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The analytic $\mu$-invariant of an elliptic curve $E$ at a prime of ordinary or multiplicative reduction is the largest integer $n$ such that $p^n$ divides the $p$-adic $L$-function of $E$. By the main conjecture for elliptic curves, this invariant should match the algebraic $\mu$-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 $\mu$-invariants: $\mu^+_p(E)$ and $\mu^-_p(E)$.

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• Review status: reviewed
• Last edited by John Jones on 2018-06-19 18:55:27
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