Jean-Benoit Bost, University of Paris 11
Gisbert Wüstholz, ETH & Universität Zürich
Shou-Wu Zhang, Princeton University
This is the second symposium of a series of three symposia devoted to
the periods of mixed motives and to the special values of their L-functions,
with an emphasis on their interaction.
Discussion topics will include some exciting recent new developments:
• Exponential motives and 1-motives, especially on the work of Fresán-Jossen and Huber-Wüstholz.
• Rational points on varieties, especially on the uniform bound result by Dimitrov-Gao-Habegger and a new proof of Mordell conjecture by Lawrence and Venkatesh.
• Arithmetic Gan-Gross-Prasad conjecture, especially the recent proof of the arithmetic fundamental lemma by Wei Zhang.
• BSD conjecture and Beilison-Bloch-Kato conjectures, especially on the work of Burungale-Skinner-Tian, Liu-Tian-Xiao-Zhang-Zhu, Jetchev-Nekovar-Skinner.
• Zeta values and multi-zeta values, especially on the work of Brown and Zudilin.
SUNDAY | 05.01.22
8:30 - 9:30 PM Welcome Dinner | The Drawing Room
MONDAY | 05.02.22
7:30 - 9:45 AM Breakfast | Strathearn Restaurant 10:00 - 11:00 AM Wadim Zudilin | Differential equations for special hyperelliptic integrals 11:00-11:30 AM Break 11:30 - 12:30 PM Annette Huber-Klawitter | Structure of periods spaces 12:30 - 1:30 PM Lunch | The American Bar 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Peter Jossen | E-functions and Geometry I 6:00 - 6:15 PM Break 6:15 - 7:15 PM Javier Frésan | E-functions and Geometry II 8:00 - 9:30 PM Dinner | The Glendevon Lounge
TUESDAY | 05.03.22
7:30 - 9:45 AM Breakfast | Strathearn Restaurant 10:00 - 11:00 AM Francis Brown | Values of L-functions and periods of elliptic curves 11:00-11:30 AM Break 11:30 - 12:30 PM Wei Zhang | Heights of the arithmetic diagonal cycles on unitary Shimura varieties 12:30 - 1:30 PM Lunch | The American Bar 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM David Loeffler | Euler systems for automorphic Galois representations 6:00 - 6:15 PM Break 6:15 - 7:15 PM Sarah Zerbes | On the Birch–Swinnerton-Dyer conjecture for abelian surfaces 8:00 - 9:30 PM Dinner | Birnam Brasserie
WEDNESDAY | 05.04.22
7:30 - 9:30 AM Breakfast at the Strathearn Restaurant 10:30 - 12:00 PM Guided Hike 12:30 - 1:30 PM Lunch | The American Bar 1:30 - 4:30 PM Discussion & Recreation* 4:30 - 5:00 PM Tea 5:00 - 6:00 PM Joseph Ayoub | On the classicality of the motivic Galois group 6:00 - 6:15 PM Break 6:15 - 7:15 PM Yunqing Tang | Applications of arithmetic holonomicity theorems 8:00 - 9:30 PM Dinner | The Drawing Room
THURSDAY | 05.05.22
7:30 - 9:45 AM Breakfast | Strathearn Restaurant 10:00 - 11:00 AM Xinwen Zhu | Isogenies of mod p CM abelian varieties via the main theorem of complex multiplication 11:00-11:30 AM Break 11:30 - 12:30 PM Jennifer Balakrishnan | Quadratic Chabauty for modular curves 12:30 - 1:30 PM Lunch | The American Bar 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Ziyang Gao | Torsion points in families of abelian varieties 6:00 - 6:15 PM Break 6:15 - 7:15 PM Olivier Benoist | Sums of squares in local fields 8:00 - 9:30 PM Dinner | Dormy Clubhouse Bar & Grill
FRIDAY | 05.06.22
7:30 - 9:45 AM Breakfast | Strathearn Restaurant 8:30 - 10:00 AM COVID Testing | The Study 10:00 - 11:00 AM Yuri Tschinkel | New invariants in equivariant birational geometry (joint with A. Kresch) 11:00-11:30 AM Break 11:30 - 12:30 PM Yohan Brunebarbe | Subpolynomial growth of integral points on varieties with large fundamental group 12:30 - 1:30 PM Lunch | The American Bar 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Antoine Chamber-Loir | Real differential forms and currents on non-archimedean spaces, reloaded 6:00 - 6:15 PM Break 6:15 - 7:15 PM Discussion 7:45 PM Transfer to Shooting Lodge 8:00 - 9:30 PM Dinner | Shooting Lodge
The Billiard RoomMEALS Various; See ScheduleTEA & DISCUSSION The Assembly RoomEXCURSION LobbySATURDAY DEPARTURE Lobby
*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.
Participants will have access to a projector and screen for computer-based talks and blackboards for those who prefer to give board-based talks.
High-speed Internet access is available.
On Wednesday symposium activities are shortened for a guided hiking tour.
If you will participate in the hike, hiking boots or shoes appropriate for traversing rocky/wet terrain, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and waterproofs (e.g. raincoat or umbrella) are strongly recommended.
A small satchel containing a light snack and water is provided, but feel free to bring your own backpack in which to carry a camera and other items that may be useful to you.
Business casual clothing should be worn during the symposium.
The weather can change very quickly; please bring warm-weather clothing appropriate for spring in Scotland.
On the classicality of the motivic Galois group
The motivic Galois group is most naturally considered as an object in spectral algebraic geometry. However, deep conjectures in the theory of motives imply that the motivic Galois group is classical, i.e., has no higher derived information. We will discuss some recent attempts to verify the classicality of the motivic Galois group.
Quadratic Chabauty for modular curves
We describe how 𝑝-adic height pairings can be used to determine the set of rational points on curves, in the spirit of Kim’s nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain modular curves. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
Sums of squares in local fields
Artin and Pfister have shown that a nonnegative real polynomial in 𝑛 variables is a sum of 2𝑛 squares of rational functions. In this talk, I will consider local variants of this statement. In particular, I will give a proof of a conjecture of Choi, Dai, Lam and Reznick: a convergent real power series in 𝑛 variables which is nonnegative near the origin is a sum of 2𝑛-1 squares of Laurent series.
Values of L-functions and periods of elliptic curves
I will begin by illustrating a simple example of how the value of the L-function of an elliptic curve at point 2 may be interpreted as a period integral associated with an extension, in accordance with Beilinson’s conjecture. However, this extension involves an additional period that has no known interpretation. In the second half of the talk, I will propose a notion of mixed L-function whose values supply this missing period.
Subpolynomial growth of integral points on varieties with large fundamental group
Answering a question asked in a recent preprint of Ellenberg, Lawrence and Venkatesh, we prove the subpolynomial growth of integral points with a bounded height of an algebraic variety over a number field whose fundamental group is large. This is joint work with Marco Maculan.
E-functions and Geometry II
Siegel introduced the notion of E-function in a landmark 1929 paper with the goal of generalizing the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers. E-functions are power series with algebraic coefficients that are solutions of a linear differential equation and satisfy some growth conditions of an arithmetic nature. Besides the exponential, examples include Bessel functions and a rich family of hypergeometric series. Siegel asked whether all E-functions are polynomial expressions in these hypergeometric series. In these two talks, we will first explain how we answered Siegel’s question in the negative. We will then try to amend it by describing how E-functions arise from geometry in the form of ”exponential period functions” and why it might seem reasonable, in the light of other conjectures, to expect that all E-functions are of this kind.
Torsion points in families of abelian varieties
Given an abelian scheme defined over 𝐐̅ and an irreducible subvariety 𝑋 which dominates the base, the Relative Manin-Mumford Conjecture (proposed by Zannier) predicts how torsion points in closed fibers lie on 𝑋. The conjecture says that if such torsion points are Zariski dense in 𝑋, then the dimension of 𝑋 is at least the relative dimension of the abelian scheme unless 𝑋 is contained in a proper subgroup scheme. In this talk, I will present proof of this conjecture. As a consequence, this gives a new proof of the Uniform Manin-Mumford Conjecture for curves (recently proved by Kühne) without using equidistribution. This is joint work with Philipp Habegger.
Structure of periods spaces
We consider the vector space generated by the periods of a single 1-motive. The weight filtration on the motive induces a bifiltration on this space. We explain the consequences of our results on the period conjecture for this filtration.
In particular, we deduce a dimension formula for the most complicated contribution (which can be understood as periods of differential forms of the third kind with respect to non-closed paths).
If time permits, we compare this to description in Tannakian language and clear up the phenomenon of deficient motives. This is a joint work with Gisbert Wüstholz.
E-functions and Geometry I
Siegel introduced the notion of 𝐸-function in a landmark 1929 paper with the goal of generalizing the Hermite-Lindemann-Weierstrass theorem on the transcendence of the values of the exponential function at algebraic numbers. 𝐸-functions are power series with algebraic coefficients that are solutions of a linear differential equation and satisfy some growth conditions of an arithmetic nature. Besides the exponential, examples include Bessel functions and a rich family of hypergeometric series. Siegel asked whether all 𝐸-functions are polynomial expressions in these hypergeometric series. In these two talks, we will first explain how we answered Siegel’s question in the negative. We will then try to amend it by describing how 𝐸-functions arise from geometry in the form of ”exponential period functions” and why it might seem reasonable, in the light of other conjectures, to expect that all 𝐸-functions are of this kind.
Euler systems for automorphic Galois representations
I will recall the notion of an Euler system, and the role these objects play in proving cases of the Bloch–Kato conjecture and Iwasawa main conjecture; and I will survey a series of recent works (joint with Sarah Zerbes and others) in which we construct Euler systems for Galois representations appearing in the cohomology of Shimura varieties for various reductive groups, including GSp(4) and GU(2, 1). In this talk, I’ll emphasize the construction of these classes and their norm-compatibility relations, while the topic of ”explicit reciprocity laws” (relating the non-triviality of Euler system classes to values of L-functions) will be treated in Sarah’s separate talk.
Real differential forms and currents on non-archimedean spaces, reloaded
I will describe some aspects of joint work with Antoine Ducros (https://arxiv.org/abs/1204.6277) where we define, for the non-archimedean analytic spaces of Berkovich, an analogue of the classical calculus of differential forms and currents on complex analytic manifolds, motivated by non-archimedean aspects of Arakelov geometry. A first version of the theory appeared on arXiv in 2012, and I will try to emphasize aspects which emerged since we started to revise this still unpublished manuscript. Besides the complex analytic picture which is used as a guide throughout our work, the theory is built on ideas from tropical geometry, a construction of A. Lagerberg on R^n, and on the presence, within non-archimedean spaces, of polyhedral real subspaces (skeleta) on which real calculus can be performed. If time permits, I will try to evoke various recent developments of this work proposed by other mathematicians.
Applications of arithmetic holonomicity theorems
In this talk, we will discuss the proof of the unbounded denominators conjecture on Fourier coefficients of SL₂(𝐙)-modular forms, and the proof of the irrationality of the 2-adic zeta value at 5. Both proofs use an arithmetic holonomicity theorem, which can be viewed as a refinement of André’s algebraicity criterion. If time permits, we will give a proof of the arithmetic holonomicity theorem via the slope method a la Bost. This is joint work with Frank Calegari and Vesselin Dimitrov.
On the Birch–Swinnerton-Dyer conjecture for abelian surfaces
As explained in David’s talk, Euler systems are one of the most powerful tools for proving cases of the Bloch–Kato conjecture and other related problems such as the Birch and Swinnerton-Dyer conjecture.
In my talk, I will explain how to relate the Euler system in the cohomology of Shimura varieties for GSp(4), which was introduced in David’s talk, to values of L-functions of genus 2 Siegel modular forms. I will then explain recent work with Loeffler, where we use this result to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and for modular elliptic curves over imaginary quadratic fields.
Heights of the arithmetic diagonal cycles on unitary Shimura varieties
For the product Shimura variety attached to the group 𝐺 = 𝑈(𝑛−2, 1) × 𝑈(𝑛, 1), there is an arithmetic diagonal cycle given by the Shimura subvariety attached to a subgroup 𝐻 = 𝑈(𝑛 − 2, 1). The arithmetic Gan-Gross-Prasad conjecture relates its height (after projection to a Hecke eigenspace) to the first central derivative of a certain L-function. We report some recent results towards this conjecture, including the proof of the arithmetic fundamental lemma (local heights at places with good reduction) by Zhang over 𝐐𝑝 and Mihatsch–Zhang over a general p-adic field and of the arithmetic transfer conjecture at certain parahoric level (local heights at places with semistable reduction) by Zhiyu Zhang.
Isogenies of mod p CM abelian varieties via the main theorem of complex multiplication
The main theorem of complex multiplication describes how automorphisms of 𝐂 act on CM abelian varieties and their torsion points. I will explain this theorem can also be used to describe 𝑝-quasi-isogenies between mod 𝑝 reductions of CM abelian varieties. Time permitting, I will explain how such a description helps us understand exotic correspondences between mod 𝑝 fibers of different Shimura varieties. Joint work with Liang Xiao.
Differential equations for special hyperelliptic integrals
I will report on an ongoing project with Mark van Hoeij and Duco van Straten, in which we explore second-order linear differential equations for hyperelliptic integrals. The equations do not reduce to the ones for the Euler-Gauss hypergeometric functions (a.k.a. elliptic integrals in these settings) and also depend on an extra parameter. In particular, they provide a 1-parameter family of counterexamples over 𝐐 to a 1990 conjecture of Dwork, which is already disproved (over specific number fields) through tough techniques involving Shimura curves and Teichmüller curves.
The meeting room will be equipped with the following:
- Computer Adapters
- Lavalier Microphone
- 4’x6′ whiteboards
The meeting room will be set classroom-style with a table and power outlets for all participants.
The foundation will coordinate and purchase air and train travel to the symposium. Travel specifications can be provided by clicking the registration link above.
Participants arriving via plane will arrive via Glasgow (GLA) or Edinburgh (EDI) International Airports. Travel time between either airport and Gleneagles Hotel is approximately 60 minutes.
The closet train station to Gleneagles Hotel is the Gleneagles Station. Travel time between the station and the hotel is approximately 5 minutes.
Should you require a visa to travel to Scotland, please contact Meghan Fazzi for an official letter of invitation to assist in the application process. The visa approval process can last several months, especially for non-residents residing in the US wishing to travel to and return from countries outside North America. We recommend attendees begin their visa application paperwork as soon as you have registered for the symposium.
Local Ground Transportation
Arrival at Glasgow and Edinburgh International Airports
All participants are required to arrive on Sunday prior to the meeting’s start and depart on Saturday. No accommodations will be made for partial participation.
The foundation will arrange for your transfer from GLA or EDI airports to Gleneagles Hotel. After claiming your luggage at baggage claim you will proceed through customs and to the exit where a uniformed driver holding a Simons Symposia sign will greet you.
Travel time from EDI or GLA to Gleneagles Hotel is 60 minutes.
Arrival at Gleneagles Train Station
The foundation will arrange for your transfer from Gleneagles train station to Gleneagles Hotel via a hotel shuttle.
Travel time from the station to the hotel is approximately 5 minutes.
Departure from Gleneagles Hotel
The day prior to your departure you will receive a departure letter advising on your departure time to the airport or train station via private car. Please meet your driver in the hotel lobby ten minutes prior to your noted departure time. If for any reason your flight departure time changes, please visit the front desk to update hotel staff.
Travel Days: Meals on travel days are reimbursable by the foundation.
Sunday Welcome Dinner: The Simons Foundation will host a welcome dinner on Sunday at 8:30 PM in The Drawing Room. All participants are expected to attend with the exception of those arriving at 10PM.
Monday through Friday: All meals occurring during the meeting week will be covered directly by the Simons Foundation. Participants are expected to be present for all foundation-hosted meals.
Breakfast is served starting at 7:00 AM at The Strathearn Restaurant. Tables are reserved for all symposium participants so that the group may dine together.
Lunch is served daily for the group in The American Bar.
Dinner is hosted around the hotel property. Locations are noted on the agenda.
Reimbursement and Participation Policy
Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the travel policy. Please review the policy and complete when submitting your expenses. Receipts are required for any expenses over $50 USD and are due within THIRTY DAYS (30) after the conclusion of the symposium. All expenses are reconciled through the foundation’s online reimbursement platform. Additional information regarding reimbursement is sent via email on the final day of the symposium.
Should you have any questions, please contact Meghan Fazzi.
Participation and Guest Policy
By registering, you agree to participate fully in the symposium program, which begins on Sunday and concludes on Saturday. No accommodations will be made for partial attendance.
It is the foundation’s policy to discourage spouses, families or others from joining participants during the symposium. Participants who choose to invite guests to come before or stay on after the symposium do so at their own expense. The Simons Foundation will not cover expenses associated with such extensions.
Should you decide to extend your stay, Gleneagles Hotel will honor the conference rate for up to two nights. Any nights beyond those associated with the conference are not reimbursable by the foundation.
If you wish to extend your stay, please contact Meghan Fazzi directly and conference rate information will be provided over email.