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The analytic $\lambda$-invariant of an elliptic curve $E$ at a prime $p$ of ordinary or multiplicative reduction is the number of zeroes of the $p$-adic L-function of $E$. Explicitly, if we write the $p$-adic L-function of $E$ as a power series in a variable $T$: $$L_p(E,T) = p^{\mu_p(E)} (a_0 + a_1 T + a_2 T^2 + ...)$$ where $\mu_p(E)$ is the $\mu$-invariant of $E$, then the $\lambda$-invariant is the first index $i$ such that $a_i$ is a $p$-adic unit. By the main conjecture for elliptic curves, this invariant should match the algebraic $\lambda$-invariant which is defined analogously in terms of the Selmer group of $E$.

When $E$ has supersingular reduction at $p$, there is a pair of $p$-adic $L$-functions: $L_p^+(E,T)$ and $L_p^-(E,T)$ and one defines analogously a pair of $\lambda$-invariants: $\lambda^+_p(E)$ and $\lambda^-_p(E)$.

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• Last edited by John Jones on 2018-06-19 00:57:42
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