BIMSA-YMSC Tsinghua Number Theory Seminar

Speaker:Qiao He (University of Wisconsin-Madison)
Organizer:Hansheng Diao, Yueke Hu, Emmanuel Lecouturier, Cezar Lupu
Time:Weekly/Biweekly
Venue:Zoom ID: 293 812 9202 Passcode: BIMSA

Note:Location & Time can change depending on the speaker's availability.


This is a research seminar on topics related to number theory and its applications which broadly can include related areas of interests such as analytic and algebraic number theory, algebra, combinatorics, algebraic and arithmetic geometry, cryptography, representation theory etc. The speakers are also encouraged to make their talk more accessible for graduate level students.

For more information,please refer to: http://www.bimsa.cn/newsinfo/647938.html.


Upcoming Talks:


Date                Speaker                     Title

Jan 17, 2023    Vesselin Dimitrov      TBA    

Jan 10, 2023    Steve Gonek              Finite Euler products and the Riemann Hypothesis    

Dec 13, 2022    Paul Nelson              Bounds for standard L-functions    

Dec 06, 2022    Dongsheng Wu        TBA    




Title: A proof of Kudla-Rapoport conjecture for Kramer models at ramified primes

Speaker: Qiao He (University of Wisconsin-Madison)

Time: Tues.,10:30-11:30am Beijing time, Nov 29, 2022

Venue: BIMSA 1131

Zoom ID: 293 812 9202  Passcode: BIMSA


Abstract:

In this talk, I will first talk about the Kudla-Rapoport conjecture, which suggests a precise identity between arithmetic intersection numbers of special cycles on Rapoport-Zink space and  derived local densities of hermitian forms. Then I will discuss how to modify the original conjecture over ramified primes and how to prove the modified conjecture. On the geometric side, we completely avoid explicit calculation of intersection number and the use of Tate’s conjecture.  On the analytic side, the key input is a surprisingly simple formula for derived primitive local density. This talk is based on joint work with Chao Li, Yousheng Shi and Tonghai Yang.




Past Talks:

Title: Generalized Paley Graphs, Finite Field Hypergeometric Functions and Modular Forms

Speaker: Dermot McCarthy (Texas Tech University)

Time: 10:30-11:30 Beijing time, Nov 22, 2022

Venue: BIMSA 1118

Zoom ID: 293 812 9202  Passcode: BIMSA


Abstract:

In 1955, Greenwood and Gleason proved that the two-color diagonal Ramsey number $R(4,4)$ equals 18. Key to their proof was constructing a self-complementary graph of order 17 which does not contain a complete subgraph of order four. This graph is one in the family of graphs now known as Paley graphs. In the 1980s, Evans, Pulham and Sheehan provided a simple closed formula for the number of complete subgraphs of order four of Paley graphs of prime order.
Since then, \emph{generalized Paley graphs} have been introduced. In this talk, we will discuss our recent work on extending the result of Evans, Pulham and Sheahan to generalized Paley graphs, using finite field hypergeometric functions. We also examine connections between our results and both multicolor diagonal Ramsey numbers and Fourier coefficients of modular forms.
This is joint work with Madeline Locus Dawsey (UT Tyler) and Mason Springfield (Texas Tech University).



Title: Quantitative weak approximation of rational points on quadrics

Speaker: Zhizhong Huang (AMSS)

Time: 16:00-17:00 Beijing time, Nov 15, 2022

Venue: W11, Ningzhai, Tsinghua University

Zoom ID: 293 812 9202  Passcode: BIMSA


Abstract:

The classical Hasse—Minkowski theorem states that rational points on quadrics (if non-empty) satisfy weak approximation. We explain how Heath-Brown’s delta circle method allows to obtain a quantitive and effective version of this theorem, namely counting rational points of bounded height on quadrics satisfying prescribed local conditions with optimal error terms. We then discuss applications in intrinsic Diophantine approximation on quadrics. This is based on joint work in progress with M. Kaesberg, D. Schindler, A. Shut.




Title: Equidistribution in Stochastic Dynamical Systems

Speaker: Bella Tobin (Oregon State University)

Time: 10:30-11:30 Beijing time, Nov 08, 2022

Venue: BIMSA 1118

Zoom ID: 293 812 9202  Passcode: BIMSA


Abstract:

In arithmetic dynamics, one typically studies the behavior and arithmetic properties of a rational map under iteration. Instead of iterating a single rational map, we will consider a countable family of rational maps, iterated according to some probability measure. We call such a system a stochastic dynamical system. As such a family can be infinite and may not be defined over a single number field, we introduce the concept of a generalized adelic measure, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rational numbers using the framework established by Allcock and Vaaler. This turns heights from sums into integrals. We prove an equidistribution result for generalized adelic measures, and in turn prove an equidistribution theorem for random backwards orbits for stochastic dynamical systems. This talk will include some background in arithmetic dynamics and will be suitable for graduate students.




Title: Slopes of modular form and ghost conjecture

Speaker: Bin Zhao (Capital Normal University)

Time: 16:00-17:00 Beijing time, Nov 1, 2022

Venue: BIMSA 1118

Zoom ID: 293 812 9202 Passcode: BIMSA


Abstract:

In 2016, Bergdall and Pollack raised a conjecture towards the computation of the p-adic slopes of Hecke cuspidal eigenforms whose associated p-adic Galois representations satisfy the assumption that their mod p reductions become reducible when restricted to the p-decomposition group. In this talk, I will report the joint work with Ruochuan Liu, Nha Truong and Liang Xiao to prove this conjecture under mild assumptions. I will start with the statement of this conjecture and the intuition behind it. Then I will explain some strategies of our proof. If time permits, I will mention some arithmetic applications of this conjecture.




Title: On $G$-isoshtukas over function fields.

Speaker: Wansu Kim

Time:  15:00-16:00 Beijing time, Oct 25, 2022

Zoom ID: 293 812 9202 Passcode: BIMSA


Abstract:

Let $F$ be a global function field, and let $G$ be a connected reductive group over $F$. In this talk, we will introduce the notion of $G$-isoshtukas, and discuss a classification result analogous to Kottwitz' classification of local and global $B(G)$. If $G=\GL_n$ then $\GL_n$-isoshtukas are nothing but $\varphi$-spaces of rank $n$ (which naturally arise as an isogeny class of rank-$n$ Drinfeld shtukas), and our classification result for $\GL_n$-isoshtukas can be read off from Drinfeld’s classification of $\varphi$-spaces. This is a joint work with Paul Hamacher.




Title: Counting polynomials with a prescribed Galois group

Speaker: Vlad Matei (Simion Stoilow Institute of Mathematics of the Romanian Academy)

Time: 15:30-16:30 Beijing time, Oct 18, 2022 (updated)

Zoom ID: 293 812 9202 Passcode: BIMSA


Abstract:

An old problem, dating back to Van der Waerden, asks about counting irreducible polynomials degree $n$ polynomials with coefficients in the box $[-H,H]$ and prescribed Galois group. Van der Waerden was the first to show that $H^n+O(H^{n-\delta})$ have Galois group $S_n$ and he conjectured that the error term can be improved to $o(H^{n-1})$.

Recently, Bhargava almost proved van der Waerden conjecture showing that there are $O(H^{n-1+\varepsilon})$ non $S_n$ extensions, while Chow and Dietmann showed that there are $O(H^{n-1.017})$ non $S_n$, non $A_n$ extensions for $n\geq 3$ and $n\neq 7,8,10$.

In joint work with Lior Bary-Soroker, and Or Ben-Porath we use a result of Hilbert to prove a lower bound for the case of $G=A_n$, and upper and lower bounds for $C_2$ wreath $S_{n/2}$ . The proof  for $A_n$ can be viewed, on the geometric side,  as constructing a morphism $\varphi$ from $A^{n/2}$ into the variety $z^2=\Delta(f)$ where each $varphi_i$ is a quadratic form.  For the upper bound for $C_2$ wreath $S_{n/2}$ we improve on the monic version of Widmer's result on counting polynomials with an imprimitive Galois group. We also pose some open problems/conjectures.




Title: Multizeta for function fields

Speaker: Dinesh Thakur (Universty of Rochester)

Time:  20:00-21:00 Beijing time, Oct 11, 2022

Zoom ID: 293 812 9202 Passcode: BIMSA


Abstract:

We will discuss multizeta values for the function field case, explain various analogies and contrasts with the rational number field case, and discuss recent developments and open questions.




Title: The plectic conjecture over local fields

Speaker: Siyan Daniel Li-Huerta

Time:  10:00-11:00 Beijing time, Sep 27, 2022

Zoom ID: 293 812 9202 Passcode: BIMSA

Room: BIMSA 1118

Affiliation: Harvard University

Host: Hansheng Diao

Abstract:

The étale cohomology of varieties over Q enjoys a Galois action. For Hilbert modular varieties, Nekovář-Scholl observed that this Galois action on the level of cohomology extends to a much larger profinite group: the plectic group. Motivated by applications to higher-rank Euler systems, they conjectured that this extension holds even on the level of complexes, as well as for more general Shimura varieties.

We present a proof of the analog of this conjecture for local Shimura varieties. Consequently, we obtain results for the basic locus of global Shimura varieties, after restricting to a decomposition group. The proof crucially uses a mixed-characteristic version of fusion due to Fargues–Scholze.




Title: The Tate conjecture over finite fields for varietes with $h^{2, 0}=1$

Speaker: Ziquan Yang

Time:  10:00-11:00 Beijing time, Sep 20, 2022

Zoom ID: 293 812 9202 Passcode: BIMSA

Room: BIMSA 1118

Abstract:

The past decade has witnessed a great advancement on the Tate conjecture for varietes with Hodge number $h^{2, 0}=1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2, 0}=1$ varietes in characteristic $0$.
In this talk, I will explain  that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2, 0}=1$ when $p>>0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p\geq 5$ the BSD conjecture holds true for height $1$ elliptic curve $\mathcal{E}$ over a function field of genus $1$, as long as $\mathcal{E}$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy  is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1, 1)$-theorem over $\mathbb{C}$ is very robust for $h^{2, 0}=1$ varietes, and works well beyond the hyperkahler world. This is a joint work with Paul Hamacher and Xiaolei Zhao.




Title: Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

Speaker: Li Lai (Tsinghua University)

Time: 16:00-17:00 Beijing time,Jul 12, 2022(updated)

Zoom ID: 361 038 6975 Passcode: BIMSA

Room: BIMSA 1110

Abstract:

In 2012, Zagier proved a formula which expresses the multiple zeta values
\[ H(a, b)=\zeta(\underbrace{2,2, \ldots, 2}_{a}, 3, \underbrace{2,2, \ldots, 2}_{b}) \]
as explicit $\mathbb{Q}$-linear combinations of products $\pi^{2m}\zeta(2n+1)$ with $2m+2n+1=2a+2b+3$. Recently, Murakami proved an odd variant of Zagier's formula for the multiple $t$-values
\[ T(a, b)=t(\underbrace{2,2, \ldots, 2}_{a}, 3, \underbrace{2,2, \ldots, 2}_{b}). \]

In this talk, we will give new and parallel proofs of these two formulas. Our proofs are elementary in the sense that they only involve the Taylor series of powers of arcsine function and certain trigonometric integrals. Thus, these formulas become more transparent from the view of analysis. This is a joint work with Cezar Lupu and Derek Orr.




Title: Spectrum of p-adic differential equations

Speaker: Tinhinane Amina Azzouz (BIMSA)

Time: 16:00-17:00 Beijing time, Jun 14, 2022

Zoom ID: 361 038 6975 Passcode: BIMSA

Room: BIMSA 1110

Abstract: In the ultrametric setting, linear differential equations present phenomena that do not appear over the complex field. Indeed, the solutions of such equations may fail to converge everywhere, even without the presence of poles. This leads to a non-trivial notion of the radius of convergence, and its knowledge permits us to obtain several interesting information about the equation. Notably, it controls the finite dimensionality of the de Rham cohomology. In practice, the radius of convergence is really hard to compute and it represents one of the most complicated features in the theory of p-adic differential equations. The radius of convergence can be expressed as the spectral norm of a specific operator and a natural notion, that refines it, is the entire spectrum of that operator, in the sense of Berkovich.
In our previous works, we introduce this invariant and compute the spectrum of differential equations over a power series field and in the p-adic case with constant coefficients.
In this talk we will discuss our last results about the shape of this spectrum for any linear differential equation, the strong link between the spectrum and all the radii of convergence, notably a decomposition theorem provided by the spectrum.




Title: Reciprocity, non-vanishing, and subconvexity of central L-values

Speaker: Subhajit Jana (MPIM)

Time: 13:30-15:00 Beijing time, May 26, 2022

Zoom ID: 844 745 8596 Passcode: 568789

Abstract: A reciprocity formula usually relates certain moments of two different families of L-functions which apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central L-values for GL(2). In this talk, we describe some instances of reciprocity formulas both in low and high rank groups and give certain applications to subconvexity and non-vanishing of central L-values. These are joint works with Nunes and Blomer--Nelson.




Title:Duals of linearized Reed-Solomon codes

Speaker: Xavier Caruso (CNRS, Université de Bordeaux)

Organiser:Emmanuel Lecouturier (BIMSA)

Time: 16:00-17:00 Friday, 2022/1/7

Zoom: 638 227 8222 PW: BIMSA

Abstract:

Errors correcting codes are a basic primitive which provides robust tools against noise in transmission. On the theoretical perspective, they are usually founded on beautiful properties of some mathematical objects. For example, one of the oldest construction of codes is due to Reed and Solomon and takes advantage of the fact the number of roots of a polynomial cannot exceed its degree. During the last decades, new problems in coding theory have emerged (e.g. secure network transmission or distributive storage) and new families of codes have been proposed. In this perspective, Martínez-Peñas has recently introduced a linearized version of Reed-Solomon codes which, roughly speaking, is obtained by replacing classical polynomials by a noncommutative version of them called Ore polynomials.

In this talk, I will revisit Martínez-Peñas' construction and give a new description of the duals of linearized Reed-Solomon codes. This will lead us to explore the fascinating world of noncommutative polynomials and notably develop a theory of residues for rational differential forms in this context.




Title:Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

Speaker: Lazar Radicevic (Maxplanck Institute, Bonn)

Organiser:Emmanuel Lecouturier (BIMSA)

Time: 16:00-17:00 Wednesday, 2021/12/15

Venue: BIMSA 1118

Zoom: 3885289728 PW: BIMSA

Abstract:

We consider the Kolyvagin cohomology classes associated to an elliptic curve E defined over ℚ from a computational point of view. We explain how to go from a model of a class as an element of (E(L)/pE(L))^Gal(L/ℚ), where p is prime and L is a dihedral extension of ℚ of degree 2p, to a geometric model as a genus one curve embedded in ℙ^(p−1). We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of E(L). Finally, we compute explicit equations for several genus one curves that represent non-trivial elements of the p-torsion part of the Tate-Shafarevich group of E, for p≤11, and hence are counterexamples to the Hasse principle.




Title:A modular construction of unramified p-extensions of $\Q(N^{1/p})$

Speaker:Jacky Lang (Philadelfia)

Organizer: Emmanuel Lecouturier (BIMSA)

Time:9:00-10:00, Nov. 19, 2021

Venue:BIMSA 1118

Zoom ID: 849 963 1368   Password: YMSC

Abstract:

In Mazur's seminal work on the Eisenstein ideal, he showed that when N and p > 3 are primes, there is a weight 2 cusp form of level N congruent to the unique weight 2 Eisenstein series of level N if and only N = 1 mod p. Calegari--Emerton, Merel, Lecouturier, and Wake--Wang-Erickson have work that relates these cuspidal-Eisenstein congruences to the p-part of the class group of $\Q(N^{1/p})$. Calegari observed that when N = -1 mod p, one can use Galois cohomology and some ideas of Wake--Wang-Erickson to show that p divides the class group of $\Q(N^{1/p})$. He asked whether there is a way to directly construct the relevant degree p everywhere unramified extension of $\Q(N^{1/p})$ in this case. After discussing some of this background, I will report of work with Preston Wake in which we give a positive answer to this question using cuspidal-Eisenstein congruences at prime-square level.




Title:The unbounded denominators conjecture

Speaker:Yunqing Tang (Princeton university)

Organizer: Emmanuel Lecouturier (BIMSA)

Time:9:30-10:30, Oct. 29, 2021

Venue:BIMSA 1118

Zoom ID: 849 963 1368   Password: YMSC

Abstract:

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. (Joint work with Frank Calegari and Vesselin Dimitrov.)




Title:Eisenstein congruences and Euler systems

Speaker:Oscar Rivero Salgado (University of Warwick)

Organizer: Emmanuel Lecouturier (BIMSA)

Time:16:00-17:00, Oct. 22, 2021

Venue:BIMSA 1118

Zoom ID: 388 528 9728   Password: BIMSA

Abstract:

Let f be a cuspidal eigenform of weight two, and let p be a prime at which f is congruent to an Eisenstein series. Beilinson constructed a class arising from the cup-product of two Siegel units and proved a relationship with the first derivative of the L-series of f at the near central point s=0. I will motivate the study of congruences between modular forms at the level of cohomology classes, and will report on a joint work with Victor Rotger where we prove two congruence formulas relating the Beilinson class with the arithmetic of circular units. The proofs make use of Galois properties satisfied by various integral lattices and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson-Kato elements and, most crucially, the work of Fukaya-Kato around Sharifi’s conjectures.




Title: Modular regulator with Rogers-Zudilin method
Speaker: Weijia Wang (ENS Lyon)
Time: 2020-7-14, 16:00 – 17:00
Abstract: Let Y (N) be the modular curve of level N and E(N) be the universal elliptic curve over Y (N). Beilinson (1986) defined the Eisenstein symbol in the motivic cohomology of Ek(N) and the work of Deninger–Scholl (1989) shows the Petersson inner product of its regulator gives us special L-values. In this talk I will present how to relate the modular regulator with L-value of quasi-modular forms by using Lanphier’s formula and Rogers–Zudilin method.
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Title: Projective bundle theorem in MW-motives

Speaker: Nanjun Yang (YMSC, Tsinghua)
Time: 2020-7-2, 10:00 – 11:00
Abstract: We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that $\widetilde{CH}^*(\mathbb{P}(E))$ is determined by $\widetilde{CH}^*(X)$ and $\widetilde{CH}^*(X\times\mathbb{P}^2)$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with odd rank. If the rank of $E$ is even, the theorem is still true under a new kind of orientability, which we call it by projective orientability. As an application, we compute the MW-motives of blow-ups over smooth centers. (arXiv 2006.11774)
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Title: Elliptic cocycle for GLN(Z) and Hecke operators

Speaker: Hao Zhang (Sorbonne Université)
Time: 2020-7-2, 16:00 – 17:00
Abstract: A classical result of Eichler, Shimura and Manin asserts that the map that assigns to a cusp form f its period polynomial r_f is a Hecke equivariant map. We propose a generalization of this result to a setting where r_f  is replaced by a family of rational function of N variables equipped with the action of GLN(Z). For this purpose, we develop a theory of Hecke operators for the elliptic cocycle recently introduced by Charollois.  In particular, when f is an eigenform, the corresponding rational function is also an eigenvector respect to Hecke operator for GLN. Finally, we give some examples for Eisenstein series and the Ramanujan Delta function.
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