Complete bipartite maps, factorisable groups and generalised Fermat curves

AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

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Gamma function identities and elliptic differentials on Fermat curves

Duke Mathematical Journal ◽

10.1215/s0012-7094-78-04507-6 ◽

1978 ◽

Vol 45 (1) ◽

pp. 87-99 ◽

Cited By ~ 5

Author(s):

Neal Koblitz

Keyword(s):

Gamma Function ◽

Fermat Curves

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Points at infinity on the Fermat curves

Inventiones mathematicae ◽

10.1007/bf01390104 ◽

1977 ◽

Vol 39 (2) ◽

pp. 95-127 ◽

Cited By ~ 15

Author(s):

David E. Rohrlich

Keyword(s):

Fermat Curves

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Rational points on Fermat curves over finite fields

Journal of Algebra and Its Applications ◽

10.1142/s0219498817500463 ◽

2017 ◽

Vol 16 (03) ◽

pp. 1750046

Author(s):

Wei Cao ◽

Shanmeng Han ◽

Ruyun Wang

Keyword(s):

Finite Fields ◽

Rational Point ◽

Rational Points ◽

Fermat Curve ◽

Fermat Curves ◽

Curves Over Finite Fields

Let [Formula: see text] be the [Formula: see text]-rational point on the Fermat curve [Formula: see text] with [Formula: see text]. It has recently been proved that if [Formula: see text] then each [Formula: see text] is a cube in [Formula: see text]. It is natural to wonder whether there is a generalization to [Formula: see text]. In this paper, we show that the result cannot be extended to [Formula: see text] in general and conjecture that each [Formula: see text] is a cube in [Formula: see text] if and only if [Formula: see text].

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