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

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 $\mu$-invariant vanishes, we obtain an explicit difference formula for the sharp/flat $\lambda$-invariants. This formula gives a supersingular analogue of Matsuno’s formula in the good ordinary case. As an application, following the method of Hatley-Ray, we obtain asymptotic lower bounds for the number of quadratic twists with prescribed sharp/flat $2$-adic Iwasawa $\lambda$-invariant.