Iwasawa invariants of sharp/flat $2$-adic $L$-functions for quadratic twists of elliptic curves

Abstract

The aim of this paper is to study the variation under quadratic twists of the analytic Iwasawa invariants of Sprung's sharp/flat 2-adic $L$-functions for elliptic curves over $\mathbb{Q}$ with good supersingular reduction at $2$. Under the hypothesis that the $μ$-invariant vanishes, we obtain an explicit formula for the sharp/flat $λ$-invariants. This formula gives a supersingular analogue of Matsuno's formula in the good ordinary case. As an application, we show that the sharp/flat $λ$-invariants can be made arbitrarily large even among quadratic twists by single primes. Moreover, using the method of Hatley-Ray, we obtain an asymptotic lower bound for the number of quadratic twists with a prescribed sharp/flat 2-adic Iwasawa $λ$-invariant.

Disclosure

“2 (log X) 3 for each i ∈ {3, 4, 5}. Methods During the preparation of this manuscript, the author used ChatGPT solely as a supple- mentary tool to identify possible inconsistencies. All mathematical arguments and conclu- sions were independently formulated and verified by the author, who takes full responsibility for the content of the manuscript.”

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2026-08-24T00:34:09.244Z

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