Periods and Lvalues of Motives (2024)
Organizers:
JeanBenoit Bost, Université ParisSud
ShouWu Zhang, Princeton University
Hotel:
The Simons Foundation will book and pay for up to six nights at the symposium hotel arriving on Sunday and departing on Saturday. All additional nights are to be paid for directly and will not be reimbursed.
Schloss Elmau
In Elmau 2, 82493 Krün, Germany
Phone: +49 8823 180
Website: https://www.schlosselmau.de/en/
Meeting Goals:
The Simons Symposium on Periods and Lvalues of Motives is devoted to the periods of motives and automorphic forms and the special values of their Lfunctions, emphasizing their interactions and relations with Diophantine geometry. Among the recent developments related to these themes, the following ones are expected to be discussed during the final symposium. We hope you can join us.
 Recent progresses in the direction of the direction of the Stark conjecture and its avatars
 Applications of padic cohomology to finiteness results in Diophantine geometry
 Arithmetic geometry of Abelian schemes and Betti maps
 Period of automorphic forms and relative Langlands Program
 Arithmetic volumes of Shimura varieties and Lfunctions, and function field analogs
Meeting Report
This symposium featured 18 lectures covering stateoftheart results on automorphic and motivic periods, heights and Diophantine problems, explicit reciprocity law, rationality, and transcendence.
Automorphic periods and relative Langlands
We had four lectures on automorphic periods and their relation to special values of automorphic Lfunctions given by Michael Harris, Birgit Speh, Chen Wan, and Raphaël BeuzartPlessis, with the following results reported:
 the recent development of Harris’ conjecture in the 1990s about the decomposition of automorphic periods, especially some progress on the relation between periods for automorphic representations for unitary groups and RankinSelberg convolution Lseries using Eisenstein cohomology and IchinoIkeda identities;
 the representation aspects of Harris’ lecture with translation functors, especially for the pairs G = U (p, q) restricted to G^{′} = U (p − 1, q);
 the new interpretation where the periods and Lvalue relation as part of farreaching conjectures in the recent relative Langlands duality formulated by BenZvi, Sakellaridis, and Venkatesh, with a series of examples using a relative trace formula;
 the recent proof of the GanGrossPrasad conjecture for U (m) ×U (n) and the exact formula in the style of IchinoIkeda.
Shimura varieties, special cycles, and motivic Hecke operators
We had four lectures on the geometry and arithmetic of Shimura varieties, special cycles, and their function fields analog given by Ananth Shankar, Zhiyu Zhang, Tony Fong, and Mattia Cavicchi, with the following results reported:
 the canonical integral models for exceptional Shimura varieties at large primes and applications to the semisimplicity of local systems and special points;
 a new formulation of an arithmetic GanGrossPrasad conjecture on Asai L functions over CM fields, with a key ingredient being a twisted arithmetic fundamental lemma with an interesting class of special cycles for nonreductive groups in the padic world;
 the construction of higher theta functions over function fields, a theory which is expected to give access to higher derivatives of automorphic Lfunctions;
 a new theory of definition for motivic Hecke algebra using modern advances in the theory of motives, which we expect to be helpful in making progress on these issues.
Rational points, CM points, and unlikely intersections
Xinyi Yuan, Marco Maculan, Mark Kisin, and Gregory Baldi gave four lectures on rational points, integral points, and CM points, with the following results reported:
 a proof of the geometric BombieriLang conjecture for the finiteness of rational points on hyperbolic varieties, admitting finite morphisms to Abelian varieties, with a new notion of partial height and a new conjecture about its nondegeneracy;
 the nondensity of integral points of moduli of subvarieties of Abelian varieties in terms of the Shafarevich conjecture;
 a proof of the Northcott property of Faltings height on each isogeny class over Q of Abelian varieties under the MumfordTate conjecture;
 the ZilberPink conjecture about orbit closures on the moduli of translation surfaces; and a new and effective approach to the finiteness results of Eskin, Filip, and Wright.
Elliptic Gamma functions and pheight pairing
Pierre Charollois, Luis García, and Jan Vonk gave three lectures on the elliptic Gamma functions and padic height pairings, with the following topics covered:
 a conjectured explicit reciprocity law for number fields with exactly one complex place by using the elliptic gamma functions, infinite products arising in mathematical physics, which is connected to observations by G. Eisenstein, with some numerical evidence for constructing units and proof of a Kronecker limit formula for complex cubic fields;
 a padic height pairing of real quadratic geodesics on modular curves, with motivation for studying this pairing coming from its relation to real quadratic (RM) singular moduli.
Green’s functions, Efunctions, and Gfunctions
We had three lectures on the rationality of L values, the algebraicity of CM values Green’s functions, and the Galois theory of E functions given by Vesselin Dimitrov, Javier Fresán, and Tonghai Yang, which included the following results:
 some refined arithmetic holonomy bounds on Gfunctions that were applied to devise a proof of the Qlinear independence of 1, ζ(2), and the Dirichlet Lfunction special value L(2, χ_{−}_{3});
 an action of the differential Galois group of the punctured affine line on the set of special values of Efunctions, which unconditionally realizes a chunk of the expected action of the motivic Galois group on exponential periods;
 a proof of a conjecture of Gross and Zagier about the algebraicity of some higher Greenfunction values at CM points.

Agenda
SUNDAY
8:30  9:30 PM Welcome Dinner @ La Salle MONDAY
7:30  9:45 AM Breakfast at La Salle 10:00  11:00 AM Michael Harris  Periods of Automorphic Forms and Motivic Periods 11:0011:30 AM Break 11:30  12:30 PM Birgit Speh  On the Restriction of Discrete Series Representations to U(p,q) to U(p1,q) 12:30  1:30 PM Lunch at La Salle 1:30  4:30 PM Discussion & Recreation* 4:30 5:00 PM Tea 5:00  6:00 PM Chen Wan  Some Examples of the Relative Langlands Duality 6:00  6:15 PM Break 6:15  7:15 PM Raphaël BeuzartPlessis  On the GanGrossPrasad Conjecture for Bessel Periods of Unitary Groups 8:00  9:30 PM Dinner @ Ganesha TUESDAY
7:30  9:45 AM Breakfast at La Salle 10:00  11:00 AM Ananth Shankar  Integral Canonical Models of Exceptional Shimura Varieties 11:0011:30 AM Break 11:30  12:30 PM Mattia Cavicchi  Automorphic Motives and the Motivic Hecke Algebra 12:30  1:30 PM Lunch at La Salle 1:30  4:30 PM Discussion & Recreation* 4:30 5:00 PM Tea 5:00  6:00 PM Tony Feng  Higher Theta Functions Over Function Fields 6:00  6:15 PM Break 6:15  7:15 PM Zhiyu Zhang  NonReductive Special Cycles and Twisted Arithmetic Fundamental Lemmas 8:00  9:30 PM Dinner @ La Salle WEDNESDAY
7:30  9:30 AM Breakfast @ La Salle 9:45  2:00 PM Guided Hike 2:00  3:00 PM Lunch at La Salle 3:00  4:30 PM Discussion & Recreation* 4:30  5:00 PM Tea 5:00  6:00 PM Xinyi Yuan  Partial Heights on Varieties Over Function Fields 6:00  6:15 PM Break 6:15  7:15 PM Marco Maculan  The Shafarevich Conjecture for Subvarieties of Abelian Varieties 8:00  9:30 PM Dinner @ La Salle THURSDAY
7:30  9:45 AM Breakfast at La Salle 10:00  11:00 AM Pierre Charollois  On Elliptic Units for Almost Totally Real Fields 11:0011:30 AM Break 11:30  12:30 PM Luis Garcia  Explicit Class Field Theory and the Elliptic Gamma Function 12:30  1:30 PM Lunch at La Salle 1:30  4:00 PM Discussion & Recreation* 4:00 4:30 PM Tea 4:30  5:30 PM Jan Vonk  padic Height Pairings of Geodesics 5:30  5:45 PM Break 5:45  6:45 PM Tonghai Yang  On Gross and Zagier’s Algebraicity Conjecture 7:00  8:00 PM Dinner @ La Salle 8:30  10:00 PM Concert: Anna Vinnitskaya, piano FRIDAY
7:30  9:45 AM Breakfast at La Salle 10:00  11:00 AM Mark Kisin  Heights in the Isogeny Class of an Abelian Variety 11:0011:30 AM Break 11:30  12:30 PM Gregory Baldi  Translation Surfaces and Periods 12:30  1:30 PM Lunch at La Salle 1:30  4:30 PM Discussion & Recreation* 4:30 5:00 PM Tea 5:00  6:00 PM Javier Fresan  A Differential Galois Action on Special Values of Efunctions 6:00  6:15 PM Break 6:15  7:15 PM Vesselin Dimitrov  Arithmetic Holonomy Bounds and Special Values of Lfunctions 8:00  9:30 PM Dinner @ Summit Pavillion LOCATIONS
SESSIONS Pavilion located at the Schloss Elmau Retreat MEALS Various, see agenda TEA & DISCUSSION Pavilion located at the Schloss Elmau Retreat EXCURSION Meet in Schloss Elmau Lobby SATURDAY DEPARTURE Meet in Schloss Elmau Lobby *Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

Abstracts & Slides
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’ ZilberPink philosophy governs the distribution of the aforementioned subvarieties, giving a new and effective approach to the finiteness results of EskinFilipWright. This is a joint work with D. Urbanik.Raphaël BeuzartPlessis
CNRS, AixMarseille Université
On the GanGrossPrasad 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 GanGrossPrasad conjecture which is a statement about the nonvanishing of such periods and the IchinoIkeda conjecture which predicts the factorization of the periods in terms of special values of global Lfunctions and local periods. BeuzartPlessis 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 noncritical 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.Vesselin Dimitrov
California Institute of Technology
Arithmetic Holonomy Bounds and Special Values of
Lfunctions
A report on a new work joint with Calegari and Tang, in which we develop refined arithmetic holonomy bounds on Gfunctions and apply them to devise a proof of the Qlinear independence of 1, $\zeta(2)$, and the Dirichlet Lfunction 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 SiegelShidlovsky theorem (transcendence sans transcendence) would yield to new linear independence proofs for special values of Gfunctions at certain special arguments of the form $x = 1/n$.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 Lfunctions. 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 Lfunctions 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 EFunctions
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 Efunctions, 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 HermiteLindemannWeierstrass theorem.Luis García
University College London
Explicit Class Field Theory and the Elliptic Gamma Function
It is a wellknown 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 Lfunctions in terms of automorphic periods — integrals of automorphic forms over grouptheoretic 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 IchinoIkeda identity for unitary groups to provide simultaneous proofs of automorphic versions of Deligne’s conjecture on critical values of RankinSelberg Lfunctions 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 IchinoIkeda identity that can be interpreted as cup products in coherent cohomology.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 MumfordTate 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
IMJPRG, 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, LawrenceVenkatesh discovered a technique to prove the nondensity of integral points, and LawrenceSawin used it to show the Shafarevich conjecture for hypersurfaces of abelian varieties. In this talk, Marco Maculan will discuss how to generalize LawrenceSawin’s result to subvarieties of arbitrary dimension. This is joint work with Krämer, and JavanpeykarKrämerLehn.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 Tateisogeny 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(p1,q)
Birgit Speh will discuss some of the results/conjectures mentioned in the M. Harris lecture in more detail. In representation theory of semisimple 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(p1,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.Jan Vonk
Leiden University
padic Height Pairings of Geodesics
Jan Vonk will discuss a certain padic 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 BenZviSakellaridisVenkatesh) 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 WisconsinMadison
On Gross and Zagier’s Algebraicity Conjecture
In their seminal paper on the GrossZagier 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.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 nondegeneracy and considered its consequence on the geometric BombieriLang conjecture. In this talk, we will introduce these terms and their relations.Zhiyu Zhang
Stanford University
NonReductive Special Cycles and Twisted Arithmetic Fundamental Lemmas
We care about arithmetic invariants of polynomial equations, e.g., Lfunctions, which (conjecturally) are often automorphic and related to cycles on Shimura varieties. Zhiyu Zhang will explain a formulation of (twisted) arithmetic GanGrossPrasad 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 nonreductive groups (in the padic world), which may be of independent interest.