We study the local behavior of special cycles on Shimura varieties for $mathbf{U}(2, 1) times mathbf{U}(1, 1)$ in the setting of the Gan-Gross-Prasad conjectures at primes $tau$ of the totally real field of definition of the unitary spaces which are
split in the corresponding totally imaginary quadratic extension. We establish a local formula for their fields of definition, and prove a distribution relation between the Galois and Hecke actions on them. This complements work of cite{jetchev:unitary} at inert primes, where the combinatorics of the formulas are reduced to calculations on the Bruhat--Tits trees, which in the split case must be replaced with higher-dimensional buildings.
Fix a prime number $ell$. Graphs of isogenies of degree a power of $ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field.
We analyse graphs of so-called $mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(ell, ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.
We give an asymptotic formula for the number of automorphic forms on the non-split norm one torus $T$ associated with an imaginary quadratic extension of $mathbb{Q}$, ordered by analytic conductor.
We consider cycles on a 3-dimensional Shimura varieties attached to a unitary group, defined over extensions of a CM field $E$, which appear in the context of the conjectures of Gan, Gross, and Prasad cite{gan-gross-prasad}. We establish a vertical d
istribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of cite{jetchev:unitary}, and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $Lambda$-module constructed from Heegner points.
