## Jerry Yu Fu : A density theorem towards p-adic monodromy

- Number Theory ( 456 Views )We investigate the $p$-adic monodromy of certain kinds of abelian varieties in $\mathcal{A}_{g}$ and prove a formal density theorem for the locus of deformations with big monodromy. Also, we prove that the small monodromy locus of the deformation space of a supersingular elliptic curve is $p$-adic nowhere dense. The approach is based on a congruence condition of $p$-divisible groups and transform of data between the Rapoport-Zink spaces and deformation spaces.

## Dick Hain : Hecke actions on loops and periods of iterated itegrals of modular forms

- Number Theory ( 314 Views )Hecke operators act on many invariants associated to modular curves and their generalizations. For example, they act on modular forms and on cohomology groups of modular curves. In each of these cases, they generate a semi-simple, commutative algebra. In the first part of this talk, I will recall (in friendly, elementary, geometric terms) what Hecke operators are and how they act on the standard invariants. I will then show that they also act on loops in modular curves (aka, conjugacy classes in modular groups). In this case, the Hecke operators generate a non-commutative subalgebra of the vector space generated by the conjugacy classes, which leads to a very natural non-commutative generalization of the classical Hecke algebra. In the second part of the talk will discuss why one might want do construct such a Hecke action. As a prelude to this, I will explain why this Hecke action commutes with the natural action of the absolute Galois group after taking profinite completions. And, in the unlikely event that I have sufficient time, I will also explain how (after taking the appropriate completion) this Hecke action is also compatible with Hodge theory.

## Aleksander Horawa : Motivic action on coherent cohomology of Hilbert modular varieties

- Number Theory ( 229 Views )A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

## Damaris Schindler : Manins conjecture for certain smooth hypersurfaces in biprojective space

- Number Theory ( 210 Views )So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture on the number of rational points of bounded anticanonical height on Fano varieties. Work of B. Birch back in 1962 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In this talk we are interested in subvarieties of biprojective space. There is not much known so far, unless the underlying polynomials are of bidegree (1,1) or (1,2). In this talk we present recent work which combines the circle method with the generalised hyperbola method developed by V. Blomer and J. Bruedern. This allows us to verify Manin's conjecture for certain smooth hypersurfaces in biprojective space of general bidegree.

## Jiuya Wang : Inductive Method in Counting Number Fields

- Number Theory ( 202 Views )We propose general frameworks to inductively count number fields building on previously known counting results and good uniformity estimates in different flavors. By this method, we prove new results in counting number fields with Galois groups ranging from direct product to wreath product. We will also mention interesting applications en route. This involves my thesis and on going project with Melanie Matchett Wood and Robert J. Lemke Oliver.

## Dan Yasaki : Modular forms and elliptic curves over the cubic field of discriminant -23

- Number Theory ( 186 Views )The cohomology of arithmetic groups is built from certain automorphic forms, allowing for explicit computation of Hecke eigenvalues using topological techniques in some cases. For modular forms attached to the general linear group over a number field F of class number one, these cohomological forms can be described in terms an associated Voronoi polyhedron coming from the study of perfect n-ary forms over F. In this talk, we describe this relationship and report on some recent computational investigations of the modularity of elliptic curves over the cubic field of discriminant -23. This is joint work with Donnelly, Gunnells, and Klages-Mundt.

## Baiying Liu : On Fourier coefficients and Arthur parameters for classical groups

- Number Theory ( 184 Views )Recently, Jiang made a conjecture investigating the connection between two fundamental invariants of an automorphic representation \pi appearing in the discrete spectrum of quasi-split classical groups G(A). The first invariant is the wave front of \pi, WF(\pi), which is the set of maximal unipotent orbits of G, such that \pi admits a non-trivial Fourier coefficients with respect to them. The second invariant is the Arthur parameter \psi of \pi to which one can associate a unipotent orbit \underline{p}(\psi) of the dual group of G. The conjecture says that in any Arthur packet associated to \psi, the Barbasch-Vogan duality of the orbit \underline{p}(\psi) is a sharp upper bound for the wave front of the representations of the packet. This is an important conjecture that vastly generalizes Shahidi's conjecture which claims that in every tempered packet there exists a generic representation. In this talk, I will review this conjecture and present some recent progress towards it. This is a joint work in progress with Dihua Jiang.

## Chung Pang Mok : Introduction to Mochizukis works on inter-universal Teichmuller theory

- Number Theory ( 183 Views )Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas.

## Hunter Brooks : Special Value Formulas for Rankin-Selberg p-adic L-Functions

- Number Theory ( 176 Views )We discuss special value formulas for a p-adic L-function L_p(f, \chi), where f is a fixed newform and \chi varies over the space of Hecke characters of a fixed imaginary quadratic field, as well as some recent applications. These formulas, first found by Bertolini, Darmon, and Prasanna, relate L_p(f, \mathbb{1}), a value which is outside the range of interpolation defining L_p, to arithmetic invariants of cycles on varieties fibered over modular curves.

## Jianqiang Zhao : Renormalizations of multiple zeta values

- Number Theory ( 159 Views )Calculating multiple zeta values at arguments of mixed signs in a way that is compatible with both the quasi-shuffle product and the meromorphic continuation, is commonly referred to as the renormalization problem for multiple zeta values. In this talk, we consider the set of all solutions to this problem and provide a framework for comparing its elements in terms of a free and transitive action of a particular subgroup of the group of characters of the quasi-shuffle Hopf algebra. This provides a transparent way of relating different solutions at non-positive values, which answers an open question in the recent literature. This is a joint work with Ebrahimi-Fard, Manchon and Singer.

## Jacek Brodzki : A generalised Julg-Valette complex for CAT(0)-cube complexes.

- Number Theory ( 149 Views )This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will extend ideas of Julg-Valette to show how this construction can be used to study K-amenability and K-homology of groups acting on CAT(0)-cube complexes. This talk is based on joint work with Erik Guentner and Nigel Higson.

## Jacob Tsimerman : Recovering elliptic curves from their p-torsion

- Number Theory ( 144 Views )(joint w/ B.Bakker) For an elliptic curve E over a field k, the p-torsion E[p] gives a 2-dimensional representation of the Galois group G_k over F_p. For k=Q and p>13, the Frey-Mazur conjecture famously states that one can recover the isogeny class of E from the representaiton E[p]. We state and prove a direct analogue of this question over function fields of complex algebraic curves. Specifically, for any complex algebraic curve C, let k(C) be its field of rational functions. Then there exists a constant A(C), such that for all primes p>A(C), isogeny classes of elliptic curves E over k(C) can be recovered from E[p]. Moreover, we show that A(C) can be made to depend only on the gonality of C, which can be thought of as the analogous notion of degree for number fields. The study of this question will lead us into the realm of moduli spaces and hyperbolic geometry. The use of the latter means that, unfortunately, these methods don't apply in finite characteristic.

## Ma Luo : Algebraic iterated integrals on the modular curve

- Number Theory ( 144 Views )In the previous talk, we discussed the algebraic de Rham theory for unipotent fundamental groups of elliptic curves. In this talk, we generalize it to a Q-de Rham theory for the relative completion of the modular group, the (orbifold) fundamental group of the modular curve. Using Chen's method of power series connections, we construct a connection on the modular curve that generalizes the elliptic KZB connection on an elliptic curve. By Tannaka duality, it can be viewed as a universal relative unipotent connection with a regular singularity at the cusp. This connection enables us to construct iterated integrals of modular forms, possibly 'of the second kind', that provide periods called 'multiple modular values' by Brown. These periods include multiple zeta values and periods of modular forms.

## Ken Ono : Special values of modular shifted convolution Dirichlet series

- Number Theory ( 139 Views )Rankin-Selberg convolution L-functions are important functions in number theory. Their properties play a central role in many of deepest works on the Ramanujan-Petersson Conjecture. In a recent paper, Hoffstein and Hulse defined generalizations of these L-functions, the so-called "shifted-convolution" L-functions. They obtained the meromorphic continuation of the functions in many cases. Here we consider symmetrizations of these L-functions, and we exactly evaluate their special values at diagonal weights for all shifts. This is joint work with Michael Mertens.

## Albert Chau : Limits of the Kahler Ricci flow and non-negatively curved Kahler manifolds

- Number Theory ( 138 Views )In this talk I will discuss joint results with L.F.Tam on the limiting behavior of the Kahler Ricci flow and its application to the structure of non-negatively curved complete non-compact Kahler manifolds.

## Sug Woo Shin : Asymptotic behavior of supercuspidal characters

- Number Theory ( 136 Views )This is joint work with Julee Kim and Nicolas Templier. Irreducible smooth representations of a p-adic reductive group are said to be supercuspidal if they do not appear in any induced representation from a proper parabolic subgroup. While it is still an open problem to obtain a precise character formula for them (apart from some special cases), I will explain that we can prove a reasonable upper bound and a limit formula as the formal degree tends to infinity, for a large class of supercuspidal representations. An expected application is an equidistribution result as well as a low-lying zero statistics for L-functions in a new kind of families of automorphic representations.

## Caroline Turnage-Butterbaugh : The Distribution of the Primes and Moments of Products of Automorphic $L$-functions

- Number Theory ( 131 Views )The prime numbers are the multiplicative building blocks of the integers, and much thought has been given towards understanding their behavior. In this talk, we will examine prime numbers from two points of view. We will first consider questions on the distribution of the primes. In particular, we will illustrate how the breakthrough work of Maynard and Tao on bounded gaps between primes settles an old problem of Erdos and Turan. Secondly, we will explore the relationship between prime numbers and zeros of the Riemann zeta-function, as a way to motivate the study of the moments of the Riemann zeta function and more general L-functions. In particular, we consider arbitrary products of L-functions attached to irreducible cuspidal automorphic representations of GL(m) over the rationals. The Langlands program suggests essentially all L-functions are of this form. Assuming some standard conjectures, I will discuss how to estimate two types of moments: the continuous moment of an arbitrary product of primitive automorphic L-functions and the discrete moment (taken over fundamental discriminants) of an arbitrary product of primitive automorphic L-functions twisted by quadratic Dirichlet characters.

## Sol Friedberg : Higher theta functions

- Number Theory ( 130 Views )Higher theta functions are the residues of Eisenstein series on covers of the adelic points of classical groups. On the one hand, they generalize the Jacobi theta function. On the other, their Whittaker-Fourier coefficients are not understood, even for covers of $GL_2$. In this talk I explain how, using methods of descent, one may establish a series of relations between the coefficients of theta functions on different groups. In the first instance, this allows us to prove a version of Patterson's famous conjecture relating the Fourier coefficient of the biquadratic theta function to quartic Gauss sums. This is based on joint work with David Ginzburg.