Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras

Butske, William

Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 38-47

Résumé

Zarhin proves that if $C$ is the curve ${{y}^{2}}\,=\,f(x)$ where $\text{Ga}{{\text{l}}_{\mathbb{Q}}}(f(x))\,=\,{{S}_{n}}$ or ${{A}_{n}}$ , then $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,=\,\mathbb{Z}$ . In seeking to examine his result in the genus $g\,=\,2$ case supposing other Galois groups, we calculate $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)\,{{\otimes }_{\mathbb{Z}}}\,{{\mathbb{F}}_{2}}$ for a genus 2 curve where $f(x)$ is irreducible. In particular, we show that unless the Galois group is ${{S}_{5}}$ or ${{A}_{5}}$ , the Galois group does not determine $\text{En}{{\text{d}}_{\overline{\mathbb{Q}}}}(J)$ .

Détail

DOI : 10.4153/CMB-2011-045-x

Mots-clés : 11G10, 20C20