Iwasawa invariants for elliptic curves over ℤ _p -extensions and Kida's formula

Abstract

This paper aims at studying the Iwasawa λ-invariant of the p-primary Selmer group. We study the growth behavior of p-primary Selmer groups in p-power degree extensions over non-cyclotomic ℤ p {\mathbb{Z}_{p}} -extensions of a number field. We prove a generalization of Kida’s formula in such a case. Unlike the cyclotomic ℤ p {\mathbb{Z}_{p}} -extension, where all primes are finitely decomposed, in the ℤ p {\mathbb{Z}_{p}} -extensions we consider primes may be infinitely decomposed. In the second part of this paper, we study the relationship of Iwasawa invariants with respect to congruences, obtaining refinements of the results of Greenberg, Vatsal and Kidwell. As an application, we provide an algorithm for constructing elliptic curves with large anticyclotomic λ-invariant. Our results are illustrated by explicit computation.

On the Distribution of Iwasawa Invariants Associated to Multigraphs

“…This allows us to prove that μ is always zero and give an upper bound on λ under the hypothesis that is larger than the number of loops in a bouquet (see Corollary 4.3 and Theorem 4.13 for the precise statements). Without this hypothesis, we show that both μ and λ can be arbitrarily large (see Lemma 2.17), which mirrors similar results in the setting of Iwasawa theory of number fields and elliptic curves (see [11], [13], [15], [18]). (b) Let t be the number of loops in a bouquet.…”

“…The derived multigraphs are again seen to be connected for all n by choosing a suitable spanning tree T in Theorem 2. 15. One can choose the spanning tree T to pass through all vertices in the following order v 1 , v t , v t−1 , .…”