p-Selmer growth in extensions of degree p

Abstract

Abstract. There is a known analogy between growth questions for class groups and for Selmer groups. If p is a prime, then the p-torsion of the ideal class group grows unboundedly in Z{pZ-extensions of a fixed number field K, so one expects the same for the p-Selmer group of a nonzero abelian variety over K. This Selmer group analogue is known in special cases and we prove it in general, along with a version for arbitrary global fields.1. Introduction 1.1. Growth of class groups and of Selmer groups. It is a classical theorem of Gauss that the 2-torsion subgroup PicpO L qr2s of the ideal class group of a quadratic number field L can be arbitrarily large. Although unboundedness of # PicpO L qrps for an odd prime p is a seemingly inaccessible conjecture, [BCH`66, VII-12, Thm. 4] explains how to extend Gauss' methods to prove that # PicpO L qrps is unbounded if L{Q ranges over the Z{pZ-extensions instead.As explained in [Čes15a], growth questions for ideal class groups and for Selmer groups of abelian varieties are often analogous. It is therefore natural to hope that for a prime p and a nonzero abelian variety A over Q, the p-Selmer group Sel p A L can be arbitrarily large when L{Q ranges over the Z{pZ-extensions. Our main result confirms this expectation. Theorem 1.2 (Theorem 5.6). Let p be a prime, K a global field, and A a nonzero abelian variety overis unbounded when L ranges over the Z{pZ-extensions of K. Remarks.1.3. In the excluded case when ArpspKq " 0 and A is not supersingular (when also char K " p and dim A ą 2), there nevertheless is an n P Z ą0 that depends on A such thatis unbounded when L ranges over the Z{p n Z-extensions of K, see Theorem 5.6.1.4. See Corollary 5.5 for a version of Theorem 1.2 for Selmer groups of arbitrary isogenies.1.5. The proof of Theorem 1.2 also reproves the unbounded growth of the p-torsion subgroup of the ideal class group in Z{pZ-extensions of K, see Corollary 5.3.

The growth of Tate–Shafarevich groups in cyclic extensions

“…The fppf cohomology was used in the original statement of [Čes17, Theorem 4.2]. Here we rephrase it in the language of Galois cohomology, due to the canonical isomorphism (see [Čes15b, p. 1661, equation (1)]), where , is the Néron model of and with the proper smooth curve over a finite field whose function field is .…”

“…Theorem 1.1 [Čes17,Theorem 1.2]. Let p be a prime number, and A be an abelian variety over a global field K. If either A[p]( K) = 0 or A is supersingular, then dim Fp Sel p (A/L) is unbounded as L varies over (Z/pZ)-extensions of K.…”

“…When fixing a (Z/nZ)-extension K/Q, Matsuno [Mat09] proved that there exist elliptic curves E/Q with the The growth of Tate-Shafarevich groups in cyclic extensions n-rank of X(E/K) [n] being arbitrarily large. Based on these results, Česnavičius proposed the following problem [Čes17,Problem 1.8].…”

“…The idea behind the proof of the above results is in fact rather simple. In [Čes17], Česnavičius found a method to construct a sequence of -extensions such that However, he did not bound . We will use the tools developed by Mazur and Rubin in [MR18] to bound the Mordell–Weil ranks.…”

Rank Jumps and Growth of Shafarevich–tate Groups for Elliptic Curves in -Extensions

“…In [Čes17], Česnavičius showed that the p -Selmer group of elliptic curves over a number field becomes arbitrarily large when varying over -extensions. This result is not surprising, as it is widely believed that the growth of ideal class groups and Selmer groups of elliptic curves are often analogous.…”