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Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras

Open Access article
 Printed: Mar 2012
  • William Butske,
    Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN 47907, U.S.A.
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Zarhin proves that if $C$ is the curve $y^2=f(x)$ where $\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his result in the genus $g=2$ case supposing other Galois groups, we calculate $\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$ 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 ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.
MSC Classifications: 11G10, 20C20 show english descriptions Abelian varieties of dimension $> 1$ [See also 14Kxx]
Modular representations and characters
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
20C20 - Modular representations and characters

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