A Markov model for Selmer ranks in families of twists

作者：Klag**run Zev^*; Mazur Barry; Rubin Karl

来源：Compositio Mathematica, 2014, 150(7): 1077-1106.

DOI：10.1112/S0010437X13007896

• elliptic curves
• quadratic twists
• Selmer groups
• Markov processes
• arithmetic statistics

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) congruent to 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 F-p-representations of the absolute Galois group of K by characters of order p.

• 出版日期2014-7