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The L-function of an Abelian variety $A$ over $\Q$ is defined as an Euler product $\prod_p L_p(p^{-s})$ over primes $p$, where the L-polynomial $L_p(T)$ is an integer polynomial of degree at most $2g$, where $g$ is the dimension of $A$; the L-function of a curve is then defined as the L-function of its Jacobian. The $L$-function is an invariant of the isogeny class of $A$.

For primes $p$ where $A$ has good reduction (all but finitely many), the polynomial $L_p(T)=T^{2g}\chi(T^{-1})$ is determined by the characteristic polynomial $\chi(T)$ of the Frobenius endomomrphism of the reduction of $A$ modulo $p$.

The Euler factors at bad primes $p$ depend on the type of bad reduction $A$ has at $p$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Cremona on 2018-06-18 12:02:25
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