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Selmer ranks of quadratic twists of elliptic curves | | Zev Klagsbrun
; Barry Mazur
; Karl Rubin
; | | Date: |
9 Nov 2011 | | Abstract: | We study the distribution of 2-Selmer ranks in the family of quadratic twists
of an arbitrary elliptic curve E over an arbitrary number field K. We first
prove that the fraction of twists (of a given elliptic curve over a fixed
number field) having even 2-Selmer rank exists as a stable limit over the
family of twists, and we compute this fraction as an explicit product of local
factors. We give an example of an elliptic curve E such that as K varies, these
fractions are dense in [0, 1]. Under the assumption that Gal(K(E[2])/K) = S_3
we also show that the density (counted in a non-standard way) of twists with
Selmer rank r exists for all positive integers r, and is given via an
equilibrium distribution, depending only on the "parity fraction" alluded to
above, 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 2-dimensional
self-dual F_p-representations of the absolute Galois group of K by characters
of order p. | | Source: | arXiv, 1111.2321 | | Services: | Forum | Review | PDF | Favorites | |
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