Periods and L-values of Motives (2024)

*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

Gregory Baldi
CNRS
Translation Surfaces and Periods
In the moduli space of translation surfaces — i.e., pairs (X,w) of a compact Riemann surface and a holomorphic one form — there are some special subvarieties known as ‘orbit closures.’ Even if their definition is far removed from algebraic geometry, their properties are closely related to Hodge theory. After recalling their main properties (following the work of Eskin, Filip, McMullen, Mirzakhani, Mohammadi, Möller, Wright and many others), we will see how an ‘enriched’ Zilber-Pink philosophy governs the distribution of the aforementioned subvarieties, giving a new and effective approach to the finiteness results of Eskin-Filip-Wright. This is a joint work with D. Urbanik.

Raphaël Beuzart-Plessis
CNRS, Aix-Marseille Université
On the Gan-Gross-Prasad Conjecture for Bessel Periods of Unitary Groups
This talk will be an account of my recent joint work, joint with P.-H. Chaudouard and M. Zydor, on the Bessel periods of cuspidal automorphic forms on unitary groups. More precisely we prove the Gan-Gross-Prasad conjecture which is a statement about the non-vanishing of such periods and the Ichino-Ikeda conjecture which predicts the factorization of the periods in terms of special values of global L-functions and local periods. Beuzart-Plessis will try to overview some of the main ingredients of the proof which is based on a comparison of relative trace formula. The main novelty is the explicit computation of some spectral contributions.

Mattia Cavicchi
Laboratoire de Mathématiques d’Orsay
Automorphic Motives and the Motivic Hecke Algebra
If f is an algebraic, cuspidal automorphic form and M(f) the motive conjecturally attached to it, then the Beilinson conjectures describe the leading term of L(f,s), at non-critical integer values of s, by means of a regulator. In order for the latter to be defined, it is necessary to dispose of an object M(f) cut out by correspondences which are idempotent modulo rational equivalence, whereas the currently available constructions of such motives only work modulo homological equivalence. The aim of the talk is to explain our recent definition, using the modern advances in the theory of motives, of a motivic Hecke algebra, that we expect to be helpful in making progress on these issues.

Pierre Charollois
Sorbonne Université
On Elliptic Units for Almost Totally Real Fields
Pierre Charollois will report on joint work with Nicolas Bergeron and Luis Garcia, as well as recent work of Pierre Morain. Together, they provide numerical and theoretical evidence supporting an emerging arithmetic theory of infinite products similar to the ones giving rise to elliptic units but now attached to a number field having exactly one complex embedding. The construction also has some connections with prospective observations by G. Eisenstein, and with mathematical physics.

Slides (PDF)

Vesselin Dimitrov
California Institute of Technology
Arithmetic Holonomy Bounds and Special Values of
L-functions
A report on a new work joint with Calegari and Tang, in which we develop refined arithmetic holonomy bounds on G-functions and apply them to devise a proof of the Q-linear independence of 1, $\zeta(2)$, and the Dirichlet L-function special value $L(2,chi_{-3})$. We raise a problem of refining the holonomy bounds integrally, which in analogy to Andre’s proof of the Siegel-Shidlovsky theorem (transcendence sans transcendence) would yield to new linear independence proofs for special values of G-functions at certain special arguments of the form $x = 1/n$.

Slides (PDF)

Tony Feng
University of California, Berkeley
Higher Theta Functions Over Function Fields
The theory of theta functions and arithmetic theta functions is used classically to access the special values and special derivatives of automorphic L-functions. In joint work with Zhiwei Yun and Wei Zhang, we have constructed a theory of higher theta functions over function fields, which we expect to give access to higher derivatives of automorphic L-functions in terms of arithmetic geometry. This has led us, also partly jointly with Adeel Khan, to unexpected and fruitful connections between theta functions, derived algebraic geometry and motivic homotopy theory.

Javier Fresán
Sorbonne Université
A Differential Galois Action on Special Values of E-Functions
Javier Fresan reports on ongoing joint work with Stéphane Fischler, where we construct an action of the differential Galois group of the punctured affine line on the set of special values of E-functions, which unconditionally realizes a chunk of the expected action of the motivic Galois group on exponential periods. Fresán will also explain how it allows for simple proofs of transcendence results such as the Hermite-Lindemann-Weierstrass theorem.

Luis García
University College London
Explicit Class Field Theory and the Elliptic Gamma Function
It is a well-known classical fact that the abelian extensions of the rational numbers and of imaginary quadratic fields are generated by special values of the exponential function and of theta functions. During the talk, Luis García 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). García 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. García will also explain the relation to Stark units and discuss a limit formula relating this function to the derivative at s=0 of partial zeta functions. This is a report on joint work with Nicolas Bergeron and Pierre Charollois.

Michael Harris
Columbia University
Periods of Automorphic Forms and Motivic Periods
Michael Harris will report on several ongoing projects with Grobner, Lin, Raghuram, Kobayashi and Speh, whose common theme is the relation between the expressions of critical values of automorphic L-functions in terms of automorphic periods — integrals of automorphic forms over group-theoretic cycles — and the conjectural expression in terms of Deligne’s motivic periods. Two related projects with Grobner, Lin and Raghuram use Eisenstein cohomology and the Ichino-Ikeda identity for unitary groups to provide simultaneous proofs of automorphic versions of Deligne’s conjecture on critical values of Rankin-Selberg L-functions and of predicted identities between periods of automorphic representations of different unitary groups with the same base change to GL(n). The project with Kobayashi and Speh characterizes the automorphic periods in the Ichino-Ikeda identity that can be interpreted as cup products in coherent cohomology.

Slides (PDF)

Mark Kisin
Harvard University
Heights in the Isogeny Class of an Abelian Variety
Let A be an abelian variety over an algebraic closure of Q. A conjecture of Mocz asserts that there are only finitely many isomorphism classes of abelian varieties isogenous to A and of height less than some fixed constant c. In this talk, Mark Kisin will sketch a proof of the conjecture when the Mumford-Tate conjecture, which is known in many cases, holds for A. This result should be compared with Faltings’ famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.

Marco Maculan
IMJ-PRG, Sorbonne Université
The Shafarevich Conjecture for Subvarieties of Abelian Varieties
In 1962, Shafarevich conjectured that over any number field $K$, there are only finitely many isomorphism classes of smooth projective curves of given genus $g\ge 2$ with good reduction outside a fixed finite set $S$ of places of $K$. This statement was proven by Faltings in 1984 as a direct consequence of the analogous finiteness statement for abelian varieties. Since then, such finiteness statements have become known as instances of the ‘Shafarevich conjecture.’ Recently, Lawrence-Venkatesh discovered a technique to prove the nondensity of integral points, and Lawrence-Sawin used it to show the Shafarevich conjecture for hypersurfaces of abelian varieties. In this talk, Marco Maculan will discuss how to generalize Lawrence-Sawin’s result to subvarieties of arbitrary dimension. This is joint work with Krämer, and Javanpeykar-Krämer-Lehn.

Ananth Shankar
Northwestern University
Integral Canonical Models of Exceptional Shimura Varieties
Ananth Shankar will speak about canonicity of integral models for exceptional Shimura varieties at large primes and will focus on applications to semisimplicity of local systems, special points modulo p and special lifts of such points, and an analogue of Tate-isogeny for many mod p points. This is joint work with Ben Bakker and Jacob Tsimerman.

Birgit Speh
Cornell University
On the Restriction of Discrete Series Representations of U(p,q) to U(p-1,q)
Birgit Speh will discuss some of the results/conjectures mentioned in the M. Harris lecture in more detail. In representation theory of semi-simple Lie groups, translation functors play an important role. Roughly speaking, they are defined by taking the tensor product of an irreducible representation with a finite dimensional representation F followed by a projection on a direct summand. Speh will consider in this talk series representations of G=U(p,q) and of G’=U(p-1,q) and discuss the translation of G’ equivariant homomorphisms between discrete series representations of G and G’, i.e., on the symmetry breaking operators between the discrete series representations of G and G’. Period integrals define such a symmetry breaking operator, and so one may ask if period integrals and translations are enough to understand all the symmetry breaking operators. An interesting example is already the special case G=U(2,2) and G’=U(1,2) . Another important problem is to understand the map induced on (p,K) cohomology. This is joint work with T. Kobayashi and M. Harris.

Slides (PDF)

Jan Vonk
Leiden University
p-adic Height Pairings of Geodesics
Jan Vonk 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. Vonk 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.

Chen Wan
Rutgers University
Some Examples of the Relative Langlands Duality
In this talk, Chen Wan will discuss some examples of the relative Langlands duality (introduced by Ben-Zvi-Sakellaridis-Venkatesh) for strongly tempered spherical varieties. In some cases, Wan will introduce a relative trace formula comparison for the models and prove the fundamental lemma/smooth transfer. This is joint work with Zhengyu Mao and Lei Zhang.

Tonghai Yang
University of Wisconsin-Madison
On Gross and Zagier’s Algebraicity Conjecture
In their seminal paper on the Gross-Zagier conjecture, Gross and Zagier also proposed a conjecture about the algebraicity of some Higher Green function values at CM points. In this talk, Tonghai Yang will reformulate and generalize their conjecture in terms of regularized theta lifting and then prove their conjecture. This is joint work with Bruinier and Yingkun Li.

Slides (PDF)

Xinyi Yuan
Peking University
Partial Heights on Varieties over Function Fields
In joint work with Junyi Xie, we have introduced a notion of partial height, conjectured its non-degeneracy and considered its consequence on the geometric Bombieri-Lang conjecture. In this talk, we will introduce these terms and their relations.

Slides (PDF)

Zhiyu Zhang
Stanford University
Non-Reductive Special Cycles and Twisted Arithmetic Fundamental Lemmas
We care about arithmetic invariants of polynomial equations, e.g., L-functions, which (conjecturally) are often automorphic and related to cycles on Shimura varieties. Zhiyu Zhang will explain a formulation of (twisted) arithmetic Gan-Gross-Prasad conjecture on Asai L functions over CM fields, via $H^1$ of unitary Shimura varieties and doubling divisors following the work of Liu. As a key ingredient to prove the conjecture, Zhang will formulate and prove a twisted arithmetic fundamental lemma. In the induction step, we find the use of an interesting class of special cycles for non-reductive groups (in the p-adic world), which may be of independent interest.