Growth of Selmer rank in nonabelian extensions of number fields
| dc.creator | Mazur, Barry | |
| dc.creator | Rubin, Karl | |
| dc.date | 2007-03-13 | |
| dc.date | 2007-09-12 | |
| dc.date.accessioned | 2026-07-07T08:28:52Z | |
| dc.date.available | 2026-07-07T08:28:52Z | |
| dc.description | 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. | |
| dc.description | Minor revisions | |
| dc.identifier | https://arxiv.org/abs/math/0703363 | |
| dc.identifier | http://arxiv.org/abs/math/0703363 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137773 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05 (Primary); 14G05, 11R23, 20C15 (Secondary) | |
| dc.title | Growth of Selmer rank in nonabelian extensions of number fields | |
| dc.type | text |