A Markov model for Selmer ranks in families of twists

2014/06/30 by Zev Klagsbrun, Barry Mazur, Karl Rubin · 3 citations
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Benford’s Law and Fraud Detection

paper · pdf · doi:10.1112/s0010437x13007896

Abstract

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 ≤ =\leqslant \let ≤ =\leqslant \let ≥ =\geqslant \let ≥ =\geqslant \def Pr \mathit Pr\def \Fr \mathit Fr\def \Rey \mathit ReE over an arbitrary number field K . Under the assumption that \rm Gal(K(E[2])/K) ≅ S₃ , 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 .

Citations

• Selmer companion curves
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques
• Disparity in Selmer ranks of quadratic twists of elliptic curves

Cited by

• p-twisted Selmer near-companion curves
• Selmer ranks under quadratic twists satisfying the Heegner hypothesis
• Constructing Jacobians of rank 1

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