Abstracts

Claudia Alfes

Title: Cycle integrals of meromorphic Hilbert modular forms.

Abstract: We establish a rationality result for linear combinations of traces of cycle integrals of certain meromorphic Hilbert modular forms. These are meromorphic counterparts to the Hilbert cusp forms omega_m(z₁,z₂), which Zagier investigated in the context of the Doi-Naganuma lift. We give an explicit formula for these cycle integrals, expressed in terms of the Fourier coefficients of harmonic Maass forms. Key elements of our proof are the construction of locally harmonic Hilbert-Maass forms, a new regularized theta lift related to the Doi-Naganuma lift, and the development of the xi-operator for Hilbert modular surfaces. This is joint work with Baptiste Depouilly, Paul Kiefer, and Markus Schwagenscheidt.

Antonio Cauchi

Title: On the arithmetic of motives with Galois group of type G₂.

Abstract: I will explain how to construct an Euler system for the Galois representation attached to cusp forms on the exceptional group G₂. This construction relies on the cohomological properties of an exceptional theta correspondence together with the use of certain Eisenstein cohomology classes for GSp(4) studied by Faltings and more recently by Sangiovanni Vincentelli and Skinner. This is joint work in progress with Joaquin Rodrigues Jacinto and Waqar Shah.

Michele Fornea

Title: A new point of view on plectic Heegner points.

Abstract: With the formulation of their plectic conjectures, Nekovar and Scholl made a strong case for studying the arithmetic of higher rank elliptic curves using CM points on higher dimensional quaternionic Shimura varieties. Even though those conjectures seem to be beyond our reach at present, in a series of joint works with Darmon, Gehrmann, Guitart and Masdeu we proposed unconditional (albeit not completely satisfying) constructions of special elements that conjecturally control Mordell-Weil groups of higher rank. The aim of this talk is to present a new construction of special elements that both uni es and generalizes plectic Heegner and mock plectic points. In the spirit of the conference we will explain how the work of Longo-Nicole on the p-adic variation of the Gross-Kohnen-Zagier (GKZ) theorem could be used to prove a GKZ-type theorem for plectic Heegner points. This is joint work with Henri Darmon.

Luis Garcia

Title: Explicit class field theory and the elliptic gamma function.

Abstract: It is a well-known classical fact that the abelian extensions of the rational numbers and of imaginary quadratic  elds are generated by special values of the exponential function and of theta functions. During the talk I will discuss the elliptic gamma function, a meromorphic function arising in mathematical physics that has been shown to have modular properties with respect to SL(3;Z). I will present numerical evidence for a conjecture stating that certain products of values of this function lie on prescribed abelian extensions of complex cubic fields and satisfy explicit reciprocity laws. I will also discuss a limit formula relating this function to the derivative at s = 0 of partial zeta functions. This is joint work with Nicolas Bergeron and Pierre Charollois.

Lennart Gehrmann

Title: The Gross-Kohnen-Zagier theorem via p-adic uniformization.

Abstract: I will sketch a new proof of the Gross-Kohnen-Zagier theorem for Shimura curves which exploits the p-adic uniformization of Cerednik-Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross-Kohnen-Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a p-adic family of positive de nite ternary theta series. If time permits, I will discuss how to adapt this strategy to the setting of plectic points. This is joint work with Lea Beneish, Henri Darmon, and Marti Roset respectively joint work in progress with Michele Fornea and Marti Roset.

Benjamin Howard

Title: An arithmetic Siegel-Weil conjecture for the spherical Hecke algebra.

Abstract: Li-Rapoport-Zhang have recently constructed Hecke correspondences on Rapoport-Zink formal schemes of type GU(n; 1), and used these to formulate a generalization of Zhang's arithmetic fundamental lemma. I will explain how to use these same Hecke correspondences to formulate a conjectural generalization of the arithmetic Siegel-Weil formula of Kudla-Rapoport, and explain a proof of the conjecture in the case of GU(1; 1).

Yingkun Li

Title: Intersections of closed geodesics and modular forms.

Abstract: In the seminal work of real quadratic analogue of singular moduli by Darmon and Vonk, the unsigned intersection number of closed geodesics played an important role. Explicit formula for these intersection numbers have been given by Rickards by studying optimal embeddings of orders into quaternion algebras. In this talk, I will give another approach to obtain these intersection numbers using modular forms and theta lifts. This is joint work in progress with Jan Bruinier and Martin Moeller.

Matteo Longo

Title: On generalized Rubin formula for Hecke characters.

Abstract: The goal of this talk is to generalize Rubin's theorem on values of Katz's p-adic Lfunction outside the range of classical interpolation from the case of characters of CM elliptic curves to more general self-dual Hecke characters. We follow the approach by Bertolini-Darmon-Prasanna, based on generalized Heegner cycles, which we extend from characters of imaginary quadratic fields of infinity type (1; 0) to characters of infinity type (1+l,-l), where l>=0 is an integer.

Keerthi Madapusi

Title: Another perspective on integral models for special cycles.

Abstract: I will talk about my attempt to find the 'correct' general context in which one expects to have modular generating series of special cycles on Shimura varieties and Rapoport-Zink spaces. I will then present some results leading to a general construction of good integral models for these cycles, at least at primes of good reduction for Shimura varieties of abelian type. These cycles are quite robust (though of an intrinsically derived nature), and can be used to write down a generating series of cycle classes on the full Schwartz space of spherical functions on a Hermitian space satisfying all the 'easy' parts of modularity. The key ideas in the construction are local, and use recent developments in p-adic Dieudonne theory due to Gardner, Mathew and myself.

Gyujin Oh

Title: A cohomological approach to harmonic Maass forms.

Abstract: We will explain how the notion of harmonic Maass forms relates to the local coherent cohomology of the modular curve, along with the analogy between the harmonic Maass forms and the overconvergent modular forms. We will see how this gives a nice interpretation of some questions regarding the algebraicity of harmonic Maass forms, and also how to define harmonic Maass forms over general Shimura varieties, focusing on the cases of Hilbert modular varieties and U(n; 1)-Shimura varieties. Finally, we will discuss the Borcherds lifts with the inputs being the harmonic Maass forms over more general Shimura varieties.

Alice Pozzi

Title: Modular generating series for Heegner objects.

Abstract: Rigid meromorphic cocycles are cocycles for p-arithmetic groups acting on p-adic symmetric spaces. Their values at special points are conjectured to belong to class fields of some suitable global fields. In previous work with Darmon and Vonk, we proved a very special instance of this conjecture, exploiting, among other tools, a modular generating series for RM values. The latter is reminiscent of the modular generating series for Heegner cycles appearing in the work of Gross and Zagier on the Birch and Swinnerton-Dyer conjecture. In this talk, we discuss modular generating series for RM values of theta cocycles in a general framework involving biquadratic extensions. This is joint work in preparation with Judith Ludwig, Isabella Negrini, Sandra Rozensztajn and Hanneke Wiersema.

Juan Esteban Rodriguez Camargo

Title: Analytic prismatization of rigid spaces.

Abstract: Motivated from the work of prismatic cohomology of Bhatt-Scholze, and the prismatization of p-adic formal schemes of Bhatt-Lurie and Drinfeld, we introduce the analytic prismatization of rigid spaces. This gadget relates different cohomology theories appearing in rigid geometry and provides a more conceptual understanding of the link between D-modules and p-adic automorphic forms initiated by Pan. This is joint work in progress with Johannes Anschutz Arthur-Cesar Le Bras and Peter Scholze.

Giovanni Rosso

Title: Hirzebruch-Zagier cycles in p-adic families and adjoint L-values.

Abstract: Let E/F be a quadratic extension of totally real fields. The embedding of the Hilbert modular variety of F inside the Hilbert modular variety of E defines a cycle, called Hirzebruch-Zagier cycle. Thanks to work of Hida and Getz-Goresky, it is known that the integral of a Hilbert modular form g for E over this cycle detects if g is the base change of a Hilbert modular form for f, and in this case the value of the integral is related to the adjoint L-function of f. In this talk we shall present joint work with Antonio Cauchi and Marc-Hubert Nicole, where we show that the Hirzebruch-Zagier cycles vary in families when one considers deeper and deeper levels at p. We shall present applications to Lambda-adic Hilbert modular forms and adjoint p-adic L-functions.

Eugenia Rosu

Title: Special cycles on compactifications of Shimura varieties.

Abstract: In joint work with Bruinier and Zemel, we construct special cycles of dimension 0 on toroidal compactifications of Shimura varieties. We show the modularity of the generating series that have these special cycles as coefficients, generalizing the open case.

Yousheng Shi

Title: A pullback formula and modularity of arithmetic theta series.

Abstract: In this talk I will present a proof of the modularity of arithmetic theta series on an arithmetic model of unitary Shimura curves. The proof is via a pullback formula and is based on a previous work of Bruinier, Howard, Kudla, Rapoport and Yang on the modularity of higher theta series on higher dimensional unitary Shimura varieties. I will explain the subtlety on the definition of the arithemtic theta series and its boundary behaviour. The talk is based on a joint work with Qiao He and Tonghai Yang.

Jan Vonk

Title: p-adic height pairings of geodesics.

Abstract: I will discuss a certain p-adic height pairing of real quadratic geodesics on modular curves. The motivation for studying this pairing comes from its relation to real quadratic (RM) singular moduli. I will discuss how the interpretation of this height pairing as a triple product period sheds light on the conjectures that were made when RM singular moduli were defined. This is joint work with Henri Darmon.

Riccardo Zuffetti

Title: The Lefschetz decomposition of the Kudla-Millson theta function.

Abstract: In the 80's Kudla and Millson introduced a theta function in two variables, nowadays known as the Kudla-Millson theta function. This behaves as a Siegel modular form with respect to one variable, and as a closed differential form on an orthogonal Shimura variety with respect to the other variable. In this talk I show that the Lefschetz decomposition of (the cohomology class of) this theta function provides simultaneously the modular decomposition in Eisenstein, Klingen and cuspidal parts. Time permitting, I will report on geometric applications. This is joint work with J. Bruinier.

SHORT TALKS

Francesco Iudica

Title:  Lambda-adic Kudla lifts.

Abstract: The Kudla lift is a classical version of the theta lift between GU(2) and GU(3). In this talk we show a p-adic interpolation property of the Kudla lift and of its adjoint lift. By works of Kudla-Millson, the latter coincides with the Cogdell lift, which is the analogue for Picard modular surfaces of the celebrated result of Hirzebruch-Zagier for Hilbert modular surfaces.

Paul Kiefer

Title: The  xi-Operator for Hilbert Modular Forms and the Doi-Naganuma-Lift.

Abstract: We introduce a di erential operator analogous to the xi-operator of Bruinier and Funke for Hilbert modular surfaces. We then construct locally harmonic Hilbert-Maassforms that are xi-preimages of Doi-Naganuma-lifts of cusp forms using a new theta-lift. As an application we derive a current equation involving cycle integrals over real-analytic cycles. This is part of joint work with Claudia Alfes, Baptiste Depouilly and Markus Schwagenscheidt.

Khai Hoan Nguyen-Dang

Title: On p-adic Uniformization Of Abelian Varieties.

Abstract: Using Fontaine integration, Iovita, Morrow, and Zaharescu introduced a new p-adic uniformization of abelian varieties with good reduction. Building on their approach, we achieve p-adic uniformization of abelian varieties in the semi-stable reduction case. This uniformization map gives rise to a category of triples, consisting of Tate modules, Lie algebras associated to Abelian varieties, and the Fontaine integration. This talk will discuss ongoing work related to this category.

Marti Roset Julia

Title: p-adic rigid cocycles for SL_n and class field theory for totally real fields.

Abstract: We outline an approach to generalize the construction of the Dedekind-Rademacher rigid analytic cocycle of Darmon, Pozzi, and Vonk to the case of SL_n. This construction departs from the Eisenstein class of a torus bundle, analyzed through the framework developed by Bergeron, Charollois, and Garcia. We conclude with speculations on the relationship between these cocycles and Gross-Stark units, as well as their connection to Fourier coefficients of p-adic families of modular forms. This is ongoing joint work with Peter Xu.