Endomorphisms of superelliptic jacobians

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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, ℤ[ζ p] the ring of integers in the pth cyclotomic field, Cf, p : yp = 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 ℤ[ζp]. The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.

Original languageEnglish (US)
Pages (from-to)691-707
Number of pages17
JournalMathematische Zeitschrift
Volume261
Issue number3
DOIs
StatePublished - Mar 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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