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Proceedings of the London Mathematical Society Advance Access published online on June 9, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp020
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© 2009 London Mathematical Society

Families of absolutely simple hyperelliptic jacobians

Yuri G. Zarhin

Department of Mathematics
Pennsylvania State University
University Park, PA 16802
USA

Received 8 July 2008. Revision received 18 March 2009.

We prove that the jacobian of a hyperelliptic curve y2 = (x – t)h(x) has no non-trivial endomorphisms over an algebraic closure of the ground field K of characteristic zero if t isin K and the Galois group of the polynomial h(x) over K is an alternating or symmetric group on deg(h) letters and deg(h) is an even number greater than 8. (The case of odd deg(h) > 3 follows easily from previous results of the author.)

2000 Mathematics Subject Classification 14H40 (primary), 14K05, 11G30, 11G10 (secondary).


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