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arXiv · 1610.09674

Computing the geometric endomorphism ring of a genus 2 Jacobian

Davide Lombardo
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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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Davide Lombardo. 2021-06-01. Computing the geometric endomorphism ring of a genus 2 Jacobian. https://doi.org/10.1090/mcom%2F3358

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