effective $\mu-\tau$ reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic CP violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for $SU(5)$ breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix.
Topologically ordered systems exhibit long-range quantum entanglement. New calculations show that a trio of looplike excitations can be braided to produce rich information about both the underlying topological order.
Topologically ordered systems exhibit long-range quantum entanglement. New calculations show that a trio of looplike excitations can be braided to produce rich information about both the underlying topological order.
We give two explicit sets of generators of the group of invertible regular functions over Q on the modular curve Y 1 (N).
We give two explicit sets of generators of the group of invertible regular functions over Q on the modular curve Y 1 (N).
For $1 < p \leq q$, a convex modular $m$ on a linear space $S$ is called an ${L_{p,q}}$ modular if ${\operatorname {Min} _{r = p,q}}{\xi ^r}m(x) \leq m(\xi x) …
For $1 < p \leq q$, a convex modular $m$ on a linear space $S$ is called an ${L_{p,q}}$ modular if ${\operatorname {Min} _{r = p,q}}{\xi ^r}m(x) \leq m(\xi x) \leq {\operatorname {Max} _{r = p,q}}{\xi ^r}m(x)$ for $\xi > 0$ and $x \in S$. We generalize the Minkowski inequality and the Hölder inequality for ${L_{p,q}}$ modulars.
We study Flipped $SU(5)\times U(1)$ Grand Unified Theories (GUTs) with $\Gamma_3\simeq A_4$ modular symmetry. We propose two models with different modular weights assignments, where the fermion mass hierarchy can arise …
We study Flipped $SU(5)\times U(1)$ Grand Unified Theories (GUTs) with $\Gamma_3\simeq A_4$ modular symmetry. We propose two models with different modular weights assignments, where the fermion mass hierarchy can arise from weighton fields. In order to relax the constraint on the Dirac neutrino Yukawa matrix we appeal to mechanisms which allow incomplete GUT representations, allowing a good fit to quark and charged lepton masses and quark mixing for a single modulus field $\tau$, with the neutrino masses and lepton mixing well determined by the type I seesaw mechanism, at the expense of some tuning. We also discuss the double seesaw possibility allowed by the extra singlets generically predicted in such string inspired theories.
In this paper we prove the functoriality of the exterior square of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow …
In this paper we prove the functoriality of the exterior square of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as automorphic forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 6"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{6}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the symmetric fourth of cusp forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as automorphic forms on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 5"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>5</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{5}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the absolute convergence of the exterior square <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 4"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Second, we prove that the fourth symmetric power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions of cuspidal representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">GL_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="three twenty-sixths"> <mml:semantics> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>26</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {3}{26}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for Hecke eigenvalues of Maass forms over any number field.
Table of contents 0. Introduction 1. Algebraic preliminaries a.The Koszul cohomology groups b.Syzygies c. Cohomology operations d.The spectral sequence relating Koszul cohomology groups of an exact complex 2. The Duality …
Table of contents 0. Introduction 1. Algebraic preliminaries a.The Koszul cohomology groups b.Syzygies c. Cohomology operations d.The spectral sequence relating Koszul cohomology groups of an exact complex 2. The Duality Theorem a. Transition to the setting of complex manifolds b.The Gaussian class c.The Duality Theorem 3. Computational techniques for Koszul cohomology a.A vanishing theorem b.The "Lefschetz Theorem" c.The K pΛ Theorem 4. Applications a.The Theorem of the Top Row b.The Arbarello-Sernesi module and Petri's analysis of the ideal of a special curve .... c.The canonical ring of a variety of general type d.The H ] Lemma, a theorem of Kϋ, and a splitting lemma e.The H° Lemma f.A holomorphic representation of the H p ' q groups of a smooth variety 5. Open problems and conjectures A. Appendix (with Robert Lazarsfeld): The nonvanishing of certain Koszul cohomology groups
A linear lower bound on the gonality of modular curves Dan Abramovich Dan Abramovich Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, …
A linear lower bound on the gonality of modular curves Dan Abramovich Dan Abramovich Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1996, Issue 20, 1996, Pages 1005–1011, https://doi.org/10.1155/S1073792896000621 Published: 01 January 1996 Article history Published: 01 January 1996 Received: 03 September 1996
Let $W(N)$ be the group of Atkin-Lehner involutions on the modular curve $X_0(N)$. The purpose of this article is to give complementary result to [7, 8, 9]; namely, we determine …
Let $W(N)$ be the group of Atkin-Lehner involutions on the modular curve $X_0(N)$. The purpose of this article is to give complementary result to [7, 8, 9]; namely, we determine trigonal curves of the form $X_0(N)/W'$, where $W'$ is a subgroup of $W(N)$ such that $2< |W'| < |W(N)|$.
Let $N$ be a positive integer, and let $\Gamma_{0}(N)=\{\left(\begin{array}{ll}a & b\\c & d\end{array}\right)\in SL_{2}(Z)|c\equiv 0(mod N)\}$ .Let $X_{0}(N)$ be the modular curve which corresponds to $\Gamma_{0}(N)$ .For each positive divisor …
Let $N$ be a positive integer, and let $\Gamma_{0}(N)=\{\left(\begin{array}{ll}a & b\\c & d\end{array}\right)\in SL_{2}(Z)|c\equiv 0(mod N)\}$ .Let $X_{0}(N)$ be the modular curve which corresponds to $\Gamma_{0}(N)$ .For each positive divisor $N^{\prime}$ of $N$ with $(N^{\prime}, N/N^{\prime})=1$ (in which case we write $N^{\prime}\Vert N$ ), $W_{N^{\prime}}=W_{N'}^{\langle N)}$ denotes the corresponding Atkin-Lehner involution on $X_{O}(N)$ . ($W_{1}$ is the identity.)It is known that the $W_{N^{\prime}}$ generate an elementary 2-abelian group, which we denote by $W(N)$ .The group $W(N)$ is of order $2^{\omega\langle N)}$ , where $\omega(N)$ is the number of distinct prime divisors of $N$ .Furthermore, these involutions are all defined over $Q:W(N)\subseteq Aut_{\Phi}(X_{0}(N))$ .Let $W^{\prime}$ be a subgroup of $W(N)$ .Then the hyperellipticity of the quotient curve $X_{0}(N)/W^{\prime}$ has been determined for two extreme cases (i.e., for $W^{\prime}=\{1\}$ or $W(N)$).THEOREM 1 ([12]).There are nineteen values ofNfor which $X_{0}(N)$ is hyperelliptic, $i.e.,$ $X_{0}(N)$ is hyperelliptic if and only if $N=22, 23,26,28-31,33,35,37,39-41,46-48,50,59,71$ .THEOREM 2 ([8] [6]).Put $X_{0}^{*}(N)=X_{0}(N)/W(N)$.There are 64 values ofNfor which $X_{0}^{*}(N)$ is hyperelliptic.(i) $X_{0}^{*}(N)$ is ofgenus two ifand only $ifN$ is in thefollowing list(57 values in total): 67, 73
Tetragonal modular curves by Daeyeol Jeon and Euisung Park (Seoul) 0. Introduction.A smooth projective curve X defined over an algebraically closed field k is called d-gonal if it admits a …
Tetragonal modular curves by Daeyeol Jeon and Euisung Park (Seoul) 0. Introduction.A smooth projective curve X defined over an algebraically closed field k is called d-gonal if it admits a map φ : X → P 1 over k of degree d.If the genus g ≥ 2 and d = 2 then X is called hyperelliptic.We will say that X is trigonal, tetragonal and pentagonal for d = 3, d = 4 and d = 5 respectively.Let N be a positive integer, and letLet X 0 (N ) denote the modular curve corresponding to Γ 0 (N ).Then Zograf [Z] gave a linear bound on the level N of d-gonal modular curves X 0 (N ).Also Nguyen and Saito [N-Sa] proved an analogue of the strong Uniform Boundedness Conjecture for elliptic curves defined over function fields of dimension one by using the connection with giving a bound on the level N of d-gonal modular curves X 0 (N ).Recently, Hasegawa and Shimura [H-S] gave a highly sharpened upper bound for 3 ≤ d ≤ 5 by trying to determine d-gonal modular curves X 0 (N ) for such d.For d = 2 it was done by Ogg [O].Actually Hasegawa and Shimura succeeded in determining all trigonal modular curves X 0 (N ) but failed for tetragonal and pentagonal X 0 (N ).The following lists N for which they did not know whether X 0 (N ) was tetragonal or not: 76, 82, 84, 88, 90, 93, 97
For a positive integer $N$, let $X_0^*(N)$ denote the quotient curve of $X_0(N)$ by the Atkin-Lehmer involutions. In this paper, we determine the trigonality of $X_0^*(N)$ for all $N$. It …
For a positive integer $N$, let $X_0^*(N)$ denote the quotient curve of $X_0(N)$ by the Atkin-Lehmer involutions. In this paper, we determine the trigonality of $X_0^*(N)$ for all $N$. It turns out that there are seven values of $N$ for which $X_0^*(N)$ is a non-trivial trigonal curve.
Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578–602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461–2484] have recently determined the quadratic points on …
Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578–602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461–2484] have recently determined the quadratic points on each modular curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus 2, 3, 4, or 5 whose Mordell–Weil group has rank 0. In this paper we do the same for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus 2, 3, 4, and 5 and positive Mordell–Weil rank. The values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are 37, 43, 53, 61, 57, 65, 67, and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell–Weil sieve. Often the quadratic points are not finite, as the degree 2 map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis right-arrow upper X 0 left-parenthesis upper N right-parenthesis Superscript plus"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)\to X_0(N)^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be a source of infinitely many such points. In such cases, we describe this map and the rational points on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis Superscript plus"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and we specify the exceptional quadratic points on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> not coming from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0 left-parenthesis upper N right-parenthesis Superscript plus"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X_0(N)^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we determine the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariants of the corresponding elliptic curves and whether they are <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:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-curves or have complex multiplication.
Abstract We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached …
Abstract We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$ . For each of these $\ell $ , we compute the directed graph of arithmetically maximal $\ell $ -power level modular curves $X_H$ , compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$ , for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$ . Aside from the $\ell $ -adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$ -isomorphism classes of elliptic curves $E/\mathbb {Q}$ , we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j -invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$ , or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $ -adic images of Galois for any elliptic curve over $\mathbb {Q}$ . In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$ -type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$ .
Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ …
Let $N\geq 1$ be a non-square free integer and let $W_N$ be a non-trivial subgroup of the group of the Atkin-Lehner involutions of $X_0(N)$ such that the modular curve $X_0(N)/W_N$ has genus at least two. We determine all pairs $(N,W_N)$ such that $X_0(N)/W_N$ is a bielliptic curve and the pairs $(N,W_N)$ such that $X_0(N)/W_N$ has an infinite number of quadratic points over $\mathbb{Q}$.
Abstract A number field K is primitive if K and $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> are the only subextensions of K . Let C be a curve defined over …
Abstract A number field K is primitive if K and $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> are the only subextensions of K . Let C be a curve defined over $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> . We call an algebraic point $$P\in C(\overline{\mathbb {Q}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> primitive if the number field $$\mathbb {Q}(P)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is primitive. We present several sets of sufficient conditions for a curve C to have finitely many primitive points of a given degree d . For example, let $$C/\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> be a hyperelliptic curve of genus g , and let $$3 \le d \le g-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>g</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Suppose that the Jacobian J of C is simple. We show that C has only finitely many primitive degree d points, and in particular it has only finitely many degree d points with Galois group $$S_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> or $$A_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> . However, for any even $$d \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , a hyperelliptic curve $$C/\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>/</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> has infinitely many imprimitive degree d points whose Galois group is a subgroup of $$S_2 \wr S_{d/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> .