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Frobenius distribution for quotients of Fermat curves of prime exponent. (English) Zbl 1400.11086
Summary: Let $$\mathcal{C}$$ denote the Fermat curve over $$\mathbb{Q}$$ of prime exponent $$\ell$$. The Jacobian $$\text{Jac}(\mathcal{C})$$ of $$\mathcal{C}$$ splits over $$\mathbb{Q}$$ as the product of Jacobians $$\text{Jac}(\mathcal{C}_k), 1\leq k\leq \ell-2$$, where $$\mathcal{C}_k$$ are curves obtained as quotients of $$\mathcal{C}$$ by certain subgroups of automorphisms of $$\mathcal{C}$$. It is well known that $$\text{Jac}(\mathcal{C}_k)$$ is the power of an absolutely simple abelian variety $$B_k$$ with complex multiplication. We call degenerate those pairs $$(\ell,k)$$ for which $$B_k$$ has degenerate CM type. For a non-degenerate pair $$(\ell,k)$$, we compute the Sato-Tate group of $$\text{Jac}(\mathcal{C}_k)$$, 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 $$(\ell,k)$$ being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $$\ell$$-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

MSC:
 11D41 Higher degree equations; Fermat’s equation 11G10 Abelian varieties of dimension $$> 1$$ 11M50 Relations with random matrices 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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