## Activities

1. YMSC Courses Mini Courses Seminars Lectures Conferences

## BIMSA-YMSC Tsinghua Number Theory Seminar

Organizer：Hansheng Diao, Yueke Hu, Emmanuel Lecouturier, Cezar Lupu
Time：Weekly/Biweekly
Venue：Zoom ID: 361 038 6975 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.

### Upcoming Talks:

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.

### Past Talks:

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 $\Q(N^{1/p})$

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.
Join Zoom Meeting
https://zoom.us/j/91653446007?pwd=QUFEUTZramJNeGpBdjVSWUV6cmpBZz09
Meeting ID: 916 5344 6007

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)
ZOOM https://zoom.us/j/91653446007?pwd=QUFEUTZramJNeGpBdjVSWUV6cmpBZz09
Meeting ID: 916 5344 6007