Iwasawa invariants

The Iwasawa invariants of elliptic curves are $\lambda$-invariants and $\mu$-invariants.

For an elliptic curve $E$ and a prime $p$ of ordinary or multiplicative reduction, the $\lambda$- and $\mu$-invariants are non-negative integers $\lambda_p(E)$ and $\mu_p(E)$ which arise as the Iwasawa invariants of the $p$-adic L-function $L_p(E)$.

If $E$ has supersingular reduction at $p$, then there are two associated $p$-adic L-functions: $L_{p,\alpha}(E)$ and $L_{p,\beta}(E)$ associated to the pair of roots $\alpha$ and $\beta$ of the Hecke polynomial $x^2-a_p(E)x+p$. However, in this case, these functions are not Iwasawa functions. Instead, we consider the pair of Iwasawa functions $L_p^+(E)$ and $L_p^-(E)$ as defined in Robert Pollack, On the p-adic L-function of a modular form at a supersingular prime, Duke Mathematical Journal, 118 (2003) no. 3, 523-558 MR:1983040 and Florian Sprung, Iwasawa theory for elliptic curves at supersingular primes: a pair of main conjectures. J. Number Theory 132 (2012), no. 7, 1483–1506 MR:2903167. We then get two $\lambda$-invariants: $\lambda^+_p(E)$ and $\lambda^-_p(E)$, and two $\mu$-invariants: $\mu^+_p(E)$ and $\mu^-_p(E)$. These invariants are computed as in section 6 of Bernadette Perrin-Riou, Arithmétique des courbes elliptiques à réduction supersingulière en p, Experimental Mathematics 12 (2003), no. 2, 155–186 [MR:2016704].

We note that if $E$ has rank 0, then its Iwasawa invariants at a prime $p>5$ are zero unless at least one of the following hold:

1. the $p$-adic valuation of $L(E,1)/\Omega_E$ is non-zero;
2. $p$ is a split multiplicative prime;
3. $a_p=1$.

Conditions (1) and (2) hold for only finitely many $p$. Condition (3) will hold for finitely many $p$ if $E$ possesses a non-trivial torsion point. In this case, we compute a prime $q$ so that for all good $p \geq q$ the Iwasawa invariants at $p$ vanish.