AGNT Ical Atom

The philosophy of arithmetic topology, first established by Mazur, gives an analogy relating arithmetic to lower dimensional topology. Under this philosophy, one gets a dictionary, relating between Number fields and 3-manifolds, primes and knots, and p-adic fields and surfaces. In the talk I will try and explain why these surprising analogies were drawn. I will also outline some recent work of the speaker, which further establishes this connection for the local case, using tools such as graphs of groups and Bass–Serre theory.

Let f:X \to Y be a morphism between smooth, geometrically irreducible Z-schemes of finite type. We study the number of solutions #{x:f(x)=y mod p^k} for prime p, positive number k, and y \in Y(Z/p^kZ), and show that the geometry and singularities of the fibers of f determine the asymptotic behavior of this quantity as p, k and y vary.

In particular, we show that f:X \to Y is flat with fibers of rational singularities, a property abbreviated (FRS), if and only if #{x:f(x)=y mod p^k}/p^{k(\dim X -\dim Y)} is uniformly bounded in p, k and y. We then consider a natural family of singularity properties, which are variants of the (FRS) property, and provide for each member of this family a number theoretic characterization using the asymptotics of #{x:f(x)=y mod p^k}/p^{k(\dim X -\dim Y)}.

To prove our results, we use model theoretic tools (and in particular the theory of motivic integration, in the sense of uniform p-adic integration) to effectively study the collection {#{x:f(x)=y mod p^k}/p^{k(\dim X -\dim Y)}. If time allows, we will discuss these methods.

Based on a joint work with Raf Cluckers and Itay Glazer.

Let f,g be two Hilbert modular newforms (functions on ’n-copies’ of the upper half-plane, satisfying properties similar to usual modular forms). Consider the L-function L(s,f \times \Sym^{2}g) (it is the degree six factor of the triple product L-function L(s,f \times g \times g)). In this talk we will give a formula for the central value of this L-function and work out its rationality in some special cases of relationships between weights of f and g. We will arrive at our formula via the refined Gan-Gross-Prasad formula for SL(2) \times \tilde(SL(2)). Our results on rationality are compatible with Deligne’s conjecture on the rationality of critical values of motivic L-functions.

Arakelov geometry offers a framework to develop an arithmetic counterpart of the usual intersection theory. For varieties defined over the ring of integers of a number field, and inspired by the geometric case, one can define a suitable notion of arithmetic Chow groups and of an arithmetic intersection product. In a joint work with Roberto Gualdi (University of Regensburg), we prove an arithmetic analogue of the classical Shioda-Tate formula, relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety to some of its geometric invariants. In doing so, we also characterize numerically trivial arithmetic divisors, confirming part of a conjecture by Gillet and Soulé.

Level raising theorems for modular forms are theorems about congruences of modular forms between different levels. These theorems play an important role in the proof of the Fermat’s last theorem by Wiles. In this talk, we will report some recent work on realizing level raising theorems for automorphic forms on GSp(4) by studying the geometry of certain quaternionic unitary Shimura variety.

A key tool in understanding (complex analytic) hypersurface singularities is to study what properties are preserved under special deformations. For example, the relationship between the Milnor number of an isolated singularity and the number of A_1 points. In this talk we will discuss the transversal discriminant of a singular hypersurfaces whose singular locus is a smooth curve, and how it can be applied in order to generalize a classical result by Siersma, Pellikaan, and de Jong regarding morsifications of such singularities. In addition, we will present some applications to the study of Yomdin-type isolated singularities.

The projective finitistic dimension of a ring is an important homological dimension which measures the complexity of homological algebra over it. The finitistic dimension conjecture which says that this invariant is finite for artin algebras is considered to be one of the most important open problems in homological algebra. In this talk we discuss this conjecture, its importance and connection to other important conjectures. We then show how by using DG-ring techniques, and the noncommutative covariant Grothendieck duality, it is possible to connect this conjecture to the global structure of injective modules in the unbounded derived category of the ring. This generalizes work of Rickard from finite dimensional algebras over a field to all noetherian rings which admit a dualizing complex.

Given a rational Hecke eigenform $f$ of weight $2$, Eichler-Shimura theory gives a construction of an elliptic curve over ${\mathbb Q}$ whose associated modular form is $f$. Mazur, van Straten, and others have asked whether there is an analogous construction for Hecke eigenforms $f$ of weight $k>2$ that produces a variety for which the Galois representation on its etale ${\mathrm H}^{k-1}$ (modulo classes of cycles if $k$ is odd) is that of $f$. In weight $3$ this is understood by work of Elkies and Sch"utt, but in higher weight it remains mysterious, despite many examples in weight $4$. In this talk I will present a new construction based on families of K3 surfaces of Picard number $19$ that recovers many existing examples in weight $4$ and produces almost $20$ new ones.

For a smooth projective hyperbolic curve Y/Q the set of rational points Y(Q) is finite by Faltings’ Theorem. Grothendieck’s section conjecture predicts that this set can be described via Galois sections of the étale fundamental group of Y. On the other hand, the non-abelian Chabauty method produces p-adic analytic functions which conjecturally cut out Y(Q) as a subset of Y(Qp). We relate the two conjectures and discuss the example of the thrice-punctured line, where non-abelian Chabauty is used to prove a local-to-glocal principle for the section conjecture.