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Endomorphisms of superelliptic jacobians

Endomorphisms of superelliptic jacobians Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, $${\mathbb{Z}[\zeta_p]}$$ the ring of integers in the pth cyclotomic field, C f, p : y p  =  f(x) the corresponding superelliptic curve and J(C f, p ) its jacobian. Assuming that either n  =  p + 1 or p does not divide n(n  −  1), we prove that the ring of all endomorphisms of J(C f, p ) coincides with $${\mathbb{Z}[\zeta_p]}$$ . The same is true if n  =  4, the Galois group of f(x) is the full symmetric group S 4 and K contains a primitive pth root of unity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Endomorphisms of superelliptic jacobians

Mathematische Zeitschrift , Volume 261 (3) – Apr 4, 2008

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References (53)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-008-0342-5
Publisher site
See Article on Publisher Site

Abstract

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, $${\mathbb{Z}[\zeta_p]}$$ the ring of integers in the pth cyclotomic field, C f, p : y p  =  f(x) the corresponding superelliptic curve and J(C f, p ) its jacobian. Assuming that either n  =  p + 1 or p does not divide n(n  −  1), we prove that the ring of all endomorphisms of J(C f, p ) coincides with $${\mathbb{Z}[\zeta_p]}$$ . The same is true if n  =  4, the Galois group of f(x) is the full symmetric group S 4 and K contains a primitive pth root of unity.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Apr 4, 2008

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