Computing the geometric endomorphism ring of a genus-2 Jacobian

Lombardo, Davide

doi:10.1090/mcom/3358

Computing the geometric endomorphism ring of a genus-2 Jacobian
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Math. Comp. 88 (2019), 889-929 Request permission

Abstract:

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname {End}_{\overline {K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname {Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus-2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.

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