Growth of Selmer rank in nonabelian extensions of number fields

Details Growth of Selmer rank in nonabelian extensions of number fields Growth of Selmer rank in nonabelian extensions of number fields

2007-03-13

arxiv.org/abs/math/0703363v3

Mathematics

Number Theory

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Scientific paper

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k 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 group of E over F. We obtain lower bounds for rk_p(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, G = Gal(F/K), G^+ is the subgroup of elements of G commuting with a choice of involution of F over k, and rk_p(E/K) is odd, then we show that (under mild hypotheses) rk_p(E/F) \ge [G:G^+]$. As a very specific example of this, suppose A is an elliptic curve over Q with a rational torsion point of order p, and with no complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p, such that every prime where both E and A have bad reduction has odd order in F_p^\times, and such that the negative of the conductor of E is not a square mod p, then there is a positive constant B, depending on A but not on E or n, such that rk_p(E/Q(A[p^n])) \ge B p^{2n} for every n.

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