• Algebraic Geometry
• Workshop: Arithmetic intersection theory and Shimura varieties

• Schedule
• Participants

To an orthogonal group of signature , or to a unitary group of any signature, one can attach a Shimura variety. The general problem is to describe the integral models of these Shimura varieties, and their reductions modulo various primes. I will give a conjectural description of the supersingular locus of the reduction in the orthogonal case, and a complete description of the supersingular locus in the case of a unitary group of signature . This is joint work with G. Pappas.

Let be a quadratic space over with signature and let be a perfect lattice I will define the Shimura variety associated to the algebraic group and describe its integral canonical model. I will then show how one can prove a conjecture of Bruinier and Yang expressing the arithmetic

intersection of small CM points and Heegner divisors on such varieties. One of the interesting aspects of the conjecture is the presence of improper intersection. This is dealt with a technique developed by Hu, which is the Arakelov theoretic analogue of the deformation to the normal cone.

This is joint work with Eyal Goren, Ben Howard and Keerthi Madapusi-Pera.

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, provided with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key to being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford’s result remains valid for line bundles on mixed Shimura varieties.

In this talk we will examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We will show that Mumford’s result can not be extended to this case and that a new interesting kind of singularities appear that are related to the phenomenon of Height jumping introduced by Hain.

We will discuss some preliminary results. This is joint work with G. Freixas, J. Kramer and U. Kühn.

In this talk I will review p-adic Arakelov theory, explain its relation with p-adic height pairings and prove a formula for the local factor at p for the p-adic height, which goes into my work with Balakrishnan and Mueller on finding integral points on hyperelliptic curves.

We introduce a notion of "infinite dimensional hermitian vector over an arithmetic curve", and we discuss some related results of "infinite dimensional geometry of number" and some of their applications, notably to the construction of vector bundles on projective varieties over number fields.

In this talk, we report on work in progress with G. Freixas generalizing the arithmetic Riemann–Roch theorem for pointed stable curves to the case where the metric is allowed to have conical singularities at the marked points. One main ingredient of the proof is a Mayer–Vietoris type formula for the singular hyperbolic metric. This formula requires the explicit computation of the regularized determinant for hyperbolic cusps and cones.

Let E be an elliptic curve, defined over the field of rational numbers, of conductor Np, where N is a positive integer and p is a prime which does not divide N. Let f be the weight 2 newform attached to E. We consider the Hida family passing through f. One can lift each classical form in the Hida family to a half-weight modular forms, by means of a generalized Kohnen-Shintani correspondence (Baruch-Mao). The resulting Fourier coefficients can be p-adically interpolated by rigid analytic functions defined over the weight space. Extending a previous work by Darmon-Tornaria, I will propose a relation between the coefficients of this formal series and certain global points on the elliptic curve E. This is a work in progress with Zhengyu Mao.

This talk explains a p-adic Beilinson formula relating the p-adic L-function associated to the Rankin convolution of two cusp forms to so-called Beilinson-Flach elements. It will then describe some applications to new cases of the Birch and Swinnerton-Dyer conjecture for elliptic curves. This is a report on work in progress with Henri Darmon and Victor Rotger.

The Gross-Zagier formula relates the heights of Heegner points on elliptic curves over Q to derivatives of L-functions ; together with the work of Kolyvagin, it implies the rank part of the Birch and Swinnerton-Dyer conjecture for curves whose L-function vanishes to order one, as well as the BSD conjectural formula up to a rational number. One way to go on and study the formula up to p-integrality is provided by the p-adic analogue of the Gross-Zagier formula proved by Perrin-Riou and Kobayashi. I will review this circle of ideas and discuss its generalisation to totally real fields.

In this talk, I will briefly describe my recent joint work with Bruinier and Howard. We define an automorphic green function for the Kudla-Rapoport divisors on a unitary Shimura variety of type (n-1, 1), and prove that its height pairing with a CM cycle is equal to the central derivative of certain Rankin-Selberg L-function.

In this talk we will discuss arithmetic properties regularized Petersson products between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight 1 modular form with integral Fourier coefficients. We prove that such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. Using the Gross-Zagier formula for local heights of Heegner poins we give an explicit factorization formula for this algebraic number. Finally, we will show that Petersson products of this kind are "building blocks" for CM-values of a wide class of regularized theta lifts.

This talk is on work in progress joint with Jan Bruinier. Kulda’s conjecture says that the generating function of special cycles in the Chow group of orthogonal Shimura varieties is a Siegel modular form. We announce a proof for cycles of codimension less than or equal to 5, which we are currently checking for correctness. Also, we briefly discuss limitations of the ideas behind it, and an extension that might lead to a proof of the conjecture in full generality.

I will define some formal moduli spaces of p-divisible groups that can be used to formulate an extension of Wei Zhang’s Arithmetic Fundamental Lemma conjecture beyond the unramified case.

Motivated by the theory of moduli spaces of p-divisible groups defined

by Rapoport and Zink, we explain the notion of affine Deligne-Lusztig varieties in the arithmetic context. Furthermore, we determine their sets of connected components. This is joint work with M. Chen and M. Kisin.

In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces. For a given compact Riemann surface X of genus g, this invariant is roughly given as minus the logarithm of the distance of the point in the moduli space of genus g curves determined by X to its boundary. In our talk, we will first give a formula for Faltings’s delta function for compact Riemann surfaces of genus g>1 in purely hyperbolic terms. This formula will then enable us to deduce effective bounds for Faltings’s delta function in terms of the smallest non-zero eigenvalue and the shortest closed geodesic of X. If time permits, we will also address a question of Parshin related to bounding the height of rational points on curves defined over number fields.

Let be the function field of a smooth curve over a finite field of characteristic . Let be a scheme, which is smooth and projective over . Suppose that the cotangent bundle is ample. We prove that if is Zariski dense in then there is a smooth and projective variety over and a finite and surjective -morphism .

We explain our joint work with Bhargav Bhatt, whose goal is to simplify the foundations of ell-adic cohomology by introducing a new site, called the pro-etale site, in which the naive definition of ell-adic sheaves and the six functors is the correct one.