No abstract

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

In this paper we prove the following results: 1 ) 1) We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. 2 ) 2) We prove that the period map associated to any pure polarized variation of integral Hodge structures V \mathbb {V} on a smooth complex quasi-projective variety S S is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. 3 ) 3) As a corollary of 2 ) 2) and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of ( S , V ) (S, \mathbb {V}) is a countable union of algebraic subvarieties of S S , a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable S L 2 SL_2 -orbit theorem of Cattani-Kaplan-Schmid.

We give a proof of the André-Oort conjecture for Ag -the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven 'averaged' version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The André-Oort conjecture then follows from previous work of Pila and the author.

Abstract. The results of Strassen [Str73] and Raz [Raz10] show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds.We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T : [n] d → F with rank at least 2n ⌊d/2⌋ + n − Θ(d lg n). This matches (over F2) or improves (all other fields) known lower bounds for d = 3 and improves (over any field) for odd d > 3.We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by counting, that there exists an order-3 permutation tensor with super-linear rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over large fields F, showing (among other things) that rank. We also show that if this upper bound is tight, then super-linear tensor rank lower bounds would follow. The second upper bound uses interpolation and only works for abelian G, showing that over any field F that rankIn either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank.We also explore monotone tensor rank. We give explicit 0/1 tensors T : [n] d → F that have tensor rank at most dn but have monotone tensor rank exactly n d−1 . This is a nearly optimal separation.