Browsing by Author "Mazur, Barry"
Now showing items 118 of 18

Average Ranks of Elliptic Curves: Tension between Data and Conjecture
Bektemirov, Baur; Mazur, Barry C.; Stein, William; Watkins, Mark (American Mathematical Society, 2007)Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational ... 
A celebration of the mathematical work of Glenn Stevens
Mazur, Barry Charles (Springer Nature, 2015) 
Complete Homogeneous Varieties via Representation Theory
Cavazzani, Francesco (20160502)Given an algebraic variety $X\subset\PP^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$translates of $X$. We define $X$ to be a \textit{homogeneous variety} if ... 
Computation of pAdic Heights and Log Convergence
Mazur, Barry C.; Stein, William; Tate, John Torrence (Universität Bielefeld, Fakultät für Mathematik, 2006)This paper is about computational and theoretical questions regarding padic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of ... 
Disparity in Selmer Ranks of Quadratic Twists of Elliptic Curves
Klagsbrun, Zev; Mazur, Barry Charles; Rubin, Karl (Princeton University, Department of Mathematics, 2013)We study the parity of 2Selmer 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 ... 
Finding Large Selmer Rank via an Arithmetic Theory of Local Constants
Mazur, Barry C.; Rubin, Karl (Princeton University, 2007)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 ... 
Growth of Selmer Rank in Nonabelian Extensions of Number Fields
Mazur, Barry C.; Rubin, Karl (Duke University Press, 2008)Let \(p\) be an odd prime number, let E be an elliptic curve over a number field \(k\), and let \(F/k\) be a Galois extension of degree twice a power of p. We study the \(Z_p\)corank \(rk_p(E/F)\) of the \(p\)power Selmer ... 
Mathematical Platonism and its Opposites
Mazur, Barry (European Mathematical Society, 2008) 
Mathematics: Controlling Our Errors
Mazur, Barry C. (Nature Publishing Group, 2006)The Sato–Tate conjecture holds that the error term occurring in many important problems in number theory conforms to a specific probability distribution. That conjecture has now been proved for a large group of cases. Even ... 
Nearly Ordinary Galois Deformations over Arbitrary Number Fields
Calegari, Frank; Mazur, Barry C. (Cambridge University Press, 2009)Let \(K\) be an arbitrary number field, and let \(\rho: Gal(K \bar/K) \rightarrow GL_2(E)\) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ... 
On Mathematics, Imagination and the Beauty of Numbers
Mazur, Barry C.; Pesic, Peter (Massachusetts Institute of Technology Press, 2005) 
Perturbations, Deformations, and Variations (and "NearMisses") in Geometry, Physics, and Number Theory
Mazur, Barry C. (American Mathematical Society, 2004) 
Pourquoi les Nombres Premiers?
Mazur, Barry C. (La Recherche, 2005)À l’image des atomes pour les molécules, les nombres premiers sont les briques élémentaires des nombres entiers. Quantité de problèmes sur les premiers sont aussi simples à énoncer que difficiles à attaquer. Un éminent ... 
Ranks of Twists of Elliptic Curves and Hilbert’s Tenth Problem
Mazur, Barry C.; Rubin, Karl (Springer Verlag, 2010)In this paper we investigate the 2Selmer 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 2Selmer ... 
Rational families of 17torsion points of elliptic curves over number fields
Derickx, Maarten; Kamienny, Sheldon; Mazur, Barry Charles (20170314)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 ... 
Refined Class Number Formulas and Kolyvagin Systems
Mazur, Barry C.; Rubin, Karl (Compositio Mathematica, 2011)We use the theory of Kolyvagin systems to prove (most of) a reﬁned class number formula conjectured by Darmon. We show that for every odd prime \(p\), each side of Darmon’s conjectured formula (indexed by positive integers ... 
Thinking about Grothendieck
Mazur, Barry Charles (2016) 
Twisting Commutative Algebraic Groups
Mazur, Barry C.; Rubin, Karl; Silverberg, Alice (Elsevier, 2007)If \(V\) is a commutative algebraic group over a field \(k\), [View the MathML] source is a commutative ring that acts on \(V\), and View the MathML source is a finitely generated free View the MathML sourcemodule with ...