A central theme in modern arithmetic geometry is to understand the distribution of special subvarieties—loci where objects acquire extra symmetry, smaller Mumford–Tate group, or atypical monodromy. Their behavior is closely tied to conjectures on unlikely intersections, and a recurring phenomenon is that these special loci, while highly structured, are notably sparse. A natural question is how one should measure this sparsity in different arithmetic contexts. In this talk, I will discuss two illustrations of this theme. The first concerns one-parameter families of abelian varieties over number fields. How often does a specialization acquire extra endomorphisms? I will explain a quantitative result showing that such “non-generic” fibers occur on a height-density zero set of rational points. The second illustration arises from the p-adic geometry of Shimura varieties. While the Hecke orbit conjecture asserts that the prime-to-p Hecke orbit of a point is Zariski dense in the central leaf containing it, one may ask how large such an orbit can be in the p-adic analytic topology. I will describe a conjectural picture—supported by new results on p-adic monodromy—suggesting that even the full Hecke orbit is p-adically nowhere dense.
Sparsity Phenomena in Arithmetic Geometry
Tue, Apr. 7 1:25pm (MATH 2…
Algebra/Logic
Nick Jamesson (CU Boulder)
X
The family of promise constraint satisfaction problems (PCSPs) is a variant of the well studied family of constraint satisfaction problems (CSPs). The computational problem of solving promise systems of equations over algebras is a variant of the problem of systems of equations over algebras. Given two algebras A and B of the same signature such that A maps homomorphically into B, one can ask if a system of equations has a solution in A or if it doesn't even have a solution in B. Given a finite algebra A, it is known that determining whether or not systems of equations over A have a solution is either solvable in polynomial time, or is NP complete (the famous dichotomy theorem for CSPs.) The question of whether or not a dichotomy holds for promise systems of equations over finite algebras is open in general. I will present some dichotomy results for promise systems of equations over various classes of algebras, including Mal'cev algebras.
We show that the number of zero dinv Fubini rankings with n competitors and at most l + 1 ties at any rank is given by the maximum number of regions of an l-dimensional cake with n-1 cuts. This is joint work with Pamela E. Harris and Alexander N. Wilson.
Zero Dinv Fubini Rankings and Cake-Cutting Geometry
Sparsity Phenomena in Arithmetic Geometry