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

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
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. 
AV
Participants will have access to a projector and screen for computerbased talks and blackboards for those who prefer to give boardbased talks.
Highspeed Internet access is available.

Participation & Funding
Funding
The foundation will arrange and pay for all air and train travel to the Simons Symposium as well as hotel accommodations and reimbursement of local expenses. Businessclass or premium economy airfare will be booked for all flights over five hours.Participation
The Simons Symposium only accommodates full inperson participation. By accepting, you agree to participate fully in the scientific program, arriving on Sunday prior to the welcome dinner and departing on the following Saturday. No accommodations will be made for partial or remote attendance.Guests & Children
It is against Simons Foundation policy for guests, including children, to accompany participants during the meeting week. Guests are welcome to join prior to or at the conclusion of the symposium at their own expense. 
Travel & Hotel
Air & Rail
The foundation will arrange and pay for roundtrip travel from a participant’s home city to the symposium. All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency. Travel arrangements not booked through the preferred agency, including triangle trips and routing/preferred airlines outside budget, must be preapproved by the Simons Foundation and a reimbursement quote must be obtained through the foundation’s travel agency.Travel specifications, including preferred airline, will be accommodated provided that these specifications are reasonable and within budget.
All costs related to changes made to ticketed travel are to be paid for by the participant and are not reimbursable. Please contact the foundation’s travel agency for further assistance.
Personal & Rental Cars
Personal car and rental car trips over 250 miles each way require prior approval from the Simons Foundation via email.Regardless of distance, rental cars must be preapproved by the Simons Foundation.

Ground Transportation
The Simons Foundation will directly arrange roundtrip ground transportation between the arriving airport and the symposium hotel.
Expenses for ground transportation outside the above arrangements will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation.

Reimbursement
Overview
Individuals will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.Expenses accrued as a result of meetings not directly related to the Simons Foundationhosted meeting will not be reimbursed by the Simons Foundation and should be paid by other sources.
Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.
Meals
The Simons Foundation will coordinate all meals during the symposium weeks. The daily meal limit for all other meals (i.e. those taken on travel days) is $125 and itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses. Meals taken on travel days are reimbursable.
 Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner) are not reimbursable.
 If a meal was not provided on a meeting day, dinner for example, that expense is reimbursable.
 Meals taken on days not associated with Simons Foundationcoordinated events are not reimbursable.
 Minibar expenses are not reimbursable.
 Meal expenses for a nonfoundation guest are not reimbursable.
Group meals consisting of fellow meeting participants paid by a single person will be reimbursed up to $65 per person per meal and the amount will count towards each individual’s $125 daily meal limit.
Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation. While in NYC, individuals are encouraged to use public transportation and not use taxi, Uber or Lyft services. 
Contacts
Travel & Hotel Assistance
Ovation Travel Team
[email protected]
www.ovationtravel.comMeeting Questions & Assistance
Meghan Fazzi
Manager, Events & Administration, MPS
[email protected]
(212) 5246080