Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

Let 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.

Remark on nondegeneracy of simple abelian varieties with many endomorphisms

We investigate a relationship between nondegeneracy of a simple abelian variety A over an algebraic closure of ℚ and of its reduction A0. We prove that under some assumptions, nondegeneracy of A implies nondegeneracy of A0.

Arithmetically defined dense subgroups of Morava stabilizer groups

For every prime p and integer n≥3 we explicitly construct an abelian variety $A/\mathbb {F}_{p^n}$ of dimension n such that for a suitable prime l the group of quasi-isogenies of $A/\mathbb {F}_{p^n}$ of l-power degree is canonically a dense subgroup of the nth Morava stabilizer group at p. We also give a variant of this result taking into account a polarization. This is motivated by the recent construction by Behrens and Lawson of topological automorphic forms which generalizes topological modular forms. For this, we prove some arithmetic results of independent interest: a result about approximation of local units in maximal orders of global skew fields which also gives a precise solution to the problem of extending automorphisms of the p-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible.

Density measure of rational points on Abelian varieties

Let be a simple Abelian variety of dimension g over ℚ, and let ℓ be the rank of the Mordell-Weil group (ℚ). Assume ℓ ≥ 1. A conjecture of Mazur asserts that the closure of (ℚ) into (ℝ) for the real topology contains the neutral component (ℝ)0 of the origin. This is known only under the extra hypothesis ℓ ≥ g2 - g + 1. We investigate here a quantitative refinement of this question: for each given positive h, the set of points in (ℚ) of Néron-Tate height ≤ h is finite, and we study how these points are distributed into the connected component (ℝ)0. More generally we consider an Abelian variety A over a number field K embedded in ℝ, and a subgroup Γ of (K) of sufficiently large rank. The effective result of density we obtain relies on an estimate of Diophantine approximation, namely a lower bound for linear combinations of determinants involving Abelian logarithms.