Consider a pair $X,\hat{L})$ of a regular and projective algebraic variety over $\mathbb{Q}$ and a semipositive adelically metrized line bundle $\hat{L}$ over $X$. Assume that all of Zhang's successive minima of the pair $X,\hat{L})$ are equal, or equivalently, that the infimum of the associated height function $h}_{\hat{L}}:X(\overline{\mathbb{Q}})\to \mathbb{R$ is reached by some Zariski-generic sequence. In that setting, I will overview some recent progress on the following question: To what extent does the Galois equidistribution of the small points of $X,\hat{L})$ persist when the test function has a logarithmic singularity along some Zariski-closed subset $Z$ of $X$?
At its core this is a Diophantine approximation problem for arithmetically small points and the fixed algebraic target set $Z$. The answer turns out to be affirmative if and only if all components of the singular set Z are either divisors of minimum height (Chambert-Loir and Thuillier), or algebraic cycles of codimension at least 2 in $X$. The main Diophantine approximation result, in its most fundamental case where the ambient arithmetic variety is a linear algebraic torus $\matbb{G}_m^d$ and the approximants are torsion points, has applications to homological torsion asymptotics in the congruence layers of infinite solvable covers of a manifold. A spinoff of the involved technique also leads to the intersection stability property of the finite unions of minimum height subvarieties of $X,\hat{L})$, giving a new approach for the Bogomolov conjecture that applies over both number fields as well as function fields of arbitrary characteristic. Lastly, and time permitting, I will describe an approach to the analogous Diophantine approximation problem in the Shimura setting, when all components of the target set have codimension 2 or higher.
Part of this work is joint work with Philipp Habegger.
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Modular forms appear in a wide variety of contexts in physics and mathematics. For example, they arise in two dimensional quantum field theories as certain observables. In algebraic topology, they emerge in the study of invariants called elliptic cohomology theories. A long-standing conjecture suggests that these two appearances of modular forms are intimately related. After explaining the ingredients, I’ll describe some recent progress.