Descent of algebraic curves
Final Report Abstract
The project gives algorithms for determining which algebraic curves can be described by means of defining equations over small fields, such as the field of rational numbers. In this case, the arithmetic properties of such a curve can be described quite completely, for example by means of the L-functions and Modular Forms Database (or LMFDB), which has made such descriptions publicly available for many algebraic curves over small fields. Provided that the descent question has an affirmative answer for a given curve, its arithmetic can therefore be studied to complete satisfaction. The main result of the project is a study of the descent problem for the class of superelliptic curves. This study gives a precise condition under which one can determine whether or not a given curve descends. If this is indeed the case, then it is also indicated how explicit defining equations over the smaller field can be obtained. All results were implemented in computer algebra systems.
Publications
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Galois descent of superelliptic curves, PhD thesis at Ulm University
Robert Slob
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Functionalities for genus 2 and 3 curves. Proceedings of MEGA 2021
Reynald Lercier, Christophe Ritzenthaler & Jeroen Sijsling
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Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication. Mathematics of Computation, 92(339), 349-383.
Dina, Bogdan; Ionica, Sorina & Sijsling, Jeroen
