| Jeroen Sijsling |  |
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Date of the talk: 15 June 2012
Descent for hyperelliptic curves
Let k be a field, and let C be a curve over the algebraic closure of k of genus g with automorphism group G that is isomorphic with all its conjugates over k . Is it then possible to find a curve C0 defined over k that is isomorphic with C (over the algebraic closure), and if one is given such a C for which the response is positive, can one then construct C0 and the isomorphism with C explicitly?It is well-known that if g = 0 or 1 or if G is trivial, one can always descend, and moreover, it is known how to do this explicitly. For g = 2 , there generically exists and obstruction to descent with was discovered by Mestre, who has also given algorithms to descend explicitly if this obstruction is trivial.
In this talk, we will consider these problems for general hyperelliptic curves. After recalling some results for "big" G by Huggins and Cardona-Quer, we will give criteria and explicit construction in the case where the order of G is at most 4 . Finally, we will give (for any G ) an arithmetic criterion for the existence of a descended curve C0 that again admits a hyperelliptic equation.
This is joint work with Reynald Lercier and Christophe Ritzenthaler.
