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 included 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.
The 17 talks of the symposium presented an excellent overview of the developments of a large domain of arithmetic geometry during the last four years. Several breakthroughs realized since the first symposium in this series, held in 2018, were notably discussed, and new directions of investigations were indicated.
A striking feature of the talks was the unifying role of the philosophy of motives and some very concrete analytic expressions, the L-values attached to motives, behind a series of, superficially, very mixed results and techniques of arithmetic geometry.
1. Among the various themes covered by the program of this conference devoted to Periods and L-values of motives, the “extreme points” have been represented by the two pairs of lectures: E-functions and geometry, by J. Fresán and Jossen, and Euler systems for automorphic Galois representations and On the Birch–Swinnerton-Dyer conjecture for abelian surfaces by D. Loeffler and S. Zerbes.
J. Fresán and P. Jossen reported on their construction of a category of exponential motives. They explained how the Tannakian approach applied to this category has allowed them to solve an old question of Siegel concerning the possible reduction of arbitrary E-functions to hypergeometric functions negatively. They also presented a speculative picture of E-functions arising from geometry and their relations with general E-functions.
The lectures by D. Loeffler and S. Zerbes provided a lucid account of the recent developments in the theory of Euler systems. Since the pioneering work of Kolyvagin, Rubin and Kato, Euler systems are known to be one essential tool of modern number theory, allowing to establish profound finiteness results on Tate-Shafarevich and Selmer groups of motives in terms of their L-values that were previously entirely out of reach.
During the first period of the development of Euler systems, every new instance of those appeared to rely on some “miraculous” insight on Shimura curves. During the last decade, a series of breakthroughs by Loeffler, Zerbes, et al., have considerably changed the situation by introducing some new construction methods for wide dimensional Shimura varieties, which turned out to be remarkably versatile. They have allowed them to establish a series of finiteness results in arithmetic geometry that would have seemed inaccessible less than ten years ago, notably concerning motives associated with automorphic representations of unitary and symplectic groups with non-vanishing central L-values.
The talks by Loeffler and Zerbes constituted both an introduction to this sophisticated circle of techniques and a presentation of their most recent results concerning applications of these to the arithmetic of abelian surfaces over Q in rank 0 case.
2. The talk by W. Zudilin, Differential equations for special hyperelliptic integrals, discussed a circle of questions in a similar spirit to the ones in the talk by Fresán and Jossen: the construction of counterexamples to a conjecture of Dwork asserting that the solutions of certain second-order linear differential equations for hyperelliptic integrals may be reduced to Euler-Gauss hypergeometric functions.
In her talk about Structure of periods spaces, A. Huber-Klawitter explained his joint work with G. Wüstholz on the period of 1-motives, focusing on the role of the weight filtration on the category of 1-motives and to its consequences on the structure of the vector space generated by the periods of a single 1-motive.
F. Brown, in his lecture about Values of L-functions and periods of elliptic curves, discussed some intriguing features of the relation between the value of the L-function of an elliptic curve at point 2 and some period integrals associated to extensions of motives. This extension involves an additional period with no previously known interpretation, and F. Brown has proposed a new notion of a mixed L-function whose values would supply this missing period.
3. W. Zhang reported on his recent work concerning some new cases concerning unitary Shimura varieties, of the arithmetic Gan-Gross-Prasad conjecture relating the height of some remarkable algebraic cycles on Shimura varieties to the central derivative of a certain L-functions. This would give evidence of Beilinson–Bloch–Kato’s conjecture in the rank 1 case. The importance of the Gan-Gross-Prasad conjecture — also visible in the lecture by D. Loeffler — in studying periods and L-values of motives associated with Shimura varieties was made particularly clear in this talk.
The arithmetic of Shimura varieties and the correspondences between these varieties was also the central theme behind the talk by X. Zhu, Isogenies of mod p CM abelian varieties via the main theorem of complex multiplication, that explained how some enhanced versions of the classical theory of CM abelian varieties was the key to an understanding of the correspondence between the reductions of different Shimura varieties over some finite field.
4. The talk of J. Balakrishnan, Y. Brunebarbe, Z. Gao and Y. Tang were devoted to recent breakthroughs on various significant problems of arithmetic geometry. All of these rely on some rather classical techniques of Diophantine geometry that very clever new geometrical insights have considerably enhanced.
J. Balakrishnan lectured on her recent works on Quadratic Chabauty for modular curves, devoted to the complete determination of the rational point on some modular curves by an elaboration of the classical method of Chabauty that relies on the use of some kind of “higher period maps,” in the spirit of Kim’s nonabelian Chabauty program.
Y. Brunebarbe discussed his recent joint result with M. Maculan on the Subpolynomial growth of integral points on varieties with large fundamental group, where some preliminary results concerning the distribution of rational points on projective varieties in the line of the classical work of Pila and Heath-Brown are combined with geometric constructions involving varieties with large fundamental groups, as introduced by Kollár in his work in complex algebraic and analytic geometry concerning the Shafarevich conjecture.
Z. Gao presented his recent results with P. Habegger on Torsion points in families of abelian varieties, where they proved a relative Manin-Mumford conjecture proposed by U. Zannier. As one application, they give an alternative proof of the Lars Kühn theorem on the uniform Manin-Mumford for curves using the O-minimality technique rather than equidistribution.
In her talk on Applications of arithmetic holonomicity theorems, Y. Tang presented a series of striking recent results with F. Calegari and V. Dimitrov, relying on some refinements of the algebraicity theorems due to Chudnovsky and André, which combined with some various clever auxiliary constructions allow them to establish the unbounded denominators conjecture on Fourier coefficients of SL2(Z) modular forms and the irrationality of the 2-adic zeta value at 5.
5. Finally, the talks by Ayoub, O. Benoist and Y. Tschinkel gave diverse illustrations of the importance of the “motivic philosophy” and how it leads to some remarkable constructions and results relating high-level mathematical structures and very concrete problems.
J. Ayoub, in his talk, On the classicality of the motivic Galois group, explained why the motivic Galois group, initially defined as an object in spectral derived algebraic geometry, is expected to contain no higher derived information and, in this sense, to be classical. This classicality is related to deep, concrete conjectures concerning algebraic cycles in algebraic varieties.
In his talk Sums of squares in local fields, O. Benoist demonstrated how it is possible to combine a whole series of recent advanced results concerning sophisticated cohomological invariants of algebraic varieties and their motives to obtain a proof of the following “elementary” result, conjectured by 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.
Y. Tschinkel presented his recent joint work with A. Kresch concerning some New invariants in equivariant birational geometry. Initially motivated by the application of motivic integration techniques to questions of rationality and by the quest for an equivariant version of these their work leads to a series of intriguing questions and results on such classical objects as the action of finite groups on low dimension algebraic varieties or finite subgroups of the Cremona group.
MONDAY | 05.02.22
10:00 - 11:00 AM Wadim Zudilin | Differential equations for special hyperelliptic integrals 11:30 - 12:30 PM Annette Huber-Klawitter | Structure of periods spaces 5:00 - 6:00 PM Peter Jossen | E-functions and Geometry I 6:15 - 7:15 PM Javier Frésan | E-functions and Geometry II
TUESDAY | 05.03.22
10:00 - 11:00 AM Francis Brown | Values of L-functions and periods of elliptic curves 11:30 - 12:30 PM Wei Zhang | Heights of the arithmetic diagonal cycles on unitary Shimura varieties 5:00 - 6:00 PM David Loeffler | Euler systems for automorphic Galois representations 6:15 - 7:15 PM Sarah Zerbes | On the Birch–Swinnerton-Dyer conjecture for abelian surfaces
WEDNESDAY | 05.04.22
5:00 - 6:00 PM Joseph Ayoub | On the classicality of the motivic Galois group 6:15 - 7:15 PM Yunqing Tang | Applications of arithmetic holonomicity theorems
THURSDAY | 05.05.22
10:00 - 11:00 AM Xinwen Zhu | Isogenies of mod p CM abelian varieties via the main theorem of complex multiplication 11:30 - 12:30 PM Jennifer Balakrishnan | Quadratic Chabauty for modular curves 5:00 - 6:00 PM Ziyang Gao | Torsion points in families of abelian varieties 6:15 - 7:15 PM Olivier Benoist | Sums of squares in local fields
FRIDAY | 05.06.22
10:00 - 11:00 AM Yuri Tschinkel | New invariants in equivariant birational geometry (joint with A. Kresch) 11:30 - 12:30 PM Yohan Brunebarbe | Subpolynomial growth of integral points on varieties with large fundamental group 5:00 - 6:00 PM Antoine Chamber-Loir | Real differential forms and currents on non-archimedean spaces, reloaded 6:15 - 7:15 PM Discussion
Abstracts & Slides
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.
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.
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.
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.
New invariants in equivariant birational geometry (joint with A. Kresch)
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.