## Non-archimedean Excursions in Probability, Number theory, Combinatorics and Geometry

- El Maazouz, Yassine
- Advisor(s): Sturmfels, Bernd

## Abstract

Similar to the field of real numbers R, which can be constructed as the completion of the rational numbers with respect the usual absolute value | · |, the field of p-adic numbers Q_p is also the completion of the rational numbers with the p-adic absolute value | · |_p. These numbers were first introduced by Kurt Hensel to harness the power of analytic tools in number theory, but nowadays non-archimedean mathematics has become a very rich and active area of research in its own right. This thesis is composed of seven chapters whose main theme is non-archimedean.

This dissertation begins with the first chapter in which we introduce some necessary back- ground and concepts that are used throughout this thesis. In particular, we recall the notion of valued fields and review some basic facts from non-archimedean algebra and analysis. We then introduce the Euclidean building associated to the reductive group PGL, which will appear throughout this thesis.

In the second chapter we study the entropy map for multivariate Gaussian distributions on a non-archimedean local field. As in the real case, the image of this map lies in the supermodular cone. Moreover, given a multivariate Gaussian measure on a local field, its image under the entropy map determines its pushforward under the valuation map. In general, this entropy map can be defined for non-archimedian valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3 and discuss non-archimedean statistical Gaussian models.

In the third chapter we give a method for sampling points from an algebraic manifold (affine or projective) over a local field with a prescribed probability distribution. In the spirit of previous work by Breiding and Marigliano on real algebraic manifolds, our method is based on slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of our sampling method and discuss a few applications. In one application, we sample from algebraic p-adic matrix groups and modular curves.

In the fourth chapter, we introduce a novel characterization of Bernoulli polynomials by circular convolution. There is a combinatorial and probabilistic model underlying this characterization which we call The Bernoulli clock. We use this model to give a probabilistic perspective to the work of Horton and Kurn [91] and the more recent work of Clifton et al. [33] on counting permutations of the multiset 1^m · · · n^m with longest continuous and increasing subsequence of length n starting from 1.

In the fifth chapter we apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the grad- uated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We also advance the ideal theory of graduated orders by introducing their ideal class polytropes. We then extend our study to bolytropes and bolytrope orders. Bolytropes are bounded subsets of an affine building that consist of all points that have distance at most r from some polytrope. We prove that the points of a bolytrope describe the set of all invariant lattices of a bolytrope order, generalizing the correspondence between polytropes and graduated orders.

In the sixth chapter, we study non-archimedean Schur representations and their invariant lattices. More precisely, fix a non-archimedean discretely valued field K and let O_K be its valuation ring. Given a vector space V of dimension n over K and a partition λ of an integer d, we study the problem of determining the invariant lattices in the Schur module S_λ(V ) under the action of the group GL(n,O_K). When K is a non-archimedean local field, our results determine the GL(n, O_K )-invariant Gaussian distributions on S_λ (V ).

Finally, in the seventh chapter, we turn to arithmetic geometry. We express the reduction types of Picard curves in terms of tropical invariants associated to binary quintics. These invariants are connected to Picard modular forms using recent work [32] of Cl ́ery and van der Geer. We furthermore give a general framework for tropical invariants associated to group actions on arbitrary varieties. The problem of describing reduction types of curves in terms of their associated invariants fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification \overline{M_{0,n}}.