# Schedule of the Workshop "Arithmetic intersection theory and Shimura varieties"

## Monday, February 3

 10:30 - 11:00 Welcome coffee 11:00 - 12:00 Ben Howard: Supersingular points on some orthogonal and unitary Shimura varieties 12:00 - 13:45 Lunch break 13:45 - 14:45 Fabrizio Andreatta: Integral canonical models of orthogonal Shimura varieties 15:00 - 16:00 José Ignacio Burgos: The singularities of the invariant metric of the sheaf of Jacobi form 16:00 - 16:30 Tea and cake 16:30 - 17:30 Amnon Besser: p-adic Arakelov theory and integral points on hyperelliptic curves

## Tuesday, February 4

 9:30 - 10:30 Jean-Benoît Bost: Infinite dimensional hermitian vector bundles and their applications 10:30 - 11:00 Coffee break 11:00 - 12:00 Anna-Maria von Pippich: An arithmetic Riemann–Roch theorem for weighted pointed curves 12:00 - 13:45 Lunch break 13:45 - 14:45 Klaus Künnemann: A tropical approach to non-archimedean Arakelov theory I 15:00 - 16:00 Walter Gubler: A tropical approach to non-archimedean Arakelov theory II 16:00 - 16:30 Tea and cake 16:30 - 17:30 Matteo Longo: Half weight modular forms and rational points on elliptic curves

## Wednesday, February 5

 9:30 - 10:30 Don Zagier: Modular embeddings of Teichmüller curves 10:30 - 11:00 Coffee break 11:00 - 12:00 Maryna Viazovska: CM values of regularized theta lifts 12:00 - 16:00 Lunch break, free afternoon 16:00 - 16:30 Tea and cake 19:00 Conference Dinner*

## Thursday, February 6

 9:30 - 10:30 Massimo Bertolini: Beilinson-Flach elements and the arithmetic of elliptic curves 10:30 - 11:00 Coffee break 11:00 - 12:00 Daniel Disegni: The p-adic Gross-Zagier formula on Shimura curves 12:00 - 13:45 Lunch break 13:45 - 14:45 Peter Scholze: The pro-etale site 15:00 - 16:00 Tonghai Yang: Heights of Kudla-Rapoport divisors and derivatives of L-functions 16:00 - 16:30 Tea and cake 16:30 - 17:30 Martin Raum: Kudla’s modularity conjecture for codimensions up to 5

## Friday, February 7

 9:30 - 10:30 Michael Rapoport: The arithmetic transfer conjecture for exotic formal moduli spaces 10:30 - 11:00 Coffee break 11:00 - 12:00 Eva Viehmann: Rapoport-Zink spaces and affine Deligne-Lusztig varieties in the arithmetic case 12:00 - 13:45 Lunch break 13:45 - 14:45 Jürg Kramer: Effective bounds for Faltings' delta function 15:00 - 16:00 Damian Rössler: Rational points of varieties with ample cotangent bundle over function fields of positive characteristic 16:00 - 16:30 Tea and cake

* There will be a conference dinner at Altes Bonn:

Gaststube Altes Bonn
Graurheindorfer Straße 1
(near Beethovenhalle)
53111 Bonn

We will leave in a group at 18h30 from the Institute, for the ones who want to walk.

Location
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# Abstracts

(Underlined titles can be clicked for the video recording)

#### Ben Howard: Supersingular points on some orthogonal and unitary Shimura varieties

To an orthogonal group of signature $(n,2)$, 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 $(2,2)$. This is joint work with G. Pappas.

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#### Fabrizio Andreatta: Integral canonical models of orthogonal Shimura varieties. A proof of a conjecture of Bruinier and Yang

Let $(V,Q)$ be a quadratic space over $Q$ with signature $(2, n)$ and let $L \subset V$ be a perfect lattice I will define the Shimura variety associated to the algebraic group $CSpin(V,Q)$ 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.

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#### José Burgos: The singularities of the invariant metric of the sheaf of Jacobi forms on the universal elliptic curve.

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.

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#### Amnon Besser: p-adic Arakelov theory and integral points on hyperelliptic curves

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.

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#### Jean-Benoît Bost: Infinite dimensional hermitian vector bundles and their applications

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.

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#### Anna von Pippich: An arithmetic Riemann–Roch theorem for weighted pointed curves

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.

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#### Matteo Longo: Half weight modular forms and rational points on elliptic curves

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.

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#### Massimo Bertolini: Beilinson-Flach elements and the arithmetic of elliptic curves

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.

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#### Daniel Disegni: The p-adic Gross-Zagier formula on Shimura curves

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.

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#### Tonghai Yang: Heights of Kudla-Rapoport divisors and derivatives of L-functions

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.

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#### Maryna Viazovska: CM values of regularized theta lifts

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.

#### Martin Raum: Kudla’s modularity conjecture for codimensions up to 5

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.

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#### Michael Rapoport: The arithmetic transfer conjecture for exotic formal moduli spaces

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.

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#### Eva Viehmann: Rapoport-Zink spaces and affine Deligne-Lusztig varieties in the arithmetic 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.

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#### Jürg Kramer: Effective bounds for Faltings’s delta function

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.

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#### Damian Rössler: Rational points of varieties with ample cotangent bundle over function fields of positive characteristic (joint with H. Gillet)

Let $K$ be the function field of a smooth curve over a finite field $k$ of characteristic $p \gt 0$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega^1_{X/K}$ is ample. We prove that if $X(K)$ is Zariski dense in $X$ then there is a smooth and projective variety $X'_0$ over $\bar{k}$ and a finite and surjective $K^\text{sep}$-morphism $X'_{0,K^\text{sep}} \rightarrow X_{K^\text{sep}}$.

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#### Peter Scholze: The pro-etale site

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.

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