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A Markov model for Selmer ranks in families of twists

Part of: Arithmetic algebraic geometry Markov processes

Published online by Cambridge University Press:  30 June 2014

Zev Klagsbrun ,
Barry Mazur  and
Karl Rubin
Zev Klagsbrun
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA email zev.klagsbrun@gmail.com Current address: Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121, USA
Barry Mazur
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA email mazur@math.harvard.edu
Karl Rubin
Affiliation:
Department of Mathematics, UC Irvine, Irvine, CA 92697, USA email krubin@math.uci.edu
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Abstract

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

Keywords

elliptic curvesquadratic twistsSelmer groupsMarkov processesarithmetic statistics

MSC classification

Secondary: 11G05: Elliptic curves over global fields 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 7 , July 2014 , pp. 1077 - 1106
Copyright
© The Author(s) 2014 

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