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Two-dimensional families of hyperelliptic Jacobians with big monodromy


Author: Yuri G. Zarhin
Journal: Trans. Amer. Math. Soc. 368 (2016), 3651-3672
MSC (2010): Primary 14H40, 14K05, 11G30, 11G10
DOI: https://doi.org/10.1090/tran/6579
Published electronically: December 3, 2015
MathSciNet review: 3451889
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Abstract: Let $ K$ be a global field of characteristic different from $ 2$ and $ u(x)\in K[x]$ be an irreducible polynomial of even degree $ 2g\ge 6$ whose Galois group over $ K$ is either the full symmetric group $ \mathbf {S}_{2g}$ or the alternating group $ \mathbf {A}_{2g}$.
We describe explicitly how to choose (infinitely many) pairs of distinct
$ t_1, t_2 \in K$ such that the $ g$-dimensional Jacobian of a hyperelliptic curve $ y^2=$
$ (x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $ K$ and has big monodromy.


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Additional Information

Yuri G. Zarhin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802; Department of Mathematics, The Weizmann Institute of Science, P.O.B. 26, Rehovot 7610001, Israel
Email: zarhin@math.psu.edu

DOI: https://doi.org/10.1090/tran/6579
Received by editor(s): October 25, 2013
Received by editor(s) in revised form: July 12, 2014
Published electronically: December 3, 2015
Additional Notes: This work was partially supported by a grant from the Simons Foundation (#246625).
Article copyright: © Copyright 2015 American Mathematical Society

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