Computing the geometric endomorphism ring of a genus-2 Jacobian

DOI10.1090/MCOM/3358zbMATH Open1410.11043arXiv1610.09674OpenAlexW2950267487MaRDI QIDQ4561378

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Publication date: 5 December 2018

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Abstract: We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $operatorname{End}_{overlineK}(A)$ 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 $operatornameJac(C)$ 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.

Full work available at URL: https://arxiv.org/abs/1610.09674

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