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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