Now showing items 1-18 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 (2016-05-02)
      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 p-Adic 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 p-adic 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 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 ...
    • 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)
    • 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 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 ...
    • Rational families of 17-torsion points of elliptic curves over number fields 

      Derickx, Maarten; Kamienny, Sheldon; Mazur, Barry Charles (2017-03-14)
      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 source-module with ...