David Schwein : Recent progress on the formal degree conjecture
- Number Theory ( 288 Views )The local Langlands correspondence is a dictionary between representations of two kinds of groups: reductive p-adic groups (such as the general linear group) and the absolute Galois groups of p-adic fields. One entry in the dictionary is a conjectural formula of Hiraga, Ichino, and Ikeda for the size of a representation of a p-adic group, its "formal degree", in terms of the corresponding representation of a Galois group. In this talk, after reviewing the broad shape of p-adic representation theory, I'll explain why the conjecture is true for almost all supercuspidals, the fundamental building blocks of the subject.
Jayce Robert Getz : Summation formula for spherical varieties
- Number Theory ( 267 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.
Manish Mishra : Self-dual cuspidal representations
- Number Theory ( 249 Views )Let F be a non-archimedean local field (such as â??_p). The Langlands philosophy says that the arithmetic of F is intimately related to the category R(G) of smooth complex representations of G(F) where G denotes a reductive F-group (for example the general linear group). The building blocks of R(G) are the "supercuspidal" representations of G(F). I will define this term in the talk. The category R(G) comes equipped with an involution - the "contragradient" or the "dual". The supercuspidal representations of G(F) which are self-dual are of considerable interest in the subject. In this talk, I will talk about a joint work with Jeff Adler about the existence of supercuspidals and self-dual supercuspidals. Specifically, we show that G(F) always admits supercuspidal representations. Under some mild hypotheses on G, we determine precisely when G(F) admits self-dual supercuspidal representations. These results are obtained from analogous results for finite reductive groups which I will also talk about.
Rafah Hajjar Munoz : On the residually indistinguishable case of Ribetâ??s lemma
- Number Theory ( 244 Views )Ribetâ??s method describes a way to construct a certain extension of fields from the existence of a suitable modular form. To do so, we consider the Galois representation of an appropriate cuspform, which gives rise to a cohomology class that cuts out our desired extension. The process of obtaining a cohomology class from such a representation is usually known as Ribetâ??s lemma. Several generalizations of this lemma have been stated and proved during the last decades, but the vast majority of them makes the assumption that the representation is residually distinguishable, meaning that the characters of its residual decomposition are non-congruent modulo the maximal ideal. However, recent applications of Ribetâ??s method, such as for the proof of the 2-part of the Brumer-Stark conjecture, have encountered the challenge that the representation we obtain does not satisfy this assumption. In my talk, I describe the limitations of the residually indistinguishable case and conjecture a new general version of Ribetâ??s lemma in this context, giving a proof in some particular cases.
Tony Feng : Steenrod operations and the Artin-Tate pairing
- Number Theory ( 240 Views )In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.
Ali Altug : Beyond Endoscopy via the Trace Formula
- Number Theory ( 230 Views )In his recent paper,\Beyond Endoscopy", Langlands proposed an approach to (ultimately) attack the general functoriality conjectures by means of the trace formula. For a (reductive algebraic) group G over a global field F and a representation of its L-group, the strategy, among other things, aims at detecting those automorphic representations of G for which the L-function, L(s;\pi ;\rho ), has a pole at s = 1. The method suggested using the the trace formula together with an averaging process to capture these poles. In this talk we will start by recalling the functoriality conjectures and brie y describe the method suggested by Langlands. Then, specializing on the group GL(2) we will discuss some recent work on Beyond Endoscopy. More precisely, we will discuss the elliptic part of the trace formula and the analytic problems caused by the volumes of tori, singularities of orbital integrals and the non-tempered terms. We will then describe how one can use an approximate functional equation in the trace formula to rewrite the elliptic part which resolves these issues. Finally, we will talk about applications of the resulting formula.
Jürgen Klüners : The negative Pell equation and the Cohen-Lenstra heuristic
- Number Theory ( 218 Views )For a (squarefree) integer d the negative Pell equation is given by: X^2 - d Y^2 = -1. It is easy to see that this equation has no solution over the integers, if d is negative or d is congruent to 3 modulo 4. In this talk we would like to study the asymptotic behavior of integers d such that this equation is solvable. This question is related to the behavior of the class group of the quadratic field generated by a square root of d. The distribution of those class groups is described by the Cohen-Lenstra heuristics.
Hiro-aki Narita : Special Bessel models with the local Maass relation and non-tempered automorphic forms on orthogonal groups
- Number Theory ( 201 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.
Wei Zhang : Selmer groups and the indivisibility of Heegner points
- Number Theory ( 190 Views )We will discuss the conjecture of Kolyvagin on the indivisibility of Heegner points and its role in constructing rational points on elliptic curves over rational numbers, particularly in the proof of a recent result of this type: "the Selmer rank being one implies that the Mordell--Weil rank being one".
Robert Rhoades : The story of a Â?strangeÂ? function
- Number Theory ( 188 Views )In a 1997 Fields Medalist Maxim Kontsevich suggested that the function F(q) = 1 + (1-q) + (1-q)(1-q^2) + (1-q)(1-q^2)(1-q^3)+Â?, defined only for q a root of unity, is similar to certain functions arising from the computation of Feynman integrals in quantum field theory. In the last sixteen years this function has been connected to interval orders in decision making theory, ascent sequences and matchings in combinatorics, and Vassiliev invariants in knot theory. Don Zagier related the asymptotic properties of this function to the Â?half-derivativesÂ? of modular forms and was led to define a notion of Â?quantum modular formÂ?. In a trilogy of papers, my collaborators (Andrews, Bryson, Ono, Pitman, Zwegers) and I have connected this function to RamanujanÂ?s mock theta functions and the combinatorics of unimodal sequences. I will tell the story of this function and these many relationships.
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.
Ken Ono : Zeta polynomials for modular forms
- Number Theory ( 182 Views )The speaker will discuss recent work on Manin's theory of zeta polynomials for modular forms. He will describe recent results which confirm Manin's speculation that there is such a theory which arises from periods of newforms. More precisely, for each even weight k>2 newform f, the speaker will describe a canonical polynomial Zf(s) which satisfies a functional equation of the form Zf(s)=Zf(1−s), and also satisfies the Riemann Hypothesis: if Zf(ρ)=0, then Re(ρ)=1/2. This zeta function is arithmetic in nature in that it encodes the moments of the critical values of L(f,s). This work builds on earlier results of many people on period polynomials of modular forms. This is joint work with Seokho Jin, Wenjun Ma, Larry Rolen, Kannan Soundararajan, and Florian Sprung.
Michael Mossinghoff : Oscillation problems in number theory
- Number Theory ( 178 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.
Rahul Dalal : Counting level-1, quaternionic automorphic representations on G2
- Number Theory ( 175 Views )Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of GL2. Like holomorphic modular forms, they are defined by having their real component be one of a particularly nice class (in this case, called quaternionic discrete series). We count quaternionic automorphic representations on the exceptional group G2 by developing a G2 version of the classical Eichler-Selberg trace formula for holomorphic modular forms. There are two main technical difficulties. First, quaternionic discrete series come in L-packets with non-quaternionic members and standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same L-packet. Using the more modern stable trace formula resolves this issue. Second, quaternionic discrete series do not satisfy a technical condition of being "regular", so the trace formula can a priori pick up unwanted contributions from automorphic representations with non-tempered components at infinity. Applying some computations of Mundy, this miraculously does not happen for our specific case of quaternionic representations on G2. Finally, we are only studying level-1 forms, so we can apply some tricks of Chenevier and Taïbi to reduce the problem to counting representations on the compact form of G2 and certain pairs of modular forms. This avoids involved computations on the geometric side of the trace formula.
Ila Varma : Counting $D_4$-quartic fields ordered by conductor
- Number Theory ( 167 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.
William Chen : Arithmetic monodromy actions on the pro-metabelian fundamental group of punctured elliptic curves
- Number Theory ( 163 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.
Matilde Lalin : The distribution of points on cyclic l-covers of genus g
- Number Theory ( 162 Views )We give an overview of a general trend of results that say that the distribution of the number of F_q-points of certain families of curves of genus g is asymptotically given by a sum of q+1 independent, identically distributed random variables as g goes to infinity. In particular, we discuss the distribution of the number of F_q-points for cyclic l-covers of genus g. (This is joint work with Bucur, David, Feigon, Kaplan, Ozman, Wood.) This work generalizes previous results in which only connected components of the moduli space were considered.
Michael Lipnowski : Torsion in the cohomology of arithmetic groups
- Number Theory ( 153 Views )The remarkable Cheeger-Muller theorem, of differential geometric origin, provides an analytic means of studying torsion in the cohomology of Riemannian manifolds. We describe how this theorem can be applied to prove a numerical form of Langlands' base change functoriality for torsion in cohomology.
Dan Goldston : Small Gaps between Zeros of the Riemann Zeta-Function
- Number Theory ( 150 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.
Adam Jacob : A special Lagrangian type equation for holomorphic line bundles
- Number Theory ( 150 Views )Consider a holomorphic line bundle L over a compact Kahler manifold. Motivated by mirror symmetry, I will define an equation on L that is the line bundle analogue of the special Lagrangian equation, which can be studied even when the base is not a Calabi-Yau manifold. I will show solutions are unique global minimizers of a positive functional. To address existence, I will introduce a line bundle analogue of the Lagrangian mean curvature flow, and prove convergence in certain cases. This is joint work with S.-T. Yau.
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.
Lillian Pierce : Burgess bounds for short mixed character sums
- Number Theory ( 147 Views )A celebrated result of Burgess proves nontrivial bounds for short multiplicative character sums. In general, bounds for short character sums have utility in a wide range of problems in number theory, and it would be highly desirable to extend BurgessÂ?s method to apply to more general character sums. This talk presents new work in this direction, joint with Roger Heath-Brown, that proves nontrivial bounds for short mixed character sums in which the additive character is evaluated at a real-valued polynomial. Our approach, via a version of the Burgess method, includes a novel application of the recent results of Wooley on the Vinogradov mean value theorem.
A. Raghuram : Special values of automorphic L-functions
- Number Theory ( 145 Views )In the first part of the talk I will describe a general context which, in some specific situations, permits us to give a cohomological interpretation to the Langlands-Shahidi theory of L-functions. In the second part of the talk, I will specialize to the context of the general linear group over a totally imaginary base field F, and discuss some recent results of mine on the special values of Rankin-Selberg L-functions for GL(n) x GL(m) over such an F. The talk is based on my preprint: https://arxiv.org/abs/2207.03393
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.
Simon Marshall : L^p norms of arithmetic eigenfunctions
- Number Theory ( 143 Views )Let M be a compact Riemannian manifold, and f an L^2 normalised Laplace eigenfunction on M. A popular question in semiclassical analysis is how well one can bound the other L^p norms of f, or its restriction to a submanifold. I will give an introduction to this problem, and describe how one can make progress on it using the additional assumptions that M is arithmetic and f is a Hecke-Maass form.
June Huh : Standard conjectures for finite vector spaces
- Number Theory ( 142 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 ( 137 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.