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Sep 3

Yifeng Liu
Beilinson-Bloch conjecture and arithmetic inner product formula

In this talk, we study the Chow group of the motive associated to a tempered global L-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification of $\pi$, if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $\pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This is a joint work with Chao Li.

Sep 10

Yufei Zhao
The joints problem for varieties

We generalize the Guth-Katz joints theorem from lines to varieties. A special case of our result says that N planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture).

Our main innovation is a new way to extend the polynomial method to higher dimensional objects. A simple, yet key step in many applications of the polynomial method is the "vanishing lemma": a single-variable degree-d polynomial has at most d zeros. In this talk, I will explain how we generalize the vanishing lemma to multivariable polynomials, for our application to the joints problem.

Joint work with Jonathan Tidor and Hung-Hsun Hans Yu (https://arxiv.org/abs/2008.01610)

Sep 17

Ziquan Yang
A Crystalline Torelli Theorem for Supersingular K3^[n]-type Varieties

In 1983, Ogus proved that a supersingular K3 surface in characteristic at least 5 is determined up to isomorphism by the Frobenius action and the Poincaré pairing on its second crystalline cohomology. This is an analogue of the classical Torelli theorem for K3's, due to Shapiro and Shafarevich, which says that a complex algebraic K3 surface is determined up to isomorphism by the Hodge structure and the Poinaré pairing on its second singular cohomology. I will explain how to re-interpret Ogus' theorem from a motivic point of view and generalize the stronger form of the theorem to a class of higher dimensional analogues of K3 surfaces, called K3^[n]-type varieties. This is also an analogue of Verbitsky's global Torelli theorem for general irreducible symplectic manifolds. A new feature in Verbitsky's theorem, which did not appear in the classical Torelli theorem for K3's, is the notion of "parallel transport operators". I will explain how to work with this notion in an arithmetic setting.

As an application, I will also present a similar crystalline Torelli theorem for supersingular cubic fourfolds, the Hodge theoretic counterpart of which is a theorem of Voisin.

Sep 24

Yousheng Shi
Kudla Rapoport conjecture over the ramified primes

In the nineties, Kudla formulated a conjecture relating central derivative of certain Eisenstein series to arithmetic intersection numbers of special cycles on Shimura varieties. Later Kudla and Rapoport formulated a local version of the conjecture which compares intersection numbers of special cycles on the unitary Rapoport Zink spaces over an inert prime of an imaginary quadratic field with derivatives of local density of hermitian forms. In this talk, I will review Kudla-Rapoport conjecture and its global motivation. Then I will talk about an attempt to formulate a Kudla-Rapoport type of conjecture over the ramified primes. In case of unitary Shimura curves, this new conjecture can be proved. This is a joint work with Qiao He and Tonghai Yang.

Oct 1

Liyang Yang
Average Central L-values on U(2,1)$\times$ U(1,1), Nonvanishing and Subconvexity

In this talk, we study an average of automorphic periods on $U(2,1)\times U(1,1).$ We also compute local factors in Ichino-Ikeda formulas for these periods to obtain an explicit asymptotic expression. Combining them together we would deduce some important properties of central $L$-values on $U(2,1)\times U(1,1)$ over certain family: the first moment, nonvanishing and subconvexity. This is joint work with Philippe Michel and Dinakar Ramakrishnan.

Oct 15

Yujie Xu
On normalization in the integral models of Shimura varieties of Hodge type

Shimura varieties are moduli spaces of abelian varieties (in characteristic zero) with extra structures. Interests in mod p points of Shimura varieties motivated the constructions of integral models of Shimura varieties by various mathematicians. In this talk, I will discuss some motivic aspects of integral models of Hodge type at hyperspecial level, constructed by Kisin. I will talk about recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models.

Oct 22

Artane Siad
Average 2-torsion in the class group of monogenic fields.

Let n greater than or equal to 3 be a fixed degree. In this talk, we prove an upper bound on the average size of the 2-torsion in the class groups of monogenised fields of degree $n$, and, conditional on a widely expected tail estimate, compute it exactly. For odd degree, we find that this average is different from the value predicted for the full family of fields by the Cohen-Lenstra-Martinet-Malle heuristic, generalising a result of Bhargava-Hanke-Shankar. For even degree at least 4, no heuristic is available about the distribution of the 2-part over the full family. In fact, it is the first time that p-torsion averages are computed for a "bad" prime in the sense of Cohen-Lenstra in degree at least 3. A corollary of our results is that in each fixed degree and signature, there are infinitely many monogenic S_n-number fields with odd class number and units of every signature.

Our proof exploits an orbit parametrisation due to Wood, clarifies the roles of genus theory in even degree, and reveals an interesting structure explaining the deviation of the odd monogenic averages from the values expected for the full family.

Oct 29

Guillermo Mantilla-Soler
A complete invariant for real S_n number fields.

In this talk we will review some of the most common invariants from algebraic number theory. We divide number fields in families, by Galois groups or signature, and study the trace form in each collection of fields. We will show how this division on number fields led us to the discovery that for real S_n number fields, with restricted ramification, the trace is a complete invariant.

Nov 5

Anup Dixit
On generalized Brauer-Siegel conjecture and Euler-Kronecker constants

For a number field K, let h_K denote its class number. It is an important theme in number theory to study how h_K varies over a family of number fields. In this context, the classical Brauer-Siegel theorem describes how the class number times the regulator varies over a family of Galois fields. An analogue of this statement for general families was conjectured by Tsfasman and Vladut in 2002. On another front, as a natural generalization of Euler's constant gamma, Ihara introduced the Euler-Kronecker constants attached to any number field. In this talk, we will discuss a connection between the generalized Brauer-Siegel conjecture and bounds on the Euler-Kronecker constants, thus proving the Brauer-Siegel conjecture in some special cases.

Nov 12

Si Ying Lee
Eichler-Shimura relations for Hodge type Shimura varieties

The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschebung. Blasius and Rogawski proposed a generalization of this result for general Shimura varieties with good reduction at $p$, and conjectured that the Frobenius satisfies a certain Hecke polynomial. I will talk about a recent proof of this conjecture for Shimura varieties of Hodge type, assuming a technical condition on the unramified sigma-conjugacy classes in the Kottwitz set.

Nov 19

Chao Li
On the Kudla-Rapoport conjecture
The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. It is a key local ingredient to establish the arithmetic Siegel-Weil formula and the arithmetic inner product formula. We will motivate this conjecture from the classical Hurwitz class number formula, explain a proof based on the uncertainty principle, and discuss global applications. This is joint work with Wei Zhang.

Dec 3

Aaron Pollack
Singular modular forms on quaternionic E_8
The exceptional group $E_{7,3}$ has a symmetric space with Hermitian tube structure. On it, Henry Kim wrote down low weight holomorphic modular forms that are "singular" in the sense that their Fourier expansion has many terms equal to zero. The symmetric space associated to the exceptional group $E_{8,4}$ does not have a Hermitian structure, but it has what might be the next best thing: a quaternionic structure and associated "modular forms". I will explain the construction of singular modular forms on $E_{8,4}$, and the proof that these special modular forms have rational Fourier expansions, in a precise sense. This builds off of work of Wee Teck Gan and uses key input from Gordan Savin.

Dec 10

Daxin Xu
Bessel F-isocrystals for reductive groups

I will first review relationship between the Bessel differential equation and the classical Kloosterman sums.

Recently, there are two generalizations of this story (corresponding to $\GL_2$-case) for reductive groups: one is due to Frenkel and Gross from the viewpoint of the Bessel differential equation; another one, due to Heinloth, Ng\^o and Yun, uses the geometric Langlands correspondence to produce $\ell$-adic sheaves.

I will report my joint work with Xinwen Zhu, where we study the p-adic aspect of this theory and unify previous two constructions.