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Let $A/\mathbb{F}_q$ be an abelian variety of dimension $g$ defined over a finite field. Its L-polynomial is the polynomial $$P(A/\mathbb{F}_q,t) = \det(1-t F_q|H^1((A_{\overline{\mathbb{F}}_q})_{et}, \mathbb{Q}_l)),$$ where $F_q$ is the inverse of Frobenius acting on cohomology. This is a polynomial of degree $2g$ with integer coefficients. By a theorem of Weil, the complex roots of this polynomial all have norm $1/\sqrt{q}$; this means that there are only finitely many L-polynomials for any fixed pair $(q,g)$.

The L-polynomial of $A$ is the reverse of the Weil polynomial.

Knowl status:
  • Review status: reviewed
  • Last edited by Kiran S. Kedlaya on 2019-09-04 17:53:55
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