Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

Abstract

Abstract. Let C denote the Fermat curve over Q of prime exponent ℓ. e Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C k ), ≤ k ≤ ℓ − , where C k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C k ) is the power of an absolutely simple abelian variety B k with complex multiplication. We call degenerate those pairs (ℓ, k) for which B k has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato-Tate group of Jac(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 (ℓ, k) being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic eld. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

On the singularity of the Demjanenko matrix of quotients of Fermat curves

“…On the other hand, combining the bound of Theorem 5 with the standard heuristic on the distribution on primes, one can derive a heuristic upper bound on ℓ s . We remark that it is shown in [FGL14] that if k satisfies the conditions of Lemma 6, that is, k ∈ K ℓ , then the rank rk ( Finally, we remark that our approach allows to study the distribution of the values of M(k, ℓ) for every ℓ.…”

“…Koblitz and Rohrlich [KR78] show that the subgroup W k,ℓ := {w ∈ (Z/ℓZ) * | wM k,ℓ = M k,ℓ } of elements stabilizing M k,ℓ has cardinality 3 or 1 depending on whether the parameter k is a primitive cubic root of unity modulo ℓ or not. The Demjanenko matrix is then defined as This is a curve of genus (ℓ − 1)/2 that may be obtained as a quotient of the Fermat curve F ℓ : Y ℓ = X ℓ + 1 by a certain subgroup of automorphisms of F ℓ ; we refer to [FGL14] for details. The fact that the rank of D k,ℓ coincides with the dimension of the Hodge group of the Jacobian of C k,ℓ has been exploited in [FGL14] to determine the distribution of Frobenius traces attached to C k,ℓ when D k,ℓ is non-singular.…”

Sato-Tate groups of 𝑦²=𝑥⁸+𝑐 and 𝑦²=𝑥⁷-𝑐𝑥.

“…is a useful tool for explicitly determining Sato-Tate groups (see [FGL16], for example, where this is exploited), but here we take a different approach that is better suited to our special situation. Our strategy is to identify an elliptic quotient of each of the curves C 1 and C 2 and then use the classification results of [FKRS12] to identify the Sato-Tate group of the complement abelian surface.…”

Sato-Tate Distributions of Catalan Curves

“…As noted in [16], it is not expected, in dimension greater than 3, that every group satisfying the Sato-Tate axioms (see Section 2.4) can be realized using Jacobians of curves. There are currently two articles that study families of Jacobian varieties of arbitrarily high genus (see [9,10]). In this article we provide a new example of an infinite family of Sato-Tate groups that can be realized by higher dimensional Jacobian varieties.…”

“…For any nondegenerate abelian variety with endomorphism given by a diagonal matrix and with CM by a cyclotomic field, this lemma can be used to determine the component group generators. This may seem like a very specific scenario, but we note that this describes the Jacobians considered in, for example, [9,10,23] and [15] (for certain values of c).…”