# Periods and L-values of Motives

Date & Time

Organizers:
Jean-Benoit Bost, University of Paris 11
Shou-Wu Zhang, Princeton University
Gisbert Wüstholz, ETH & Universität Zürich

Motives over number fields provide a modern organizing framework for Diophantine geometry | a core subject in mathematics since Greek antiquity, that still plays a central role in pure mathematics. Motives can be viewed as an abstract version of an algebraic variety. To each motive over a number field one can attached well-defined periods (that are complex numbers) and L-functions (that are meromorphic functions of one complex variable). Two basic – although still largely conjectural – principles appear to govern the relations between periods of motives, special values of their L-functions, and their arithmetic geometry:

• special values of L-functions may be expressed in terms of arithmetic intersection theory;
• algebraic relations between periods of a projective variety X and between special values of L-functions are explained by geometry, i.e., by algebraic cycles on X and its powers.

During the last two decades, important new developments in Diophantine geometry, mathematical physics, and automorphic representations have led to new perspectives on these two principles and on their relations to diverse central problems of arithmetic geometry. The symposia would bring together some of the experts of these developments, and stimulate possible new applications to Diophantine geometry.

In view of these recent developments, the symposium on Periods and L-values of motives, will bring together experts from different areas to discuss state-of-the-art mathematics that lies at the interface of motives over global fields, periods, and L-functions and to initiate new collaborations.

### Meeting Report

1. The study of motives attached to varieties over number fields by means of their periods and of the relation between these and special values of their $$L$$-functions has a long history and constitutes a central and extremely active domain of algebraic geometry and number theory.

A notable feature of this circle of questions is the array of conjectures it has motivated during the last decades, including notably Langlands conjectures relating $$L$$-functions of motives and automorphic forms and the Bloch-Beilinson conjectures. These conjectures that have an amazingly accurate predictive power and partial progresses on them have had a considerable impact on the recent developments of of arithmetic geometry, although a definitive understanding of these conjectures still appear out of reach.

Another remarkable feature of the domain is the diversity of techniques involved in these recent progresses and the way they require some combination of “big general machinery” and of in-depth understanding of some very specific classes of algebraic varieties, such as Shimura varieties or suitable moduli spaces.

The talks in of the symposium and its schedule have been devised both to reflect the multiplicity of the recent advances in this field and to ensure some thematic continuity between its various aspects.

The organizers of the symposium are very grateful to the participants for the quality of their lectures. They included presentations of major advances recently obtained by the participants, emphasized the interrelations between periods, motives and $$L$$-functions in a format accessible to a public of arithmetic geometers with diverse expertises and indicated some new connections between these themes.

2. The first talk of the symposium, by Joseph Ayoub, has been an overview of his recently announced proof of the Conservativity Conjecture, concerning classical (e.g. de Rham) realizations of Voevodsky motives in characteristic zero. This sensational result aroused a considerable interest and led to many discussions (an even to some informal additional talks by Ayoub during the following days) involving the participants with a special expertise in the theory of algebraic cycles.

The following talks on Monday, April 29 were devoted to some recent works in Diophantine geometry where periods of Abelian varieties and 1-motives did play a central role.

In his lecture Fabrizio Andreatta discussed a question of Grothendieck about the comparison of the de Rham and the Betti cohomology of a scheme $$X$$ and applied transcendence theory to give an answer in a particular case.

Yves André reported on his recent work with Corvaja and Zannier, that emphasizes the importance of Betti coordinates of sections of Abelian schemes for “unlikely intersections” problems.

A different way to study rational points was explained by Philipp Habegger. It uses the Néron-Tate height of an abelian variety in the case of a variation of abelian varieties over a curve and here the monodromy action on the period lattice enters the picture. As an application new cases of the Bogomolov Conjecture in the generic fiber are obtained.

3. The first two talks on Tuesday, April 30, have been devoted to various aspects of polylogarithms, seen as periods of mixed Tate motives and their avatars.

Tomohide Terasoma presented his recent description of the Aomoto polylogarithms, associated to arrangements of hyperplanes, in terms of the Bloch-Kriz formalism.

Chang Chieh-Yu gave an overview of multiple zeta values in positive characteristic, including his recent work with Y. Mishiba describing their linear relations, based on the theory of $$t$$-modules.

Alexander Goncharov presented his striking recent results on the Zagier Conjecture, expressing the value at 4 of Dedekind zeta-functions of number fields in terms of tetralogarithms, based on the cluster variety structure of the moduli spaces $$\mathcal{M}_{0,n}.$$

Apéry’s proof of the irrationality of $$\zeta(3)$$ initiated a study over years of the geometry underlying his proof. In his talk, Matt Kerr explained a very new approach how limits of higher normal functions can explain Apéry numbers of some Fano varieties. He obtains a uniform proof of the irrationality of $$\zeta(3), \zeta(2)$$ and $$\zeta(1+k^{-n})$$ for positive integers $$k$$.

4. The six lectures on Wednesday afternoon and Thursday (May 1st and 2nd) were devoted the recent progresses on the relations among periods of motives, algebraic cycles and special values of $$L$$-series, as predicted by the conjectures of Tate, Deligne, Birch and Swinnerton–Dyer and Beilinson–Block–Kato, by the arithmetic Gan–Gross–Prasad conjecture and by the arithmetic Siegel–Weil formula.

The talk by Jan Bruinier was about his joint work with Tonghai Yang on some local arithmetic Siegel–Weil formula in Kudla’s program for zero cycles in both archimedean and non-archimedean cases. (The previous work on Kudla’s program was mostly focused on codimension one cycles.)

The talk by Jie Lin was about her joint work with Grobner and Harris on some factorization formula for automorphic periods that is compatible with similar formula for motivic periods predicted by conjectures of Deligne and Tate. They proved the formula by using some recent advance on the Gan–Gross–Prasad conjecture by Wei Zhang et al.

The talks by Yifeng Liu and Yichao Liu were about their joint work with Liang Xiao, Wei Zhang and Xinwen Zhu on the Beilinson-Bloch-Kato conjecture in the rank zero case for Rankin–Selberg motives attached to $$GL(n)\times GL(n+1)$$. They proved many cases of the conjecture by using some recent work on Tate’s conjecture on special fibers of Shimura varieties and the Gan–Gross–Prasad conjecture by some subsets of authors.

The talk by Michael Rapoport was devoted to his joint work with B. Smithling and W. Zhang on some local arithmetic Gan–Gross–Prasad conjecture on some Shimura varieties with smooth integral models. They reduced the conjecture to the arithmetic fundamental lemma by Wei Zhang.

The talk by Ye Tian was about his joint work with Ashay Burungdale on the converse theorem to the Gross–Zagier and Kolyvagin in CM cases. (In the non CM case, a similar results were already obtained by C. Skinner and W. Zhang.)

5. The talks on Friday, May 3, were devoted to periods of local systems over algebraic varieties defined over number fields.

Hélène Esnault gave on overview of her recent work with M. Groechenig establishing the Grothendieck $$p$$-curvature conjecture and Simpson’s integrality conjecture for cohomologically rigid local systems, that ultimately relies on the Langlands correspondance for function fields.

Transcendence theory à la Lindemann-Weierstrass-Siegel-Shidlowski and motives were brought together in the contribution of Peter Jossen. Here exponential motives were introduced and discussed. In some cases periods of such motives can be expressed as special values of $$E$$-functions.

Exponential motives also played a role in the talk of Javier Frèsan. He constructed a Klostermann motive as an object in the category of exponential motives and then showed that it is actually in the category of motives. A key role is played by the Klostermann connection. Among the periods one finds the modified Bessel function of the first kind.

In the lecture about rational points on a smooth variety $$X$$ over a number field $$K$$, Minhyong Kim explained how one can use non-abelian class field theory with coefficients in the nilpotent completion of $$X$$ to derive defining equations for $$X(K)\subset X(\mathbb{A}_K)$$. As an example he explained how one could apply such methods to modular curves of level 13, a distinguished case for the levels.

• Agenda

#### Monday

 10:00 - 11:00 AM Joseph Ayoub | Proof of the Conservativity Conjecture 11:30 - 12:30 PM Fabrizio Andreatta | On Periods of 1-Motives and Applications 5:00 - 6:00 PM Yves André | On the Betti Coordinates of Sections of Abelian Schemes 6:00 -7:00 PM Philipp Habegger | Variation of the Néron-Tate Height in a Family of Abelian Varieties and the Bogomolov Conjecture for Function Fields

#### Tuesday

 10:00 - 11:00 AM Tomohide Terasoma | Motivic Bar Comodules Associated to Polylogarithms and Aomoto Polylogarithms 11:30 - 12:30 PM Chieh-Yu Chang | Logarithmic Interpretation of Multiple Zeta Values in Positive Characteristic 5:00 - 6:00 PM Alexander Goncharov | Cluster Polylogarithms and Zagier's Conjecture on Zeta(F,4) 6:00 -7:00 PM Matt Kerr | Apéry Extensions

#### Wednesday

 5:00 - 6:00 PM Yifeng Liu | Beilinson-Bloch-Kato Conjecture for Rankin-Selberg Motives 6:00 -7:00 PM Yichao Tian | Beilinson-Bloch-Kato Conjecture for Rankin-Selberg Motives (Case of GL(2)*GL(3))

#### Thursday

 10:00 - 11:00 AM Michael Rapoport | The Arithmetic Intersection Conjecture 11:30 - 12:30 PM Jie Lin | L-functions and Periods of Automorphic Motives 5:00 - 6:00 PM Jan Bruinier | Arithmetic Degrees of Special Cycles and Derivatives of Siegel Eisenstein Series 6:00 -7:00 PM Ye Tian | A Converse to a Theorem of Gross-Zagier and Kolyvagin: CM Case

#### Friday

 10:00 - 11:00 AM Hélène Esnault | p-curvature of Connections, Geometricity and Integrality of Local Systems 11:30 - 12:30 PM Peter Jossen | Special Values of E-functions as Exponential Periods 5:00 - 6:00 PM Javier Fresan | Hodge Theory of Kloosterman Connections 6:00 -7:00 PM Minhyong Kim | Non-Abelian Reciprocity, Period Maps, and Diophantine Geometry
• Abstracts and Lecture Reports

#### Yves Andre

Institut de Math. de Jussieu

##### On the Betti Coordinates of Sections of Abelian Schemes

A point of a complex abelian variety can be located by its (Betti) coordinates w.r.t. its period lattice. In a family, the Betti coordinates of a section define multivalued real-analytic functions on the base. We shall report on its differential-theoretic properties, which play an important role in an unlikely intersection of problems. (Joint with P. Corvaja, U. Zannier, and partly Z. Gao.)

#### Fabrizio Andreatta

Universita Statale di Milano

##### On Periods of 1-motives and Applications

We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers, we formulate a period conjecture, generalizing Grothendieck period conjecture, by saying that this period regulator is surjective. By proving that a suitable Betti–de Rham realization of 1-motives is fully faithful we can verify the period conjecture in several cases.

#### Joseph Ayoub

Universität Zürich

##### Proof of the Conservativity Conjecture

We will overview the proof that the classical realizations are conservative on the category of Voevodsky motives in characteristic zero. The proof relies on a new model of the $$T$$-spectrum representing algebraic de Rham cohomology and is achieved by analyzing the homotopy limit of the Cech cosimplicial motive associated to the latter $$T$$-spectrum.

#### Jan Bruinier

##### Arithmetic Degrees of Special Cycles and Derivatives of Siegel Eisenstein Series

Let $$V$$ be a rational quadratic space of signature $$(m,2)$$. A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with $$\S O(V)$$ to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus $$m+1$$. We report on recent joint work with T. Yang proving this conjecture for the coefficients of non-singular index $$T$$ in a large class of cases. To this end we establish some new local arithmetic Siegel-Weil formulas at the Archimedean and non-Archimedean places.

#### Chang Chieh-Yu

National Tsing Hua University

##### Logarithmic Interpretation of Multiple Zeta Values in Positive Characteristic

In the classical theory, there is a conjecture asserting that p-adic multiple zeta values (MZV’s) satisfy the same linear relations that the corresponding real-valued MZV’s satisfy. The main result in this talk is to verify the analogue of the conjecture in the function field setting (joint work with Y. Mishiba). The key strategy of the proof is to relate the MZV in question to certain coordinate of the logarithm of certain t-module at a rational point, and then apply Yu’s sub-t-module theorem that plays the function field analogue of Wuestholz’s analytic subgroup theorem.

#### Hélène Esnault

FU Berlin

##### p-curvature of Connections, Geometricity and Integrality of Local Systems

We’ll survey the general conjectures on how to recognize the complex connections which are geometric (p-curvature conjecture, Simpson’s conjecture), and discuss some recent progresses on some aspects of them (based on recent joint works with Tomoyuki Abe, Marc Kisin and Michael Groechenig).

#### Javier Fresán

École Polytechnique, Palaiseau

##### Hodge Theory of Kloosterman Connections

Recently, Broadhurst and Roberts studied the L-functions associated with symmetric powers of Kloosterman sums and conjectured a functional equation after extensive numerical computations. By the work of Yun, these L-functions correspond to “usual” motives over Q which, in low degree, are known to be modular. For the purpose of computing the Hodge numbers or relating the L-functions to periods, it is however more convenient to change gears and work with exponential motives. Fresán will construct the relevant motives and show how the irregular Hodge filtration allows one to explain the gamma factors at infinity in the functional equation. Based on work in progress with Claude Sabbah and Jeng-Daw Yu.

#### Alexander Goncharov

Yale University

##### Cluster Polylogarithms and Zagier’s Conjecture on Zeta(F,4)

We prove Zagier’s conjecture on the special value of the Dedekind zeta function at s=4, which tells that zeta(F,4) can be expressed via classical tetralogarithms. We also give a substantial evidence for the Freeness Conjecture in the weight 4, which describes periods of mixed Tate motives. Our main tools are motivic correlators and cluster variety structure of the moduli space $$M_{0,n}$$. This is a joint work with Daniil Rudenko (UChicago).

#### Philipp Habegger

University of Basel

##### Variation of the Néron-Tate Height in a Family of Abelian Varieties and the Bogomolov Conjecture for Function Fields

The Néron-Tate height attached to a polarized abelian variety defined over a number field can detect points of finite order. The distribution of points of small height with respect to the Zariski topology is governed by Bogomolov’s Conjecture, proved by Ullmo and Zhang. At the center of this talk is the variation of the Néron-Tate height in a family of abelian varieties when the base is a curve defined over a number field. I will explain how the monodromy action on the period lattice can be used to recover a relation between the Néron-Tate height and the Weil height on a subvariety of the family. This result extends aspects of Silverman’s Specialization Theorem to higher dimension. As an application of our work we obtain new cases of the Bogomolov Conjecture in the generic fiber of the family. This is joint work with Ziyang Gao.

#### Peter Jossen

ETH Zürich

##### Special values of E-functions as exponential periods

We give a few concrete examples of exponential motives M such that at least some of the periods of M are special values of E-functions. From the Siegel-Shidlovskii theorem we obtain thus a lower bound on the transcendence degree of the period algebra of M, and we can compare this lower bound with what is predicted by the period conjecture.

#### Matt Kerr

Washington University

##### Apéry Extensions

The title refers to certain extensions in the category of admissible variations of mixed Hodge structure over a multiply-punctured Riemann sphere, which are canonically associated to a Landau-Ginzburg model. We will define these extensions, describe how to extract well-defined “Apéry numbers” from them, discuss some examples, and finally explain the relation to Apéry’s irrationality proofs (and our preprint arXiv:1708.03836).

#### Minhyong Kim

Oxford University

##### Non-Abelian Reciprocity, Period Maps, and Diophantine Geometry

This lecture will survey recent developments in the use of fundamental groups, non-abelian cohomology, and non-abelian period isomorphisms to the construction of non-abelian reciprocity maps and their applications to Diophantine geometry.

#### Jie Lin

IHES

##### L-functions and Periods of Automorphic Motives

A conjecture of Deligne predicts a relation between motivic L-functions and geometric periods. In this talk, we will explain an approach towards this conjecture for automorphic motives. This is a joint work with Harald Grobner and Michael Harris.

#### Yifeng Liu

Northwestern University

##### Beilinson-Bloch-Kato Conjecture for Rankin-Selberg Motives

This is the first part of joint report by Yichao and Liu on their ongoing collaboration with Liang Xiao, Wei Zhang, and Xinwen Zhu. Their project concerns the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin-Selberg motives for GL(n) x GL(n+1). In his talk, Liu will introduce our project in the general aspect, and explain results, some ideas and examples. Liu will mainly focus on the case where $$n>2$$, and leave the case $$n=2$$ with more details to Yichao in which we obtain better and finer results.

#### Michael Rapoport

Mathematiches Institut Universitat Bonn

##### The Arithmetic Intersection Conjecture

The Gan-Gross-Prasad conjecture relates the non-vanishing of a special value of the derivative of an L- function to the non-triviality of a certain functional on the Chow group of a Shimura variety. Beyond the one-dimensional case, there is little hope for proving this conjecture. I will explain a variant of this conjecture (suggested by Wei Zhang) which seems more accessible and report on progress on it. This is joint work with B. Smithling and W. Zhang.

#### Tomohide Terasoma

University of Tokyo

##### Motivic Bar Comodules Associated to Polylogarithms and Aomoto Polylogarithms

In a paper by Bloch-Kriz, they defined the category of motives by that of comodules over a Hopf algebra, which is called the Bloch-Kriz Hopf algebra. In this talk, we will give a homotopical method to construct the comodules over the Bloch-Kriz Hopf algebra associated to the polylogarithms which yeilds the algebraic cycles used by Bloch-Totaro. We apply the same methode to obtain the explicit description of the comodulees over the Bloch-Kriz Hopf algebra associated to Aomoto polylogarithm after Beinson-Goncharov-Schehchtman-Varchenko.

#### Ye Tian

Academy of Mathematics & Systems Science

##### A Converse to a Theorem of Gross-Zagier and Kolyvagin: CM Case

For a CM elliptic curve over rationals and a prime p, under certain condition we show that if the co-rank of p-Selmer of the elliptic curve is one than the vanishing order of its L-function at center is also one. This is a joint work with Ashay Burungale.

#### Yichao Tian

Morningside Center for Mathematics