TY - CHAP

T1 - Endomorphisms of ordinary superelliptic jacobians

AU - Zarhin, Yuri G.

N1 - Funding Information:
2020 Mathematics Subject Classification. Primary 14H40, 14K05, 11G30, 11G10. Key words and phrases. Ordinary abelian varieties, superelliptic Jacobians, endomorphisms of abelian varieties. The author was partially supported by Simons Foundation Collaboration grant #585711. Part of this work was done during the author’s stay at the Institute Henri Poincaré(June - July 2019) and the Weizmann Institute of Science (December 2019 - January 2020), whose hospitality and support are gratefully acknowledged.
Publisher Copyright:
© 2021 American Mathematical Society.

PY - 2021

Y1 - 2021

N2 - Let K be a field of prime characteristic p, n ≥ 5 an integer, f (x) an irreducible polynomial over K of degree n, whose Galois group is either the full symmetric group Sn or the alternating group An . Let ℓ be an odd prime different from p, Z[ζℓ ] the ring of integers in the ℓth cyclotomic field, Cf,ℓ: yℓ = f (x) the corresponding superelliptic curve and J(Cf,ℓ ) its Jacobian. We prove that the ring of all ¯K-endomorphisms of J(Cf,ℓ ) coincides with Z[ζℓ ] if J(Cf,ℓ ) is an ordinary abelian variety and (ℓ, n) ≠ (5, 5).

AB - Let K be a field of prime characteristic p, n ≥ 5 an integer, f (x) an irreducible polynomial over K of degree n, whose Galois group is either the full symmetric group Sn or the alternating group An . Let ℓ be an odd prime different from p, Z[ζℓ ] the ring of integers in the ℓth cyclotomic field, Cf,ℓ: yℓ = f (x) the corresponding superelliptic curve and J(Cf,ℓ ) its Jacobian. We prove that the ring of all ¯K-endomorphisms of J(Cf,ℓ ) coincides with Z[ζℓ ] if J(Cf,ℓ ) is an ordinary abelian variety and (ℓ, n) ≠ (5, 5).

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U2 - 10.1090/conm/767/15397

DO - 10.1090/conm/767/15397

M3 - Chapter

AN - SCOPUS:85107284679

T3 - Contemporary Mathematics

SP - 51

EP - 69

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -