Shrenik Shah (Columbia U)
Class number formulae for some Shimura varieties of low dimension
The class number formula connects the residue of the Dedekind zeta function at s=1 to the regulator, which measures the covolume of the lattice generated by logarithms of units. Beilinson defined a generalized regulator morphism and conjectural class number formula in the "motivic" setting. His formula provides arithmetic meaning to the orders of "trivial" zeroes of L-functions at integer points as well as the value of the first nonzero derivative at these points.
We study this conjecture for the middle degree cohomology of the Shimura varieties associated to unitary groups of signature (2,1) and (2,2) over Q. We construct explicit "Beilinson-Flach elements" in the motivic cohomology of these varieties and compute their regulator. This is joint work with Aaron Pollack.
Nicole Looper (Northwestern U)
A lower bound on the canonical height for polynomials
The canonical height associated to a rational function defined over a number field measures arithmetic information about the forward orbits of points under that function. Silverman conjectured that given any number field K and degree d at least 2, there is a uniform lower bound on the canonical heights associated to degree d rational functions defined over K, evaluated at points of K having infinite forward orbit. I will discuss a proof of such a lower bound across large families of polynomials. I will also discuss related questions, both in the setting of elliptic curves and that of dynamical systems.
Liang Xiao (U Connecticut)
Cycles on the special fiber of some Shimura varieties and Tate conjecture
We describe the irreducible components of the basic locus of Shimura varieties of Hodge type at a place with good reduction, when the basic locus is of middle dimension. Under certain genericity condition, we show that they generate the Tate classes of the special fiber of the Shimura varieties. This is a joint work with Xinwen Zhu.
Lue Pan (Princeton U)
Fontaine–Mazur conjecture in the residually reducible case
We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q under certain hypothesis on the residual representation at p. To do this, we generalize Emerton's local-global compatibility and devise a patching argument for completed homology in this setting.
Lucia Mocz (Princeton U)
A new Northcott property for Faltings height
The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.
Daniel Kriz (Princeton U)
A new p-adic Maass-Shimura operator and supersingular Rankin-Selberg p-adic L-functions
We introduce a new p-adic Maass-Shimura operator acting on a space of "generalized p-adic modular forms" (extending Katz's notion of p-adic modular forms) defined on the p-adic (preperfectoid) universal cover of Shimura curves. Using this operator, we construct new p-adic L-functions in the style of Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang for Rankin-Selberg families over imaginary quadratic fields K in the case where p is inert or ramified in K. We also establish new p-adic Waldspurger formulas, relating p-adic logarithms of elliptic units and Heegner points to special values of these p-adic L-functions.
Jingren Chi (U Chicago)
Geometry of generalized affine Springer fibers
This talk is about certain generalization of affine Springer fibers that arises naturally in the study of orbital integrals of spherical Hecke functions on a p-adic group. We compare basic geometric properties of these generalized affine Springer fibers with their classical Lie algebra analogue. In particular, we will formulate a conjecture relating the number of irreducible components of such varieties to certain weight multiplicities defined by the Langlands dual group and discuss known cases of this conjecture.