Rigorous computation of the endomorphism ring of a Jacobian. (English) Zbl 1484.11135

Costa, Edgar; Mascot, Nicolas; Sijsling, Jeroen; Voight, John

Math. Comput. 88, No. 317, 1303-1339 (2019).

Let \(F\) be a number field with algebraic closure \(F^{\text{al}}\). Given a nice curve \(X\) over \(F\), denote by \(J\) and \(J^{\text{al}}\) the Jacobian of \(X\) and its base change to \(F^{\text{al}}\). The practical computations of the geometric endomorphism ring \(\text{End}(J^{\text{al}})\) for the curve \(X\) of gens \(\geq 2\) is the main concern of the paper under review, which is done by several improvements and generalization of the existing algorithms and methods.
In Section 2, after setting up some notation and background, the authors discussed the representations of endomorphisms in bits. Sections 3-5 are devoted to describing the algorithms and methods of numerically computing the group law of the Jacobian by developing the methods of [K. Khuri-Makdisi, Math. Comput. 73, No. 245, 333–357 (2004; Zbl 1095.14057)]. The key point of their methods is to use the Newton and Puiseux lifts to numerical inversion of the Abel-Jacobi map by working infinitesimally. Then, in Section 6, they prove the correctness of their methods and algorithm.
In Section 7, the upper bounds on the dimension of endomorphism algebra as a \(\mathbb Q\)-vector space has been considered. They showed how determining Frobenius action on \(X\) for a large set of primes often quickly leads to sharp upper bounds. The last section of the paper includes several worked examples of curves with genus \(\geq 2\), where the algorithms and methods are examined practically.

Reviewer: Sajad Salami (Rio de Janeiro)

MSC:

11G10	Abelian varieties of dimension \(> 1\)
11Y99	Computational number theory
14H40	Jacobians, Prym varieties
14K15	Arithmetic ground fields for abelian varieties
14Q05	Computational aspects of algebraic curves

Keywords:

nice curve; Jacobian; endomorphism ring

Citations:

Zbl 1095.14057

Software:

arithmetic-geometric_mean; LMFDB; GitHub; endomorphisms; Magma

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