Michael Harris : L-functions and the local Langlands correspondence
- Number Theory ( 164 Views )Henniart derived the following theorem from his numerical local Langlands correspondence: If $F$ is a non-archimedean local field and if $\pi$ is an irreducible representation of $GL(n,F)$, then, after a finite series of cyclic base changes, the image of $\pi$ contains a fixed vector under an Iwahori subgroup. This result was indispensable in all demonstrations of the local correspondence. Scholze gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower (and this result also appears, in a somewhat different form, in proofs based on the global correspondence for function fields). An analogous theorem should be valid for every reductive group, but the known proofs only work for GL(n). I will sketch a different proof, based on properties of L-functions and assuming the existence of cyclic base change, that also applies to classical groups; I will also explain how the analogous result for a general reductive group is related to the local parametrization of Genestier-Lafforgue.
Asif Zaman : Moments of other random multiplicative functions
- Number Theory ( 155 Views )Random multiplicative functions naturally serve as models for number theoretic objects such as the Mobius function. After fixing a particular model, there are many interesting questions one can ask. For example, what is the distribution of their partial sums? Harper has recently made remarkable progress for partial sums of certain random multiplicative functions with values that lie on the complex unit circle. He settled the correct order of magnitude for their low moments and surprisingly established that one expects better than square-root cancellation in their partial sums. I will discuss an extension of Harper's analysis to a wider class of multiplicative functions such as those modeling the coefficients of automorphic $L$-functions.
Raphael Beuzart-Plessis : Recent progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures for unitary groups
- Number Theory ( 155 Views )In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the non-vanishing of central values of certain Rankin-Selberg L-functions to the non-vanishing of certain explicit integrals of automorphic forms, called 'automorphic periods', on classical groups. They have been subsequently refined by Ichino-Ikeda and Neal Harris into precise conjectural identities relating these two invariants thus generalizing a famous result of Waldspurger for toric periods on GL(2). In the case of unitary groups, those have been established by Wei Zhang under some local restrictions. I will review the current state of the art on this and in particular how certain results in local harmonic analysis allow to remove almost all the local restrictions made by Zhang.
Samit Dasgupta : The Brumer-Stark Conjecture
- Number Theory ( 194 Views )I will give a very informal talk on some work I am doing now with Mahesh Kakde. We hope to make progress on the Brumer-Stark conjecture using the theory of group-ring families of modular forms. I will motivate and state the conjecture, and describe the flavor of our approach.
Hiro-aki Narita : Special Bessel models with the local Maass relation and non-tempered automorphic forms on orthogonal groups
- Number Theory ( 186 Views )I will provide some general class of automorphic forms or representations on a general orthogonal group, having a non-tempered non-archimedean local component. We call them non-tempered automorphic forms or representations. It is a fundamental problem to find non-tempered cusp forms, which are nothing but counterexamples to the Ramanujan conjecture. The general class above includes the cusp forms given by the Oda-Rallis-Schiffmann lifting to O(2,m) and non-holomorphic lifting to O(1,8n+1) recently given by the joint work with Yingkun Li and Ameya Pitale. Such general class is given by means of the notion of the special Bessel model and the local Maass relation.
Mahesh Kakde : Congruences between derivatives of geometric L-series
- Number Theory ( 194 Views )I will present a formulation of equivariant Tamagawa number conjecture for flat smooth sheaves on separated schemes of finite type over a finite field. After sketching a proof of this I will give application to Chinburg’s conjectures in Galois module theory and tower of fields conjecture. If time permits I will also give an application towards equivariant BSD for abelian varieties defined over global function fields. This is a joint work with David Burns.
June Huh : Standard conjectures for finite vector spaces
- Number Theory ( 133 Views )I will build a commutative ring that satisfies "standard conjectures", starting from a finite field. What is this ring? What does it say about the finite field? This talk will be elementary: No background beyond the first year graduate algebra will be necessary. Joint with Mats Boij, Bill Huang, and Greg Smith.
Spencer Leslie : Whittaker functions and connections to crystal graphs
- Number Theory ( 128 Views )Whittaker functions are certain special functions that play a central role in automorphic representation theory. When dealing with automorphic forms on covering groups, new methods are needed to compute these functions. In this talk, I will outline how the study of this problem has uncovered connections with geometric representation theory and crystal graphs. I also explain my work in making this connection practical, allowing for new computations of spherical Whittaker functions for covering groups.
Jiuya Wang : Inductive Method in Counting Number Fields
- Number Theory ( 189 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.
Jessica Fintzen : Representations of p-adic groups
- Number Theory ( 195 Views )In the 1990s Moy and Prasad revolutionized p-adic representation theory by showing how to use Bruhat-Tits theory to assign invariants to p-adic representations. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.
Michal Zydor : Relative trace formula of Jacquet-Rallis, recent progress
- Number Theory ( 178 Views )I will discuss the relative trace formula approach to the global Gan-Gross-Prasad conjectures for unitary groups. The focus will be on the spectral side. I will present the various terms that appear in the spectral development of the relative trace formula and discuss what is still missing. This is a joint work with Pierre-Henri Chaudouard.
Jayce Robert Getz : Summation formula for spherical varieties
- Number Theory ( 245 Views )Braverman and Kazhdan, L. Lafforgue, Ngo, and Sakellaridis have pursued a set of conjectures asserting that analogues of the Poisson summation formula are valid for all spherical varieties. If proven, these conjectures imply the analytic continuation and functional equations of quite general Langlands L-functions (and thus, by converse theory, much of Langlands functoriality). I will explain techniques for proving the conjectures in special cases that include the first known case where the underlying spherical variety is not a generalized flag variety.
Ma Luo : Algebraic iterated integrals on the modular curve
- Number Theory ( 126 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.
Dan Goldston : Small Gaps between Zeros of the Riemann Zeta-Function
- Number Theory ( 135 Views )We consider the complex zeros of the Riemann zeta-function &rho = &beta + i &gamma, &gamma > 0. The Riemann Hypothesis (RH) is that &beta = 1/2. If we consider the vertical distribution of these zeros, then the average vertical spacing between zeros at height T is 2&pi / log T. We expect theoretically and find numerically that the distribution of the lengths of these gaps follows a certain continuous GUE distribution where both very small and very large multiples of the average spacing occur. In contrast to this, the existence of a Landau Siegel-zero would force all the gaps in a certain large range to never be closer than half the average spacing, and also have even more bizarre and unlikely properties. There are three methods that have been developed to prove something about small gaps. First, Selberg in the 1940's using moments for the number of zeros in short intervals, was able to prove unconditionally that there are some gaps larger than the average spacing and others smaller than the average spacing. Next assuming RH Montgomery in 1972 introduced a pair correlation method for zeros and produced small gaps less than 0.67 times the average spacing. Finally, in 1981 Montgomery-Odlyzko assuming RH introduced a Dirichlet polynomial weighted method which found small gaps less then 0.5179 times the average spacing. (This method was further developed by Conrey, Ghosh, and Gonek.) These methods all exhibit the presumed barrier at 1/2 times the average spacing for small gaps. I will talk about two projects that are work in progress. The first is joint with Hugh Montgomery and is motivated by the observations that all the results mentioned above do not exclude the possibility that the small gaps found are all coming from multiple zeros and thus gaps of length zero, and at present we do not know if there are any non-zero gaps that are shorter then the average spacing. While we have not yet be able to prove there are any smaller than average non-zero gaps, we can quantify the relationship between non-zero gaps and multiple zeros and show there is a positive proportion of one or the other. The second project is joint work with Caroline Turnage-Butterbaugh where we have developed a Dirichlet Polynomial Weighted Pair Correlation Method which potentially can be applied to a number of questions on zeros.
Ma Luo : Algebraic iterated integrals on elliptic curves
- Number Theory ( 132 Views )I will give two talks on algebraic iterated integrals. In this first one, I will focus on the case of once punctured elliptic curves over a field of characteristic zero, and describe an algebraic de Rham theory for their unipotent fundamental groups by using the elliptic KZB connection. This connection is explicitly expressed by algebraic 1-forms, which are used to construct algebraic iterated integrals on elliptic curves. It also gives an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.
William Chen : Arithmetic monodromy actions on the pro-metabelian fundamental group of punctured elliptic curves
- Number Theory ( 154 Views )For a finite 2-generated group G, one can consider the moduli of elliptic curves equipped with G-structures, which is roughly a G-Galois cover of the elliptic curve, unramified away from the origin. The resulting moduli spaces are quotients of the upper half plane by possibly noncongruence subgroups of SL(2,Z). When G is abelian, it is easy to see that such level structures are equivalent to classical congruence level structures, but in general it is difficult to classify the groups G which yield congruence level structures. In this talk I will focus on a recent joint result with Pierre Deligne, where we show that for any metabelian G, G-structures are congruence in an arithmetic sense. We do this by studying the monodromy action of the fundamental group of the moduli stack of elliptic curves (over Q) on the pro-metabelian fundamental group of a punctured elliptic curve.
Eitan Tadmor : Emergent behavior in self-organized dynamics: from consensus to hydrodynamic flocking
- Number Theory ( 124 Views )A fascinating aspect in collective dynamics is self-organization: ants form colonies, birds flock, mobile networks coordinate a rendezvous and human crowds reach a consensus. We discuss the large-time, large-crowd behavior of different models for collective dynamics. The models are driven by different rules of engagement which quantify how each member of the crowd interacts with its immediate neighbors.
We address two related questions.
(i) How short-range interactions lead, over time, to the emergence of long-range patterns;
(ii) How the flocking behavior of large crowds is captured by their hydrodynamic description.
Samit Dasgupta : Starks Conjectures and Hilberts 12th Problem
- Number Theory ( 110 Views )In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe my recent proof of the Gross-Stark conjecture, a p-adic version of Stark's Conjecture that relates the leading term of the Deligne-Ribet p-adic L-function to a determinant of p-adic logarithms of p-units in abelian extensions. Next I will state my refinement of the Gross-Stark conjecture that gives an exact formula for Gross-Stark units. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a p-adic solution to Hilbert's 12th problem.
Ma Luo : Algebraic de Rham theory for relative completion of $\mathrm{SL}_2(\mathbb{Z})$
- Number Theory ( 166 Views )In this talk, I will first review relative (unipotent) completions of discrete groups in general, and $\mathrm{SL}_2(\mathbb{Z})$ in particular. We then develop an explicit $\mathbb{Q}$-de Rham theory for the relative completion of $\mathrm{SL}_2(\mathbb{Z})$, which enables us to construct iterated integrals of modular forms of the second kind that provide its periods. Following Francis Brown, these periods are called `multiple modular values'. They contain periods of modular forms.
Michael Mossinghoff : Oscillation problems in number theory
- Number Theory ( 159 Views )The Liouville function λ(n) is the completely multiplicative arithmetic function defined by λ(p) = −1 for each prime p. Pólya investigated its summatory function L(x) = Σn≤x λ(n), and showed for instance that the Riemann hypothesis would follow if L(x) never changed sign for large x. While it has been known since the work of Haselgrove in 1958 that L(x) changes sign infinitely often, oscillations in L(x) and related functions remain of interest due to their connections to the Riemann hypothesis and other questions in number theory. We describe some connections between the zeta function and a number of oscillation problems, including Pólya's question and some of its weighted relatives, and, in joint work with T. Trudgian, describe a method involving substantial computation that establishes new lower bounds on the size of these oscillations.
W. Spencer Leslie : A new lifting via higher theta functions
- Number Theory ( 169 Views )Theta functions are automorphic forms on the double cover of symplectic groups and are important for constructing automorphic liftings. For higher-degree covers of symplectic groups, there are generalized theta representations and it is natural to ask if these ``higher'' theta functions play a similar role in the theory of metaplectic forms. In this talk, I will discuss new lifting of automorphic representations on the 4-fold cover of symplectic groups using such theta functions. A key feature is that this lift produces counterexamples of the generalized Ramanujan conjecture, which motivates a connection to the emerging ``Langlands program for covering groups'' by way of Arthur parameters. The crucial fact allowing this lift to work is that theta functions for the 4-fold cover still have few non-vanishing Fourier coefficients, which fails for higher-degree covers.
Stephen Kudla : Theta integrals and generalized error functions
- Number Theory ( 115 Views )Recently Alexandrov, Banerjee, Manschot and Pioline [ABMP] constructed generalizations of Zwegers theta functions for lattices of signature (n-2,2). They also suggested a generalization to the case of arbitrary signature (n-q,q) and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\CC$ of negative vectors, are obtained by `completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. In joint work with Jens Funke, we show that their completed modular series arises as integrals of the q-form valued theta functions, defined in old joint work of the author and John Millson, over a certain singular $q$-cube determined by the data $\CC$. This gives an alternative construction of such series and a conceptual basis for their modularity. If time permits, I will discuss the simplicial case and a curious `convexity' problem for Grassmannians that arises in this context.
Abhishek Parab : Absolute convergence of the twisted Arthur-Selberg trace formula
- Number Theory ( 103 Views )We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven among other cases, when the group is split. This result extends the work of Finis-Lapid (and Muller, spectral side) in the non-twisted setting. In the end, we will give an application towards residues of Rankin-Selberg L-functions suggested by J. Getz.
Rahul Krishna : A New Approach to Waldspurgers Formula.
- Number Theory ( 287 Views )I will present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on $PGL_2$. The method is motivated by interpreting Waldspurger's result as a period relation on $SO_2 \times SO_3$, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, and discuss some small progress towards extending these results to high rank orthogonal groups.
Shuichiro Takeda : The Langlands quotient theorem for symmetric spaces
- Number Theory ( 192 Views )We will discuss how to generalize the Langlands quotient theorem to symmetric spaces. The key idea is to generalize so-called Casselmans criterion for temperedness to the context of symmetric spaces by using the work of Kato-Takano.
Silas Johnson : Counting Functions, Mass Formulas, and Heuristics for Number Fields
- Number Theory ( 97 Views )The Malle-Bhargava heuristics give asymptotic predictions for the density of number fields of bounded discriminant with a given Galois group G, in terms of the number of G-extensions of p-adic fields Q_p. These heuristics can also be applied when the discriminant is replaced by any of a wide variety of other counting functions. Ill discuss how some of these alternate counting functions are built, the idea of global mass formulas, and some cases in which the heuristic predictions can be compared to known results.
Ila Varma : Counting $D_4$-quartic fields ordered by conductor
- Number Theory ( 154 Views )We consider the family of $D_4$-quartic fields ordered by the Artin conductors of the corresponding 2-dimensional irreducible Galois representations. In this talk, I will describe ways to compute the number of such $D_4$ fields with bounded conductor. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination of geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of $D_4$ fields since counting quartic fields containing a quadratic subfield of large discriminant is difficult. However, when ordering by conductor, these techniques can be utilized due to additional algebraic structure that the Galois closures of such quartic fields have, arising from the outer automorphism of $D_4$. This result is joint work with Ali Altug, Arul Shankar, and Kevin Wilson.