Schedule
 Jan. 20  Jan. 21  Jan. 22  Jan. 23  Jan. 24 
8:3010:00  Jing Yu  Ngaiming Mok  Philipp Habegger  Ye Tian  Pietro Corvaja 
10:2011:50  Rafael Kanel  Xu Shen  Ziyang Gao  Xinyi Yuan  Gisbert Wüstholz 
14:0015:30  Yifeng Liu  Yongquan Hu 
 Annette HuberKlawitter 

16:0017:30  Daniel Kriz  Chuangxun Cheng  Peter Jossen 
 Name  Institution  Title  Abstract 
1  Chuangxun Cheng  Nanjing University  Bad reductions of Shimura curves  In this talk, I will explain the structure of reductions of Shimura curves with Gamma_0 level structure and construct two exact sequences of Galois modules. Then I will explain some applications of these two sequences related to the level part of Serre’s modularity conjecture and the multiplicity part of the BuzzardDiamondJarvis conjecture. 
2  Pietro Corvaja  Università degli Studi di Udine  On the Betti map for sections of abelian schemes  Given an abelian scheme with a section, expressing its abelian logarithm in terms of periods gives rise to the socalled Betti map, a real analytic map on the universal cover of the base. The investigation of the Betti map is linked with classical issues like Manin's theorem of the kernel and 'Poncelet games'. In a recent work with Y. André and U. Zannier (partially with Z. Gao) we investigated the rank of the Betti map and its relations with the KodairaSpencer map. 
3  Ziyang Gao  Princeton University  Application of mixed AxSchanuel to bounding the number of rational points on curves  With Philipp Habegger we recently proved a height inequality, using which one can bound the number of rational points on 1parameter families of curves in terms of the genus, the degree of the number field and the MordellWeil rank (but no dependence on the Faltings height). In this talk I will give a blueprint to generalize this method to arbitrary curves. In particular I will focus on: (1) how establishing a criterion for the Betti map to be immersive leads to the desired bound; (2) how to apply mixed AxSchanuel to establish such a criterion. This is work in progress, partly joint with Vesselin Dimitrov and Philipp Habegger. 
4  Philipp Habegger  Universität Basel  No Singular Modulus is a Unit  A singular modulus is the jinvariant of an elliptic curve with complex multiplication. It is a classical fact that singular moduli are algebraic numbers and even algebraic integers. Bilu, Masser, and Zannier asked whether there exists a singular modulus that is an algebraic unit. In earlier work I was able to show that at most finitely many singular moduli are units. But this proof was ineffective due to trouble coming from a possible Siegel zero. In joint work with Bilu and Kühne, we give a new proof and show that no singular modulus is a unit. 
5  Yongquan Hu  MCM  Mod p cohomology of Shimura curves  At present, the mod $p$ (and padic) local Langlands correspondence is only well understood for the group $\mathrm{GL}_2(\mathbb{Q}_p)$, but remains mysterious even for $\mathrm{GL}_2$ of an unramified extension of $\mathbb{Q}_p$. However, the BuzzardDiamondJarvis conjecture and the mod $p$ localglobal compatibility for $\mathrm{GL}_2/\mathbb{Q}$ suggest that this hypothetical correspondence may be realized in the cohomology of Shimura curves with characteristic $p$ coefficients (cut out by some modular residual global representation). In the talk, I will report some results on the mod $p$ cohomology of Shimura curves from the point of view of the mod $p$ Langlands program. This is a joint work (in progress) with Haoran Wang. 
6  Annette HuberKlawitter  AlbertLudwigsUniversität Freiburg  A generalisation of Baker's theorem  We report on joint work with Gisbert Wuestholz. Logarithms of algebraic numbers are known to be transcendental. Baker gave a description of the dimension of the $\bar{\mathbf{Q}$vector space spanned by the logarithms of the algebraic numbers $\alpha_1,\dots,\alpha_n$: it is the rank of the abelian group generated by the $\alpha_i$ in $\mathbf{C}$. We generalise this to arbitrary $1$dimensional periods, i.e., complex numbers obtained by integrating algebraic $1$forms on an algebraic curve defined over $\bar{\mathbf{Q}}$ over paths with endpoints in algebraic points. The proof relies on a reinterpretation of the periods as periods of Deligne $1$motives and a generalisation of Wuestholz's analytic subgroup theory to $1$motives. 
7  Peter Jossen  ETH Zürich  Exponential Periods and Exponential Motives  I begin by explaining Nori's formalism and how to use it to construct abelian categories of motives. Then, following ideas of Katz and Kontsevich, I show how to construct a tannakian category of "exponential motives" by applying Nori's formalism to rapid decay cohomology, which one thinks of as the Betti realisation. This category of exponential motives contains the classical mixed motives à la Nori. We then introduce the de Rham realisation, as well as a comparison isomorphism with the Betti realisation. When k = IQ, this comparison isomorphism yields a class of complex numbers, "exponential periods", which includes special values of the gamma and the Bessel functions, the EulerMascheroni constant, and other interesting numbers which are not expected to be periods of classical motives. In particular, we attach to exponential motives a Galois group which conjecturally governs all algebraic relations among their periods. 
8  Rafael von Kanel  Tsinghua University  On the representability of moduli problems on Hilbert moduli stacks  It is often fundamental for the study of a moduli space to know whether the underlying moduli problem is representable. In the first part of this talk we discuss explicit representability criteria for moduli problems on the Hilbert moduli stacks of Rapoport and DelignePappas. Our criteria also apply over the bad primes and they are optimal in many situations of interest. In the second part, we consider applications of our criteria to the study of integral points on Hilbert modular varieties and we explain how the work of MasserWustholz is used in the proofs. This is joint work with Arno Kret. 
9  Daniel Kriz  Massachusetts Institute of Technology  A new padic MaassShimura operator and supersingular RankinSelberg padic Lfunctions  We introduce a new padic Maass–Shimura operator acting on a space of “generalized padic modular forms” (extending Katz’s notion of padic modular forms), defined on the padic (preperfectoid) universal cover of a Shimura curve. Using this operator, we construct new padic Lfunctions in the style of Katz, Bertolini–Darmon–Prasanna and Liu–Zhang–Zhang for Rankin–Selberg families over imaginary quadratic fields K, in the ”supersingular” case where p is inert or ramified in K. We also establish new padic Waldspurger formulas, relating padic logarithms of elliptic units and Heegner points to special values of these padic Lfunctions. If time permits, we will discuss some applications to the arithmetic of abelian varieties. 
10  Yifeng Liu  Northwestern University  Mixed arithmetic theta lifting  In this talk, we will extend the construction of arithmetic generating functions and arithmetic theta liftings to unitary groups of odd ranks (on the source). We also formulate the conjectural arithmetic inner product formula (for central Lderivative) for unitary groups of odd ranks. 
11  Ngaimin Mok  Hong Kong University  Universal Covering Maps from Bounded Symmetric Domains to Their FiniteVolume Quotients  By the Uniformization Theorem a compact Riemann surface other than the Riemann Sphere or an elliptic curve is uniformized by the unit disk and equivalently by the upper half plane. The upper half plane is also the universal covering space of the moduli space of elliptic curves equipped with a suitable level structure. In higher dimensions the Siegel upper half plane (which is biholomorphic to a bounded symmetric domain) is an analogue of the upper half plane, and it is the universal covering space of moduli spaces of polarized Abelian varieties with appropriate level structures. In general, finitevolume quotients of bounded symmetric domains, which are naturally quasiprojective varieties, are of immense interest to Several Complex Variables, Algebraic Geometry and Number Theory, and an important object of study is the universal covering map $\pi_\Gamma: \Omega \to X_\Gamma$ from a bounded symmetric domain $\Omega$ onto its quotient $X_\Gamma := \Omega/\Gamma$ by a torsionfree discrete lattice $\Gamma \subset {\rm Aut}(\Omega)$. We will explain an approach from the perspectives of Complex Differential Geometry and Several Complex Variables to the study of the universal covering map revolving around the notion of asymptotic curvature behavior, rescaling arguments and the use of meromorphic foliations, and illustrate how this approach using transcendental techniques leads to solutions of various problems from Functional Transcendence Theory concerning totally geodesic subvarieties of finitevolume quotients without the assumption of arithmeticity. 
12  Xu Shen  MCM  padic periods and the FarguesRapoport conjecture  In his 1970 ICM report, Grothendieck asked the question to describe the padic analogues of Griffiths period domains. In this talk, we will review some constructions for these padic period domains, following recent developments in padic Hodge theory. We will then explain some ideas in a proof of the FarguesRapoport conjecture about the structure of certain padic period domains. This is joint work with Miaofen Chen and Laurent Fargues. 
13  Ye Tian  MCM  PerrinRiou conjecture and padic periods  We give a proof of PerrinRiou conjecture, which relates padic logarithm of BeilinsonKato elements to Lfunctions, and we will also discuss its application to BSD formula. This is joint work with Burungale and Skinner. 
14  Gisbert Wüstholz  ETH Zurich  Geodesic Billiards 

15  Jing Yu  National Taiwan University  A Garden in Positive Characteristic  I will give survey talk on progress of characteristic p>0 transcendence theory, in particular in the direction of algebraic independence. The impact of the seminal paper by Wuestholz in 1980's opens a door for us, so that this part of arithmetic geometry of positive characteristic can go far further than the classical theory. I will focus on: Drinfeld modules, Multiple zeta values, and Gamma values. 
16  Xinyi Yuan  UC Berkeley  Weak Lefschetz theorems for Brauer groups  In this talk, we introduce some Lefschetztype theorems for Brauer groups of hyperplane sections of smooth projective varieties. This is more or less known when the dimension of the hyperplane section is at least 3, but we will also introduce a version which lowers the dimension from 3 to 2. As a consequence, we reduce the Tate conjecture for divisors on smooth projective varieties from general dimensions to dimension 2, and thus proves a results of Morrow by a different method. 