Barry Mazur

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Author Description

Barry Mazur is a distinguished American mathematician known for his profound contributions to number theory, algebraic geometry, and topology. Born on December 19, 1937, Mazur has been a prominent figure in mathematics, particularly recognized for his work at Harvard University, where he has served as a professor.

Key Contributions:

Topology: Early in his career, Mazur made significant advances in geometric topology. Notably, he proved the Mazur Swindle, a technique in topology, and worked on the proof of the generalized Schoenflies theorem in higher dimensions.
Number Theory and Algebraic Geometry: Mazur’s work in number theory is highly influential. He is well-known for Mazur’s Torsion Theorem, which classifies the possible torsion subgroups of elliptic curves over the rational numbers. This result has been fundamental in the study of elliptic curves and has implications in Diophantine equations.
Galois Deformation Theory: He introduced important concepts in Galois deformation theory, which have played a crucial role in modern number theory, including Wiles’s proof of Fermat’s Last Theorem.
Iwasawa Theory and Modular Forms: Mazur has contributed extensively to Iwasawa theory and the study of modular forms, exploring deep connections between these areas and elliptic curves.
Educational Impact: Beyond his research, Mazur is celebrated for his mentorship and exposition. He has guided numerous students and collaborators, fostering developments in mathematics through teaching and collaborative work.

Publications:

Mazur has authored and co-authored numerous influential papers and books. His writings often illuminate complex mathematical ideas, making them accessible to a broader audience.

Recognition:

Throughout his career, Mazur has received several awards and honors in recognition of his mathematical achievements, including membership in esteemed institutions like the National Academy of Sciences.

Barry Mazur’s work has had a lasting impact on mathematics, contributing foundational results and inspiring ongoing research in several mathematical fields.

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All published works (183)

Ogg’s torsion conjecture: Fifty years later

2025-02-12

Jennifer S. Balakrishnan , Barry Mazur

Andrew Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has … Andrew Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg’s celebrated torsion conjecture—as it relates to modular curves—can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg’s torsion conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.

Shadow line distributions

2024-09-01

Jennifer S. Balakrishnan , Mirela Çiperiani , Barry Mazur +1 more

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the … Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the Heegner hypothesis, there is (subject to the Shafarevich--Tate conjecture) a line, i.e., a free $\mathbb{Z}_p$-submodule of rank $1$, in $ E(K)\otimes \mathbb{Z}_p$ given by universal norms coming from the Mordell--Weil groups of subfields of the anticyclotomic $\mathbb{Z}_p$-extension of $K$; we call it the {\it shadow line}. When the twist of $E$ by $K$ has analytic rank $1$, the shadow line is conjectured to lie in $E(\mathbb{Q})\otimes\mathbb{Z}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_p$ as $K$ varies, framing conjectures based on the computations we have made.

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Dimension of the deformation space of ordinary representations in the cyclotomic limit

2024-06-04

Ashay Burungale , Laurent Clozel , Barry Mazur

The weight two ordinary deformations are unobstructed in the cyclotomic limit under certain assumptions. We show that such an ordinary deformation ring over the cyclotomic tower can have arbitrarily large … The weight two ordinary deformations are unobstructed in the cyclotomic limit under certain assumptions. We show that such an ordinary deformation ring over the cyclotomic tower can have arbitrarily large dimension.

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Corrigendum to “Existential definability and diophantine stability” [J. Number Theory 254 (2024) 1–64]

2024-05-21

Barry Mazur , Karl Rubin , Alexandra Shlapentokh

Derived class field theory

2024-01-01

Tony Feng , M. P. Harris , Barry Mazur

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Defining $\mathbb Z$ using unit groups

2024-01-01

Barry Mazur , Karl Rubin , Alexandra Shlapentokh

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of Q, paying attention to the uniformity of definitions.The uniformity follows from the simplicity of our … We consider first-order definability and decidability questions over rings of integers of algebraic extensions of Q, paying attention to the uniformity of definitions.The uniformity follows from the simplicity of our first-order definition of Z. Namely, we prove that for a large collection of algebraic extensions K/Q,where O K denotes the ring of integers of K.One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948.

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Erratum to "Disparity in Selmer ranks of quadratic twists of elliptic curves"

2023-08-31

Zev Klagsbrun , Barry Mazur , Karl Rubin

Existential definability and diophantine stability

2023-08-02

Barry Mazur , Karl Rubin , Alexandra Shlapentokh

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Defining $\mathbb Z$ using unit groups

2023-01-01

Barry Mazur , Karl Rubin , Alexandra Shlapentokh

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\mathbb Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity … We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\mathbb Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $\mathbb Z$. Namely, we prove that for a large collection of algebraic extensions $K/\mathbb Q$, $$ \{x \in {\mathcal O}_K : \text{$\forall \varepsilon \in {\mathcal O}_K^\times \;\exists \delta \in {\mathcal O}_K^\times$ such that $\delta-1 \equiv (\varepsilon-1)x \pmod{(\varepsilon-1)^2}$}\} = \mathbb Z $$ where ${\mathcal O}_K$ denotes the ring of integers of $K$.

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Derived class field theory

2023-01-01

Tony Feng , M. P. Harris , Barry Mazur

We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit. We sketch the construction of a derived enhancement of the reciprocity isomorphism of class field theory. Details will appear in a forthcoming joint paper of the authors with A. Raksit.

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Ogg's Torsion conjecture: Fifty years later

2023-01-01

Jennifer S. Balakrishnan , Barry Mazur

Ogg's celebrated Torsion conjecture -- as it relates to modular curves -- can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic … Ogg's celebrated Torsion conjecture -- as it relates to modular curves -- can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason. We give a survey of the field over the last five decades, inspired by Ogg's results and conjectures.

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Uniformity of rational points: an up-date and corrections

2022-03-30

Lucia Caporaso , Joe Harris , Barry Mazur

This paper corrects an error in the authors' earlier work, by proving stronger forms of the basic lemmas This paper corrects an error in the authors' earlier work, by proving stronger forms of the basic lemmas

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Existential definability and diophantine stability

2022-01-01

Barry Mazur , Karl Rubin , Alexandra Shlapentokh

Let $K$ be a number field, let $L$ be an algebraic (possibly infinite degree) extension of $K$, and let $O_K$ $\subset$ $O_L$ be their rings of integers. Suppose $A$ is … Let $K$ be a number field, let $L$ be an algebraic (possibly infinite degree) extension of $K$, and let $O_K$ $\subset$ $O_L$ be their rings of integers. Suppose $A$ is an abelian variety defined over $K$ such that $A(K)$ is infinite and $A(L)/A(K)$ is a torsion group. If at least one of the following conditions is satisfied: 1. $L$ is a number field, 2. $L$ is totally real, 3. $L$ is a quadratic extension of a totally real field, then $O_K$ has a diophantine definition over $O_L$.

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Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols

2021-12-11

Barry Mazur , Karl Rubin

Suppose E is an elliptic curve over Q and χ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via the … Suppose E is an elliptic curve over Q and χ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell–Weil rank grows in abelian extensions of Q. Using this heuristic, we find a large class of infinite abelian extensions F where we expect E(F) to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

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Corrections to {\it Uniformity of rational points} and further comments

2020-12-28

Lucia Caporaso , Joe Harris , Barry Mazur

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Big fields that are not large

2020-11-13

Barry Mazur , Karl Rubin

A subfield <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> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> … A subfield <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> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {\mathbb {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>large</italic> if every smooth curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <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> with a <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>-rational point has infinitely many <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>-rational points. A subfield <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> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {\mathbb {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <italic>big</italic> if for every 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>, <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> contains a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper F colon double-struck upper Q right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[F:\mathbb {Q}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> divisible by <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>. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.

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Big fields that are not large

2020-04-19

Barry Mazur , Karl Rubin

A subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$ over $K$ with a rational point has infinitely many rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is $big$ … A subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$ over $K$ with a rational point has infinitely many rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is $big$ if for every positive integer $n$, $K$ contains a number field $F$ with $[F:\mathbb{Q}]$ divisible by $n$. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.

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The Consolation of Math in Plague Time

2020-01-01

Barry Mazur

Big fields that are not large

2020-01-01

Barry Mazur , Karl Rubin

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Arithmetic conjectures suggested by the statistical behavior of modular symbols

2019-01-01

Barry Mazur , Karl Rubin

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. … Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell-Weil rank grows in abelian extensions of $\mathbb{Q}$. Using this heuristic we find a large class of infinite abelian extensions $F$ where we expect $E(F)$ to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

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Explicit Universal Deformations of Galois Representations

2018-06-20

Nigel Boston , Barry Mazur

To K. Jwasawa on the occasion of his 70th birthdayGiven a continuous absolutely irreducible representation and a finite set of primes S which contains the primes of ramification for p … To K. Jwasawa on the occasion of his 70th birthdayGiven a continuous absolutely irreducible representation and a finite set of primes S which contains the primes of ramification for p and the prime number p, the notion of universal deformation for (]i, S) was discussed in [M].It was shown in [M] that there exists a complete noetherian local ring R with residue field F 1 ,, uniquely determined up to canonical isomorphism, and a lifting of p (unique up to strict equivalence-see § 3.1 below) which is unramified outside S, and satisfies a universal property vis a vis all liftings of p to GLld) which are unramified outside S, where d ranges through the category of complete local noetherian rings with residue field Fp.For S={p} and a class of representations p ("special dihedral representations") the universal deformation ring R was shown to be a power series ring in 3 variables over ZP.If X is the "universal deformation space", i.e., the space of continuous homomorphisms from R to Zp, then Xis a 3-dimensional analytic manifold over QP and for each x e X specialization of p yields a Galois representation (determined up to strict equivalence) which is a lifting of p and is unramified outside S. One of the aims of [M] was to embark on a systematic study of certain "natural subspaces" in X: loci of points x e X such that

Afterword to the article “Arithmetic on curves”

2018-04-19

Barry Mazur

In this afterword the author discusses his previously published article “Arithmetic on curves”, which appeared in the <italic>Bulletin of the American Mathematical Society (N.S.)</italic> <bold>14</bold> (1986), no. 2, 207–259. In this afterword the author discusses his previously published article “Arithmetic on curves”, which appeared in the <italic>Bulletin of the American Mathematical Society (N.S.)</italic> <bold>14</bold> (1986), no. 2, 207–259.

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Rational families of 17-torsion points of elliptic curves over number fields

2018-01-01

Maarten Derickx , Barry Mazur , S. Kamienny

Fumiyuki Momose is very much missed.He was a generous warm human being, with immense energy and generosity of spirit, and an extremely gifted mathematician.One of his abiding interests was rational … Fumiyuki Momose is very much missed.He was a generous warm human being, with immense energy and generosity of spirit, and an extremely gifted mathematician.One of his abiding interests was rational torsion on elliptic curves over number fields, as in [32], [24].

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Diophantine stability

2018-01-01

Barry Mazur , Karl Rubin , Michael Larsen

If $V$ is an irreducible algebraic variety over a number field $K$, and $L$ is a field containing $K$, we say that $V$ is {\it diophantine-stable} for $L/K$ if $V(L)=V(K)$. … If $V$ is an irreducible algebraic variety over a number field $K$, and $L$ is a field containing $K$, we say that $V$ is {\it diophantine-stable} for $L/K$ if $V(L)=V(K)$. We prove that if $V$ is either a simple abelian variety, or a curve of genus at least one, then under mild hypotheses there is a set $S$ of rational primes with positive density such that for every $\ell\in S$ and every $n\ge 1$, there are infinitely many cyclic extensions $L/K$ of degree $\ell^n$ for which $V$ is diophantine-stable. We use this result to study the collection of finite extensions of $K$ generated by points in $V(\bar{K})$.

A celebration of the mathematical work of Glenn Stevens

2016-01-08

Barry Mazur

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Refined class number formulas for \mathbb{G}_m

2016-01-01

Barry Mazur , Karl Rubin

We formulate a generalization of a "refined class number formula" of Darmon. Our conjecture deals with Stickelberger-type elements formed from generalized Stark units, and has two parts: the "order of … We formulate a generalization of a "refined class number formula" of Darmon. Our conjecture deals with Stickelberger-type elements formed from generalized Stark units, and has two parts: the "order of vanishing" and the "leading term". Using the theory of Kolyvagin systems we prove a large part of this conjecture when the order of vanishing of the corresponding complex L-function is 1.

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Controlling Selmer groups in the higher core rank case

2016-01-01

Barry Mazur , Karl Rubin

We define Kolyvagin systems and Stark systems attached to p-adic representations in the case of arbitrary “core rank” (the core rank is a measure of the generic Selmer rank in … We define Kolyvagin systems and Stark systems attached to p-adic representations in the case of arbitrary “core rank” (the core rank is a measure of the generic Selmer rank in a family of Selmer groups). Previous work dealt only with the case of core rank one, where the Kolyvagin and Stark systems are collections of cohomology classes. For general core rank, they are collections of elements of exterior powers of cohomology groups. We show under mild hypotheses that for general core rank these systems still control the size and structure of Selmer groups, and that the module of all Kolyvagin (or Stark) systems is free of rank one.

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Fourier Transforms: Second Visit

2015-12-31

Barry Mazur , William Stein

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Building π(X) from the Spectrum

2015-12-31

Barry Mazur , William Stein

A View of|Li(X)-π(X)|

2015-12-31

Barry Mazur , William E. Stein

Companions to the Zeta Function

2015-12-31

Barry Mazur , William Stein

Thoughts About Numbers

2015-12-31

Barry Mazur , William Stein

A summary is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ … A summary is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

From the Riemann Spectrum to Primes

2015-12-31

Barry Mazur , William Stein

How Many Primes are There?

2015-12-31

Barry Mazur , William Stein

What is Riemann's Hypothesis?

2015-12-31

Barry Mazur , William Stein

Spectra and Trigonometric Sums

2015-12-31

Barry Mazur , William Stein

From Primes to the Riemann Spectrum

2015-12-31

Barry Mazur , William Stein

Prime Numbers Viewed from a Distance

2015-12-31

Barry Mazur , William Stein

The Prime Number Theorem

2015-12-31

Barry Mazur , William Stein

Further Questions About Primes

2015-12-31

Barry Mazur , William Stein

Further Questions About the Riemann Spectrum

2015-12-31

Barry Mazur , William Stein

Tinkering with the Staircase of Primes

2015-12-31

Barry Mazur , William E. Stein

For starters, notice that all the (vertical) risers of this staircase (Figure 18.1 above) have unit height. That is, they contain no numerical information except for their placement on the … For starters, notice that all the (vertical) risers of this staircase (Figure 18.1 above) have unit height. That is, they contain no numerical information except for their placement on the x-axis. So, we could distort our staircase by changing (in any way we please) the height of each riser; and as long as we haven't brought new risers into – or old risers out of – existence, and have not modified their position over the x-axis, we have retained all the information of our original staircase.

The Spectrum and the Staircase of Primes

2015-12-31

Barry Mazur , William Stein

To Our Readers of Part I

2015-12-31

Barry Mazur , William Stein

The statement of the Riemann Hypothesis – admittedly as elusive as before – has, at least, been expressed elegantly and more simply, given our new staircase that approximates (conjecturally with … The statement of the Riemann Hypothesis – admittedly as elusive as before – has, at least, been expressed elegantly and more simply, given our new staircase that approximates (conjecturally with essential square root accuracy) a 45 degree straight line.

As Riemann Envisioned It

2015-12-31

Barry Mazur , William Stein

What is a “Good Approximation”?

2015-12-31

Barry Mazur , William Stein

If you are trying to estimate a number, say, around ten thousand, and you get it right to within a hundred, let us celebrate this kind of accuracy by saying … If you are trying to estimate a number, say, around ten thousand, and you get it right to within a hundred, let us celebrate this kind of accuracy by saying that you have made an approximation with square-root error (√10,000 = 100). Of course, we should really use the more clumsy phrase “an approximation with at worst square-root error.” Sometimes we'll simply refer to such approximations as good approximations. If you are trying to estimate a number in the millions, and you get it right to within a thousand, let's agree that – again – you have made an approximation with square-root error (√1,000,000 = 1,000). Again, for short, call this a good approximation. So, when Gauss thought his curve missed by 226 in estimating the number of primes less than three million, it was well within the margin we have given for a “good approximation.”

The Staircase of Primes

2015-12-31

Barry Mazur , William Stein

What are Prime Numbers?

2015-12-31

Barry Mazur , William Stein

“Named” Prime Numbers

2015-12-31

Barry Mazur , William Stein

A Probabilistic First Guess

2015-12-31

Barry Mazur , William E. Stein

Common Coauthors

Coauthor	Papers Together
Karl Rubin	38
William Stein	24
Michael Artin	8
Joe Harris	6
John Tate	6
William E. Stein	5
Alexandra Shlapentokh	5
Lucia Caporaso	4
Kenneth A. Ribet	4
Zev Klagsbrun	3
Andrew Wiles	3
Fernando Gouvêa	3
Jason Starr	3
Jennifer S. Balakrishnan	3
Glenn Stevens	2
Fernando Q. Gouvêa	2
Matthew Baker	2
Joseph Harris	2
Alice Silverberg	2
Tony Feng	2
John Friedlander	2
M. P. Harris	2
Frank Calegari	2
David Eisenbud	2
Nicholas M. Katz	2
T.M. Graber	2
Tom Graber	2
William Messing	2
Eric M. Friedlander	2
Ivar Ekeland	1
Maarten Derickx	1
Harsh Pittie	1
J. E. Cremona	1
Dorian Goldfeld	1
Ashay Burungale	1
Jacques Tilouine	1
Je Remercie	1
B Oriat	1
J-P Serre	1
V Ennola	1
R Gillard	1
G Gras	1
M Hirabayashi	1
Swinnerton-Dyer Conjecture	1
Mark E. Watkins	1
Karl Rubin	1
H. W. Lenstra	1
Carsten Schmidt	1
Robert F. Coleman	1
Jean-Pierre Serre	1

Commonly Cited References

Rational points of abelian varieties with values in towers of number fields

1972-12-01

Barry Mazur

Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

1971-12-01

Jean-Pierre Serre

The Arithmetic of Elliptic Curves

1986-01-01

Joseph H. Silverman

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Introduction to the Arithmetic Theory of Automorphic Functions

1972-10-01

Marvin Tretkoff , Goro Shimura

Abelian l-Adic Representations and Elliptic Curves

1997-11-15

Jean-Pierre Serre

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on … This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Rational isogenies of prime degree

1978-06-01

Barry Mazur , Dorian Goldfeld

Good Reduction of Abelian Varieties

1968-11-01

Jean-Pierre Serre , John Tate

Finding large Selmer rank via an arithmetic theory of local constants

2007-09-01

Barry Mazur , Karl Rubin

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined … We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields.Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime.Let K -denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K -/K) by inversion).We prove (under mild hypotheses on p) that if the Z p -rank of the pro-p Selmer group S p (E/K) is odd, then rank Zp S p (E/F ) ≥ [F : K] for every finite extension F of K in K -.

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Modular curves and the eisenstein ideal

1977-12-01

Barry Mazur

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Canonical Height Pairings via Biextensions

1983-01-01

Barry Mazur , John Tate

The object of this paper is to present the foundations of a theory of p-adic-valued height pairings * % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaabm … The object of this paper is to present the foundations of a theory of p-adic-valued height pairings * % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaabm % aabaGaam4saaGaayjkaiaawMcaaiabgEna0kqadgeagaqbamaabmaa % baGaam4saaGaayjkaiaawMcaaiabgkziUkaadgfadaWgaaWcbaGaam % iCaaqabaaaaa!4239! $$A\left( K \right) \times A'\left( K \right) \to {Q_p}$$ , where Λ is a abelian variety over a global field K, and A′ is its dual. We say "pairings" in the plural because, in contrast to the classical theory of ℝ-valued) canonical height, there may be many canonical p-adic valued pairings: as we explain in § 4, up to nontrivial scalar multiple, they are in one-to-one correspondence with ℤ p -extensions L/K whose ramified primes are finite in number and are primes of ordinary reduction (1.1) for A.

Arithmetic of Weil curves

1974-03-01

Barry Mazur , Peter Swinnerton‐Dyer

Algebraic number theory

1999-11-22

Kazuya Katô , Nobushige Kurokawa , Takeshi Saito

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background … This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

Hecke operators on ?0(m)

1970-06-01

A. O. L. Atkin , Joseph Lehner

Ranks of twists of elliptic curves and Hilbert’s tenth problem

2010-05-18

Barry Mazur , Karl Rubin

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that … In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth Problem has a negative answer over the ring of integers of every number field.

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P-adic properties of modular schemes and modular forms

2006-11-23

Nicholas M. Katz

Mazur’s conjecture on higher Heegner points

2002-06-01

Christophe Cornut

On the parity of ranks of Selmer groups

2000-01-01

Jan Nekovář , Andrew J. Plater

Introduction.Let / = X)n>i a n(f)Q n € Sko (To{N)) be a normalized newform of even weight fco > 2. Let F be the number field generated by the coefficients … Introduction.Let / = X)n>i a n(f)Q n € Sko (To{N)) be a normalized newform of even weight fco > 2. Let F be the number field generated by the coefficients of / and p a prime of F lying above a rational prime p.There is a two-dimensional representation V(f) of GQ = Gal (Q/Q) over Fp associated to /, characterized by the conditions It (FWgeomW/)) =<*/(/)det(FV(^) geom |y(/)) = ^-1 for all primes £ { piV.The Tate twist Vk 0 = V(f)(ko/2) is self dual: there is a skew-symmetric bilinear formThe complex L-function L 00 (f, s) = ^2n>1 cin(/)^~s satisfies the functional equation Aoo(/, s) := ( -J r(s)L 00 (/, s) = Woo(/)Aoo(/, fco -5), where w 00 (f) = ±1 = (-l) eo0 for eoo = 0 or 1. Bloch and Kato [BI-Ka] defined a generalized "Selmer group" Hj(Q,V ko ) C i? 1 (Q, Vk 0 ) and conjectured that ord^/aLooCf,*) = dim Fp H}(Q, Vi 0 ).We are interested in a (mod 2) version of this conjecture:The Parity Conjecture for ranks of Selmer groups ord a=ibo/ 2Loo(/,s) = dim Fp if}(Q,^0) (mod 2).Assume that p > 3 and that / is ordinary at p, i.e. that ap(f) 6 Fp is a p-adic unit.According to Hida's theory, there is a p-adic family of ordinary modular forms of varying weights containing / (we ignore the phenomenon of "p-stabilization" in this Introduction).In concrete terms, this means that there is an integer c > 0 such that for every integer k > 2 satisfying k = ko (mod (p -l)p c ), there is an ordinary newform fk of weight k on ro(iV) such that /fc 0 = / and .k = k' (mod (p -l)p n+c ) implies /* = /*' (modp"). LetAT, k = 2, a p (f) = 1, f 1 ifp\ \ 0 other otherwise.

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Arithmetic of p-adic Modular Forms

1988-01-01

Fernando Q. Gouvêa

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The size of Selmer groups for the congruent number problem

1993-12-01

D. R. Heath‐Brown

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Rational points on certain elliptic modular curves

1973-01-01

A. P. Ogg

Schémas en groupes de type $(p,\ldots,p)$

1974-01-01

Michel Raynaud

Schémas en F-vectorielsDans toute la suite, p désigne un nombre premier.Soient r un entier ^ 1, q == T/, F un corps fini à q éléments, et F* le groupe … Schémas en F-vectorielsDans toute la suite, p désigne un nombre premier.Soient r un entier ^ 1, q == T/, F un corps fini à q éléments, et F* le groupe multiplicatif des éléments non nuls de F.Pour tout entier n ^ 1, tout schéma S, on note (1^)5, ou simplement ^, le 5'-schéma en groupes des racines /î-ièmes de l'unité.

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Twisting commutative algebraic groups

2007-03-22

Barry Mazur , Karl Rubin , Alice Silverberg

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Sur les représentations modulaires de degré 2 de Gal(Q¯/Q)

1987-01-01

Jean-Pierre Serre

Galois representations into GL2 (Z p [[X]]) attached to ordinary cusp forms

1986-10-01

Haruzo Hida

The Heegner point Kolyvagin system

2004-10-15

Benjamin Howard

In Bull. Soc. Math. France 115 (1987), 399–456, Perrin-Riou formulates a form of the Iwasawa main conjecture which relates Heegner points to the Selmer group of an elliptic curve defined … In Bull. Soc. Math. France 115 (1987), 399–456, Perrin-Riou formulates a form of the Iwasawa main conjecture which relates Heegner points to the Selmer group of an elliptic curve defined over -extension of a quadratic imaginary field K. Building on the earlier work of Bertolini on this conjecture, and making use of the recent work of Mazur and Rubin on Kolyvagin's theory of Euler systems, we prove one divisibility of Perrin-Riou's conjectured equality. As a consequence, one obtains an upper bound on the rank of the Mordell–Weil group E(K) in terms of Heegner points.

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Uniformity of rational points

1997-01-01

Lucia Caporaso , Joe Harris , Barry Mazur

degree in any projective space P n for n ≥ 3 that are known to satisfy it.)In any event, since the main geometric results of this paper are birational in … degree in any projective space P n for n ≥ 3 that are known to satisfy it.)In any event, since the main geometric results of this paper are birational in nature, we will not be able to use it.There is, however, a conjecture that combines the two, a sort of fiber product of them.This is what we call the Strong Lang Conjecture: Conjecture B (Strong Lang Conjecture).Let X be any variety of general type, defined over a number field K.There exists a proper closed subvariety Ξ ⊂ X such that for any number field L containing K, the set of L-rational points of X lying outside of Ξ is finite.It is worth remarking that this is actually implied by Conjecture A together with the purely geometric conjecture: Geometric Lang Conjecture.If X is any variety of general type, the union of all irreducible, positive-dimensional subvarieties of X not of general type is a proper, closed subvariety Ξ ⊂ X.The Strong Lang Conjecture is known to hold for arbitrary subvarieties of abelian varieties ([F]).The Geometric Lang Conjecture has been proved for all surfaces with c 2 1 > c 2 ([B]), and has recently been announced for all surfaces ([LM]).1.1.2.The main arithmetic results of this paper.The first statement we will prove is simply that the Weak Lang Conjecture implies the Uniformity Conjecture; that is, the Theorem 1.1 (Uniform Bound).If Conjecture A is true, then for every number field K and integer g ≥ 2 there exists an integer B(K, g) such that no smooth curve defined over K has more than B(K, g) rational points.We can deduce much more if we assume the stronger Conjecture B: remarkably, we can say that the number B(g, K) depends only on g and not on K! -allowing, as we must, a finite list of exceptions for each K. Precisely, we will prove Theorem 1.2 (Universal Generic Bound).The Strong Lang Conjecture implies that for any g ≥ 2, there exists an integer N(g) such that for any number field K there are only finitely many smooth curves of genus g defined over K with more than N(g) K-rational points.Note that the presence of the finite set of exceptions to the inequality |X(K)| ≤ N(g) is necessary: we can take any given curve X and give it as many K-rational points as we like simply by enlarging the number field K.We can similarly do this for any finite set of curves X 1 , . . ., X n simultaneously, so that the set of exceptions will necessarily grow as K does.What the existence of a universal generic bound N(g) is saying, though, is that even though we can make up towers of number fieldsany such tower is specific to X in the sense that for each i all but finitely many other curves will continue to have N(g) or fewer K i -rational points.The proofs of Theorems 1.1 and 1.2 are based on our main geometric statement, the Correlation Theorem 1.3.This will be the subject of 1.1.3below.

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Uniform distribution of Heegner points

2002-04-01

Vinayak Vatsal

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Arithmetic of elliptic curves upon quadratic extension

1981-01-01

Kenneth Kramer

This paper is a study of variations in the rank of the Mordell-Weil 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> … This paper is a study of variations in the rank of the Mordell-Weil 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> defined over a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as one passes to quadratic extensions <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> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Selmer group for multiplication by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In analogy with genus theory, we describe <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of various objects defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the local norm indices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i Subscript upsilon Baseline equals dimension Subscript bold upper F 2 Baseline upper E left-parenthesis upper F Subscript upsilon Baseline right-parenthesis slash Norm left-brace upper E left-parenthesis upper K Subscript w Baseline right-parenthesis right-brace"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>υ</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>υ</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mtext>Norm</mml:mtext> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{i_\upsilon } = {\dim _{{{\mathbf {F}}_2}}}E({F_\upsilon })/\text {Norm} \{ E({K_w})\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each completion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript upsilon"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>υ</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_\upsilon }</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 F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular we show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension upper S left-parenthesis upper K right-parenthesis plus dimension upper E left-parenthesis upper K right-parenthesis Subscript 2"> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>dim</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\dim S(K) + \dim E{(K)_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the same parity as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma i Subscript upsilon"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>υ</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Sigma {i_\upsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We compute <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i Subscript upsilon"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>υ</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{i_\upsilon }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <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> has good or multiplicative reduction modulo <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upsilon"> <mml:semantics> <mml:mi>υ</mml:mi> <mml:annotation encoding="application/x-tex">\upsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-primary component of the Tate-Shafarevitch group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Ш left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>Ш</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Sha (K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite, as conjectured, we obtain the parity of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For semistable elliptic curves defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbf {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.

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Disparity in Selmer ranks of quadratic twists of elliptic curves

2013-04-04

Zev Klagsbrun , Barry Mazur , Karl Rubin

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction … We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

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Algebraic Number Theory

1994-01-01

Serge Lang

On $p$-adic analytic families of Galois representations

1986-01-01

Barry Mazur , Andrew Wiles

The arithmetic of elliptic curves

1974-09-01

John Tate

Points of order 13 on elliptic curves

1973-03-01

Barry Mazur , John Tate

Galois theory, elliptic curves, and root numbers

1996-01-01

David E. Rohrlich

The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has … The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Given an irreducible complex representation τ of Gal(K/Q), does there exist an elliptic curve E over Q such that τ occurs in the natural representation of Gal(K/Q) on C ⊗Z E(K)? The present paper does not attempt to investigate this question directly. Instead we adopt Greenberg’s point of view in his remarks on nonabelian Iwasawa theory [5] and consider a related question about root numbers. Let ρE denote the representation of Gal(K/Q) on C ⊗Z E(K) and 〈τ, ρE〉 the multiplicity of τ in ρE , and write L(E, τ, s) for the tensor product L-function associated to E and τ . The conjectures of Birch-Swinnerton-Dyer and Deligne-Gross imply that

Diophantine Equations with Special Reference To Elliptic Curves

1966-01-01

J. W. S. Cassels

Diophantine Sets over Some Rings of Algebraic Integers

1978-12-01

Jan Denef , Leonard Lipshitz

Studying the growth of Mordell-Weil

2003-01-01

Barry Mazur , Karl Rubin

We study the growth of the Mordell-Weil groups $E(K\_n)$ of an elliptic curve $E$ as $K\_n$ runs through the intermediate fields of a $\Bbb Z\_p$-extension. We describe those $\Bbb Z\_p$-extensions … We study the growth of the Mordell-Weil groups $E(K\_n)$ of an elliptic curve $E$ as $K\_n$ runs through the intermediate fields of a $\Bbb Z\_p$-extension. We describe those $\Bbb Z\_p$-extensions $K\_\infty/K$ where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms.

On the parity of ranks of Selmer groups II

2001-02-01

Jan Nekovář

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On modular representations of $$(\bar Q/Q)$$ arising from modular forms

1990-12-01

Kenneth A. Ribet

Universal Bounds on the Torsion of Elliptic Curves

1976-09-01

Daniel S. Kubert

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Organizing the arithmetic of elliptic curves

2005-10-24

Barry Mazur , Karl Rubin

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Courbes Elliptiques et Symboles Modulaires

1973-01-01

Barry Mazur

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Class fields of abelian extensions of Q

1984-06-01

Barry Mazur , Andrew Wiles

Twists of newforms and pseudo-eigenvalues ofW-operators

1978-10-01

A. O. L. Atkin , W Li

Modèles minimaux des variétés abéliennes sur les corps locaux et globaux

1964-12-01

André Néron

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The irreducibility of the space of curves of given genus

1969-01-01

Pierre Deligne , D. Mumford

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A modular construction of unramifiedp-extensions ofQ(? p )

1976-10-01

Kenneth A. Ribet

Rings of algebraic numbers in infinite extensions of $${\mathbb {Q}}$$ and elliptic curves retaining their rank

2009-01-05

Alexandra Shlapentokh

Families of rationally connected varieties

2002-07-29

Tom Graber , Joe Harris , Jason Starr

We prove that every one-parameter family of complex rationally connected varieties has a section. We prove that every one-parameter family of complex rationally connected varieties has a section.

Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer.

1965-01-01

J. W. S. Cassels

Article Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. was published on January 1, 1965 in the journal Journal für die reine und angewandte Mathematik … Article Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. was published on January 1, 1965 in the journal Journal für die reine und angewandte Mathematik (volume 1965, issue 217).