Classical Iwasawa theory and infinite descent on a family of abelian varieties

TL;DR

This paper explores the Iwasawa theory of a specific family of CM elliptic curves, confirming conjectural predictions and revealing phenomena related to Greenberg's conjecture, using elementary methods for primes congruent to 7 mod 16.

Contribution

It provides new elementary proofs and results on the Iwasawa theory of Gross's elliptic curves, aligning with BSD conjecture predictions and addressing Greenberg's conjecture.

Findings

Strong results on Iwasawa theory for primes q ≡ 7 mod 16

Confirmation of BSD conjecture predictions for these curves

New insights into Greenberg's conjecture phenomena

Abstract

For primes $q \equiv 7 \mod 16$, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field $K = \mathbb{Q}(\sqrt{-q})$, which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.