Hyperelliptic jacobians with real multiplication

Details Hyperelliptic jacobians with real multiplication Hyperelliptic jacobians with real multiplication

2004-03-31

arxiv.org/abs/math/0403553v5

Mathematics

Algebraic Geometry

Corrected typos; clarified proofs; added more examples in positive characteristic

Scientific paper

Let $K$ be a field of characteristic $p \neq 2$, and let $f(x)$ be a sextic polynomial irreducible over $K$ with no repeated roots, whose Galois group is isomorphic to $\A_5$. If the jacobian $J(C)$ of the hyperelliptic curve $C:y^2=f(x)$ admits real multiplication over the ground field from an order of a real quadratic field $D$, then either its endomorphism algebra is isomorphic to $D$, or $p > 0$ and $J(C)$ is a supersingular abelian variety. The supersingular outcome cannot occur when $p$ splits in $D$.

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