A bound on the $\mu $-invariants of supersingular elliptic curves

Gajek-Leonard, Rylan 

Canadian mathematical bulletin, Tome 68 (2025) no. 3, pp. 787-796

Résumé

Let $E/\mathbb {Q}$ be an elliptic curve and let p be a prime of good supersingular reduction. Attached to E are pairs of Iwasawa invariants $\mu _p^\pm $ and $\lambda _p^\pm $ which encode arithmetic properties of E along the cyclotomic $\mathbb {Z}_p$-extension of $\mathbb {Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $\mu _p^\pm =0$. We provide support for this conjecture by proving that for any $\ell \geq 0$, we have $\mu _p^\pm \leq 1$ for all but finitely many primes p with $\lambda _p^\pm =\ell $. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $\mu _p^\pm \leq 1$ holds on a density 1 set of good supersingular primes for E.

Détail

DOI : 10.4153/S000843952500013X

Mots-clés : Iwasawa theory, elliptic curves, p-adic L-functions