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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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