BCMIT Joint Number Theory Abstracts
department of mathematics
Boston College and MIT will join forces again this coming year to create a Number Theory Seminar series. The seminar will meet six times per year with two speakers each time, with three meetings at BC and three meetings at MIT. The first talk each session will begin at 3:00 p.m. The goal is to create a seminar series that will attract number theorists from the greater Boston area and to feature important advances in modern number theory.
The organizers are Ben Howard and David Geraghty at BC, and Bjorn Poonen at MIT.
Note: Parking for all seminars is available. Commonwealth Avenue offers areas with free parking. In some places the restriction of parking to one hour ends at 3:00 p.m., so if you park after 2:00 p.m. you may stay until evening. However, please be sure to read all signs carefully as time limits are enforced. Paid parking for visitors is available in the Beacon Street Parking Garage. For more information please see BC Visitor Parking.
20142015  
September 16, 2014 at BC, McGuinn 334 
3:004:00 pm Liang Xiao, University of Connecticut, Storrs Title: Slopes of modular forms Abstract: An interesting computation of Buzzard and Kilford suggested that the slope distribution of the Up operators on the space of overconvergent modular forms tends to form an arithmetic progression, when the Nybentypus character is highly divisible by p. Unfortunately, this was only verified for very small prime p and small tame level. I will explain a joint work with Daqing Wan and Jun Zhang, in which we work with overconvergent automorphic forms for a definite quaternion algebra instead. We prove certain weak version of this expectation for general p and general tame level structure. 4:305:30 pm Kiran Kedlaya, University of California, San Diego Title: Cohomology of local systems on rigid analytic spaces Abstract: Let K be a finite extension of Q_p. The notion of the etale fundamental group of a rigid analytic space has been introduced by de Jong. It is highly noncompact; consider for example Tate's uniformization of elliptic curves. Let X be a smooth proper connected rigid analytic space over K. One then has a RiemannHilbertstyle identification of continuous representations of the etale fundamental group of X on finitedimensional Q_pvector spaces with locally constant sheaves of finitedimensional Q_pvector spaces with respect to Scholze's proetale topology; these are what we call "etale Q_plocal systems" on X. We prove that the cohomology groups of such a sheaf are finitedimensional Q_pvector spaces. The proof uses an extension of padic Hodge theory, especially the theory of (phi, Gamma)modules, to the setting of etale fundamental groups, in order to transform the problem into something resembling the finiteness of cohomology of coherent sheaves on X (proved by Kiehl). Joint work with Ruochuan Liu (Beijing). 
October 14 at MIT Room 4163 
3:004:00 pm Wei Zhang (Columbia U.) Title: ATC, special parahorics and exotic good reduction Abstract: I will report a joint work with M. Rapoport and B. Smithling, on an 4:305:30 pm Martin Olsson, University of California, Berkeley Title: Motivic invariants of ladic sheaves Abstract: I will give an overview of a project aimed at understanding the motivic nature of ladic sheaves. I will survey motivating questions about independence of l and past results of Lafforgue, Drinfeld, and others. I will then discuss how to incorporate correspondences into the theory, recent results, and open questions. 
November 18 at BC McGuinn 521 
3:004:00 p.m. Laurent Fargues, Directeur de Recherche CNRS, Institut de Mathématiques de Jussieu Title: Gbundles on the curve Abstract: In my joint work with Fontaine, we have defined and studied a "curve" linked to padic 4:305:30 p.m. Joseph H. Silverman, Brown University Title: Canonical heights and nef divisors on abelian varieties, with an application to arithmetic dynamics Abstract: Let A/K be an abelian variety defined over a number field, and let D be a divisor on A. The NeronTate height q_D(P) = lim h_D(nP)/n^2 is a quadratic form q_D : A(K) > R, and if D is ample, then q_D is positive definite on A(K) modulo torsion. I will discuss an extension of this theorem to the case that D is only assumed to be a nef divisor and will give, as an application, a proof for abelian varieties of the following conjecture in arithmetic dynamics: Let f : X > X be a dominant rational selfmap of a smooth projective variety, all defined over a number field. Let P be an algebraic point of X whose forward orbit by iterates of f is welldefined and Zariski dense in X. Then the forbit of P has maximal arithmetic complexity. (Joint work with Shu Kawaguchi) 
March 10, 2015 at BC Cushing 209 
3:00 p.m. Ben Brubaker (U. Minnesota) Title: "Matrix coefficients for padic groups and Hecke algebra modules" Abstract: I'll report on continuing work to categorize important matrix coefficients for representations of padic groups (includingspherical, Whittaker, and Bessel functionals among others) in terms of representations of IwahoriHecke algebras. We'll showhow these characterizations lead to simple computations of the functionals on distinguished test vectors (e.g. the spherical vector), which has applications to global constructions in automorphic forms. This is based on joint work with Bump, Friedberg, and Licata. 4:30 p.m. David Harbater (U Penn) Title: Localglobal principles for torsors Abstract: Many algebraic structures can be classified by torsors, i.e. by principal homogenous spaces for algebraic groups. This holds, for example, for quadratic forms and central simple algebras. Localglobal principles for such algebraic structures can then be obtained from localglobal principles for torsors. This has been studied classically in the case of torsors over global fields. This talk, on recent and continuing work with Julia Hartmann and Daniel Krashen, will focus on the case of semiglobal fields (onevariable function fields over complete discretely valued fields). 
April 7, 2015 at MIT Room E25111 (Rescheduled from Feb. 10) 
3:00 p.m. Francis Brown (IHES) Title: "Irrationality proofs, moduli spaces and dinner parties" Abstract: After introducing an elementary criterion for a real number to be irrational, I will discuss Apery's famous result proving the irrationality of zeta(3). Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game. Bruno Klingler (Jussieu) Title: "The hyperbolic AxLindemannWeierstrass conjecture" Abstract: The hyperbolic AxLindemannWeierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this talk I will describe the conjecture, its role and its proof (joint work with E. Ullmo and A. Yafaev). 
April 14, 2015 at MIT Room 4237 
3:00 p.m. Alexander Gamburd (The Graduate Center, CUNY) After defining the terms in the title and describing relations between them I will discuss recent joint work with Bourgain and Sarnak on Markoff graphs (obtained by reducing Markoff tree modulo q). Our work yields results on diophantine properties of Markoff numbers, including, in particular, a theorem asserting that almost all of them are composite. 4:30 p.m. Dinesh S. Thakur (University of Rochester) "Special values scenario in function field arithmetic" We will give a survey of results, conjectures and techniques dealing with the arithmetic of special values in function field arithmetic. 
20132014  
September 17, 2013 MIT, room 4163 
Melanie Matchett Wood (Wisconsin) 3:004:00 p.m. In 1984, Cohen and Lenstra gave conjectures predicting the Bianca Viray (Brown) 4:305:30 p.m. In 1970, Manin showed that the Brauer group can obstruct the existence of global rational points, even when there exist points everywhere locally. Conjecturally, this BrauerManin obstruction explains all failures of the localtoglobal principle in the case of (geometrically) rational surfaces. However, beyond rational surfaces, for example in the case of Enriques surfaces and K3 surfaces, there is little evidence in support (or against) such a conjecture. This lack of evidence stems from a difficulty in computing the BrauerManin obstruction for such surfaces, specifically in computing the socalled transcendental part of the Brauer group. In this talk, we explain how to compute the transcendental Brauer element on any Enriques surface, and, more generally, how to compute the 2torsion Brauer classes on any double cover of a ruled surface. This is joint work with Brendan Creutz. 
October 22, 2013 at BC, Fulton 230 Directions 
3:004:00 p.m. Kirsten Eisentrager (Penn State) Title: "Hilbert's Tenth Problem for function fields of positive characteristic" Abstract: Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. In 1970 Matiyasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by asking the same question for polynomial equations with coefficients and solutions in other commutative rings. In this talk we will discuss some recent undecidability results for function fields of positive characteristic. 4:305:30 p.m. Thomas Tucker (Rochester) Title: "Integral points in twoparameter orbits" Abstract: Let K be a number field, let f : P^1 > P^1 be a nonconstant rational map of degree greater than 1 that is not conjugate to a powering map, let S be a finite set of places of K, and suppose that u,w ∈ P^1(K) are not preperiodic under f. We prove that the set of (m,n) ∈ N^2 such that f^m(u) is Sintegral relative to f^n(w) is finite and effectively computable. This may be thought of as a twoparameter analog of a result of Silverman on integral points in orbits of rational maps. We also discuss socalled BangZsigmondy variants of this question. This represents joint work Corvaja, Sookdeo, and Zannier. 
November 19, 2013 at MIT, room 4163 
3:004:00 p.m. Roman Holowinsky (The Ohio State University) "Hybrid subconvexity bounds for RankinSelberg convolutions" We'll discuss several recent results concerning the subconvexity problem for Lfunctions. In all of the cases we shall consider, one benefits analytically from the particular size or form of the Lfunction's conductor. The results presented are joint with either Ritabrata Munshi, Nicolas Templier or Zhi Qi. 4:305:30 p.m. Michael Larsen (Indiana University) "Type A images of Galois representations and maximality" I will talk about recent joint work with Chun Yin Hui about images of motivic Galois representations of "Type A", i.e., for which every simple composition factor of the Zariski closure of the image is of type A in the CartanKilling classification. 
February 11, 2014 at BC Cushing 209 Directions 
3:004:00 p.m. Tasho Kaletha (Princeton) Title: Rigid inner forms and endoscopy Abstract: The basic version of the local Langlands conjecture predicts a correspondence between Langlands parameters and packets of representations of a given reductive group G defined over a local field F. The more refined version enhances the space of parameters by including representations of certain finite groups and then predicts a correspondence between enhanced parameters and individual representations of G. This refinement is needed in many application, one example being the multiplicity formula for discrete automorphic representations. While the basic version is easy to state for any G, a precise statement of the refined version was not known for general reductive groups over padic fields (including classical groups like special linear, symplectic, and special orthogonal groups over division algebras). In this talk, we will propose a uniform and precise statement of the refined local Langlands conjecture for arbitrary connected reductive groups over local fields of characteristic zero. It is based on the construction of a canonical gerb over such a field, whose arithmetic properties lead to a normalization of the endoscopic objects involved in the local Langlands conjecture. Time permitting, we will discuss evidence for the validity of this statement, which includes the case of real groups, as well as certain classes of representations of padic groups. 4:305:30 p.m. Florian Herzig (Toronto) Title: On mod p localglobal compatibility for GL3 in the ordinary case Abstract: Suppose that rhobar : G_{Qp} > GL3(Fpbar) is a maximally nonsplit, ordinary, FontaineLaffaille Galois representation. Then its "extension class" is determined by an invariant in Fpbar. In a global situation, under suitable hypotheses, we show that this invariant can be calculated using GL3(Qp)representation theory. This is joint work in progress with Stefano Morra. 
March 11, 2014 at MIT 
Junecue Suh (Harvard) Eric Urban (Columbia) 
April 1, 2014 3:00 PM BC Cushing 209 Directions 
Nicolas Templier (Princeton) Title: Families of Lfunctions and their Symmetry Abstract: We consider families of automorphic representations on reductive groups. We establish a quantitative Plancherel equidistribution theorem for the Satake parameters of these families. We proceed via the trace formula by proving uniform bounds for orbital integrals and for trace characters of discrete series representations. The theorem is strong enough to apply to the KatzSarnak heuristics on zeros of Lfunctions making it possible to conjecture the universality class of the distribution of zeros in families. Joint works with S.W.Shin, P.Sarnak and J.L.Kim. Zhiwei Yun (Stanford) Title: Epipelagic representations and rigid local systems Abstract: Reeder and Yu have constructed in a uniform way certain supercuspidal representations of padic groups called "epipelagic representations", using invariant theory studied by Vinberg et al. In the function field case, we will realize these epipelagic representations as local components of automorphic representations, and construct the corresponding Langlands parameters, i.e., local systems over P^1 minus two points. These local systems can be computed explicitly for classical groups, and they give new families of local systems (with monodromy in all types of groups, classical or exceptional) that are expected to be rigid. 
20122013  
October 16, 2012 at BC 9 Lake Street, room 100 Directions 
3:00–4:00 p.m. 4:30–5:30 p.m. Abstract: The noncommutative main conjectures in Iwasawa theory predicts certain congruences between special padic Lvalues. We shall discuss the noncommutative main conjecture along with evidence for the predicted congruences, arising from modular forms. 
November 13, 2012 at MIT, room 4163 
3:00–4:00 p.m. Abstract: Artin introduced his nonabelian Lfunctions for representations of the Galois group in a series of papers in 19231931. He was able to define the local Euler factors for all primes and define the Artin conductor that appears in the functional equation, but the Artin root number remained mysterious. It was factored by Deligne in 1971 as part of his proof of the existence of the local epsilonfactors that appear in the functional equation of the Artin Lfunctions. One way too understand these Lfunctions and epsilonfactors is to find a corresponding analytic object, and automorphic form, whose Lfunction and epsilonfactors match the arithmetic ones. This is the content of the local Langlands correspondence. This correspondence should be robust and be preserved under various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In collaboration with F. Shahidi and TL. Tsai, we have recently showed that indeed the correspondence between the local arithmetic and analytic epsilon—factors is preserved under these operations. The proof is an application of local/global techniques and the stability of these factors under highly ramified twists. In this talk I will attempt to explain a bit about these objects and the techniques we use in our proof. 4:30–5:30 p.m. Abstract: Geometric Igusa integrals appear as important technical tools in the study of rational and integral points on algebraic varieties. I will describe some of these applications (joint work with A. ChambertLoir). 
December 4, 2012 
3:00–4:00 p.m. Abstract: To any essentially selfdual, regular algebraic (ie cohomological) automorphic representation of GL(n) over a CM field one knows how to associate a compatible system of ladic representations. These ladic representations occur (perhaps slightly twisted) in the cohomology of a Shimura variety. Recently Harris, Lan, Thorne and myself have constructed ladic representations without the `essentially selfdual' hypothesis'. In this case the ladic representations do not occur in the cohomology of any Shimura variety. Rather we construct them using a congruence argument. In this talk I will describe this theorem and sketch the proof. 4:30–5:30 p.m. Abstract: There has been a burst of recent activity on supersingular K3 surfaces and their moduli. I will describe some interrelated results on the Tate conjecture, the derived category, and rational curves in the moduli space. 
February 5, 2013 at MIT room 10250 
3:00–4:00 p.m. Abstract: There are no abelian schemes over Spec(Z), by a theorem of Fontaine and Abrashkin. On the other hand, there are several examples of elliptic curves over quadratic fields with good reduction everywhere, the first being given by Tate and Serre over the field Q(sqrt(29)). I will report on recent joint work with Lassina Dembele, in which we produce several examples of abelian surfaces with real multiplication, defined over a real quadratic field and having good reduction everywhere. It relies on recent explicit equations for Hilbert modular surfaces (due to Elkies and the speaker) and also on efficient computation of Hecke eigenvalues for spaces of Hilbert modular forms (due to Dembele, Donnelly, Voight and Greenberg). Our work also provides some evidence for the EichlerShimura conjecture for Hilbert modular forms. 4:30–5:30 p.m. Abstract: We will discuss the finite descent obstruction for the Hasse principle in the moduli space of abelian varieties and how it is related to a localglobal condition on Galois representations to come from abelian varieties. We will discuss some cases where this is the only obstruction and some cases where it isn't. Time permitting, we will discuss how this is related to anabelian geometry. 
March 19, 2013 at BC, Fulton 220 
3:00–4:00 p.m. Abstract: In 1859 Riemann published a 10 page monograph in which he showed how an understanding of the distribution of prime numbers can be achieved through the study of the zeros of the Riemann zeta function, which all occur in the domain of analytic continuation, and his extraordinary approach has dominated the subject ever since. Riemann starts by using Perron's formula to create an identity for the (weighted) count of the number of primes up to x. He then pulls this contour integral to the left, into the region of analytic continuation (which is fraught with difficulties). What if, instead, we stick in the natural region of definition of the zeta function, and attempt to better understand the original contour integral? In this talk we discuss the tentative beginnings of such an approach, and its (farreaching) consequences. This is joint work with Adam Harper and K. Soundararajan. 4:30–5:30 p.m. Abstract: I will discuss properties of functions from a number field K to itself obtained by evaluating a polynomial. In particular, I will show that for any f(x) in Q[x], the induced function on Q is at most 6to1 outside a finite set. I will also describe the pairs of polynomials in Q[x] whose images have infinite intersection, and describe connections with other topics in number theory, algebraic geometry, and Nevanlinna theory. The proofs use a range of tools, including an intricate application of the classification of finite simple groups. 
April 9, 2013 at MIT, room 32144 
3:00–4:00 p.m. Abstract: The homology of SL_N(Z) in a fixed degree d turns out to be independent of N for sufficiently large N. This phenomenon is known as *stability*. The actual homology groups themselves turn out to be intimately related to the algebraic Ktheory of the integers. Borel computed the rational Ktheory of Z using automorphic forms, and the integral Kgroups of Z are now essentially known due to work of many people. In this talk, we address the following question: what does the homology of congruence subgroups of SL_N(Z) in small degree look like for large N? This is already an open question for H_2. 4:30–5:30 p.m. Abstract: An important fact about a principally polarized abelian variety A defined over an algebraically closed field k of characteristic p > 0 is that the multiplicationbyp morphism of A is inseparable. The prank is the number r such that p^r is the number of ptorsion points of A. The EkedahlOort type is a finer invariant which characterizes the isomorphism class of the ptorsion group scheme A[p]. There are many deep results about the stratifications of the moduli space of principally polarized abelian varieties by prank and EkedahlOort type. Much less is known about the analogous stratifications of the Torelli locus. In this talk, I will discuss the geometry of the prank strata of the moduli space of curves. If time permits, I will also discuss EkedahlOort strata of low codimension, for which the Jacobians are close to being ordinary. 
20112012  
September 20, 2011 at BC  9 Lake Street, Room 035 (north of Commonwealth Avenue near the "B" line) Directions 
3:00–4:00 p.m. MarieFrance Vigneras (Jussieu) Title: "From $p$adic Galois representations to $G$equivariant sheaves on the flag variety $G/P$" Abstract: This is joint work with P.Schneider and Z. Zabradi. We associate to the (phi,Gamma)module D of a padic representation of the Galois group of Q_p and to a linear action on D of the dominant kernel of a simple root of T in P=NT, a Gequivariant sheaf on G/P. For G=GL(2, Q_p) this sheaf is due to Colmez. 4:30–5:30 p.m. Abstract: This talk will give an overview of work being done to understand the arithmetic intersection of certain divisors on the Siegel moduli space with CM cycles attached to primitive quartic CM fields. This work has important applications to generating genus 2 curves for cryptography, and to Stark’s conjectures. 
October 18, 2011 at MIT, Room 2132 
3:00–4:00 p.m. Abstract: I will describe some generalities of the motives of the title and then focus on those of weight zero, which give rise to Artin Lfunctions. The main example will be the case where the corresponding Galois group is the (a subgroup of) the Weyl group of F_4. This group has order 1152 and a natural irreducible representation of dimension 4. I will discuss how we may explicitly compute the associated degree four Lfunctions and their relation to the lines in certain affine cubic surfaces. This is joint work with H. Cohen. 4:30–5:30 p.m. Abstract: In this talk I will introduce a formula between the height of the GrossSchoen diagonal cycle constructed from Shimura curves and the central derivative of the triple product Lfunction. It is a joint work with Wei Zhang and Shouwu Zhang. 
November 15, 2011 at BC, McGuinn 521 
3:00–4:00 p.m. Brian Conrey (AIM) Title: "A reciprocity formula for a cotangent sum" Abstract: The function c(x) defined on rationals x = h/k with (h, k) = 1 and k > 0 by There is a connection with the period functions of Eisenstein series introduced by Lewis and Zagier. In this circle of ideas is an extension of Voronoi’s summation formula and a new exact formula for a second weighted moment of the Riemann zetafunction. 4:30–5:30 p.m. Abstract: Fourier coefficients of automorphic forms on classical groups are used in a wide variety of contexts, e.g., integral representations of Lfunctions and multiplicity one theorems. I'll discuss some recent work with Wilfried Schmid on decay estimates for these unipotent periods, with applications to Lfunctions. There is a particularly rich structure of Fourier coefficients for automorphic realizations of small representations of exceptional groups. I'll explain a physics construction of such realizations for some small representations of E7 and E8 (joint with Michael Green and Pierre Vanhove) developed from graviton scattering amplitudes, as well as how some intricate mathematics of supersymmetry can be observed in the fine structure of these coefficients (verifying string theory predictions) 
February 14, 2012 at MIT, Room 3333 
3:00–4:00 p.m. Abstract: I will discuss the construction in terms of integral operators of cuspidal automorphic forms on classical groups and their relations to the Langlands functorality and endoscopy transfers. These constructions can be viewed as a combination of the automorphic descents of Ginzburg RallisSoudry and the classical theta correspondences. It is work in progress with Ginzburg and Soudry. 4:30–5:30 p.m. Abstract: It is wellknown that the closed geodesics on the modular surface X, when collected according to the discriminants, are equidistributed with respect to the hyperbolic invariant measure. This is originally the Linnik problem, solved by Duke via bounding the Fourier coefficients of halfintegral weight modular forms. We study and evaluate asymptotically the variance of this distribution on the unit tangent bundle of X, and show it is equal to the classic variance of the geodesic flow a la Ratner, but twisted by an intriguing arithmetic invariant, the central value of certain Lfunction. Our approach makes use of the work of Shintani on Weil representation and the theta correspondence. We also obtain analogous result for the variance in the Linnik distribution of integer points on spheres, via JacquetLanglands correspondence and Yoshida lift. This talk is partly based on my joint work with Z. Rudnick and P. Sarnak. 
March 20, 2012 at BC, McGuinn 521 
3:00–4:00 p.m. Abstract: I will discuss work on the value distribution and moments of families of Lfunctions. We will start with values to the right of the critical line, where the problem can be well modeled by random Euler products. This fails on the critical line, and the Lvalues here have a different flavor with Selberg's theorem on lognormality being a representative result. I will discuss here work on upper and lower bounds for moments of Lfunctions, and also recent work (with Conrey and Iwaniec) on the asymptotic large sieve which provides an asymptotic formula in a new case. 4:30–5:30 p.m. Abstract: We present a new construction of the padic Lfunctions associated to ray class characters of totally real fields. We define a certain measurevalued cohomology class on GL_n(Z) that we call the Eisenstein class. We have two separate constructions of cocycles representing this class: one is obtained by enacting a "smoothing operation" on the Eisenstein cocycle defined by Sczech, and the other is obtained by smoothing the cocycles arising from Shintani's method as studied by Solomon, Hill, and Colmez. For each ray class of a totally real field we define an associated cycle such that the natural cap product with our class yields the desired padic Lfunction. As a corollary of our construction and a result of Spiess, we prove that the order of vanishing at s=0 of the padic Lfunction is at least equal to the expected one, as conjectured by Gross. This result was known from Wiles' proof of the Iwasawa Main Conjecture under an auxiliary assumption that is not necessary via our method. We also discuss refinements of Gross's conjecture arising from our construction that yield exact formulas for Stark units. This is joint work with Pierre Charollois. 
April 3, 2012 at MIT, Room 3333 
3:00–4:00 p.m. Abstract: The understanding of the arithmetic of modular forms for noncongruence subgroups pales when compared to that of congruence subgroups. This is primarily due to the lack of effective Hecke operators, as conjectured by Atkin. The first pioneering work on noncongruence modular forms was done by Atkin and SwinnertonDyer in 1971, who proposed 3term padic congruence relations replacing the usual 3term recursive relations satisfied by eigenforms of the Hecke operator at p.In 1985 Scholl attached to noncongruence forms a family of Galois representations, and establishedpadic congruences of ASD type. In this talk we shall survey historical development of the subject and present recent progress with emphasis on the ASD congruences and modularity of Scholl representations. 4:30–5:30 p.m. Abstract: It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasiMonte Carlo methods for multidimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "lowlying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamic formalism. This is joint work with Jean Bourgain. 
20102011  
Tuesday, September 21 
3:00 p.m. Title: "Vanishing theorems for torsion automorphic sheaves" Abstract: In this talk, I will explain my joint work with Junecue Suh on when and why the cohomology of Shimura varieties (with nontrivial integral coefficients) has no torsion, based on certain vanishing theorems we proved recently. (All conditions involved will be explicit, independent of level, and effectively computable.) 4:30 p.m. Title: "The LanglandsKottwitz method for the zeta function, beyond the parahoric case" Abstract: The LKmethod is counting the points mod p of a Shimura variety, at a place of good reduction. It has subsequently been extended to some parahoric cases of bad reduction (Haines, Ngo). In this talk I will report on joint work with Haines which concerns some examples of slightly deeper level structure, and will perhaps also mention work of Scholze on examples of arbitrarily deep level structure. 
Tuesday, October 19 
3:00 p.m. Title: “Canonical subgroups over Hilbert modular varieties” Abstract: I shall discuss recent joint work with Payman Kassaei (King's College) on the canonical subgroup problem for abelian varieties with real multiplication. We give an explicit description of the region over which the canonical subgroup exists, which is in a sense the maximal possible, and obtain analogues of the whole range of theorems well known for modular curves. Our method is based on the rigid geometry of the moduli spaces and makes strong use of their local models and geometry of the special fibres. 4:30 p.m. Title: “On the padic local Langlands correspondence for GL2” 
Tuesday, November 16 
3:00 p.m. Title: “On some local models for Shimura varieties” Abstract: A basic problem in the theory of Shimura varieties is the definition and subsequent study of "good" integral models. For PEL Shimura varieties whose level structures have pcomponent of parahoric type, Rapoport and Zink have defined natural integral models and reduced their local study to their _local models_, which are defined purely in terms of linear algebra. Unfortunately, the RapoportZink integral Shimura models and local models can fail one of the most basic tests of reasonableness, namely they need not be flat. In a recent paper, Pappas and Rapoport have proposed corrections to the definitions of local models attached to even orthogonal groups and ramified unitary groups, which they conjecture to give flat schemes. I will report on the proof of a version of their conjecture. A key ingredient is the enumeration of certain Schubert varieties in affine flag varieties. 4:30 p.m. Title: “Tropical and Berkovich analytic curves” Abstract: We will discuss the relationship between a Berkovich analytic curve over a complete and algebraically closed nonArchimedean field and its tropicalizations, paying special attention to the natural metric structure on both sides. This is joint work with Sam Payne and Joe Rabinoff. 
Tuesday, February 8 
3:00 p.m. Title: "elladic properties of the partition function" Abstract: Ramanujan's partition congruences modulo powers of 5,7 and 11 imply that certain sequences of partition generating functions tend elladically to 0. Little is known about the elladic behavior of these sequences for primes ell > 11. We show that modulo powers of primes ell at least 5, these sequences of generating functions elladically converge to linear combinations of a(n explicitly given) finite number of special qseries. We use our general result to reveal a theory of "multiplicative partition congruences" anticipated by Atkin in the 1960s with respect to primes ell between 13 and 31. This is joint work with Zachary Kent and Ken Ono. 4:30 p.m. Title: "Expander graphs, gonality, and Galois representations" Abstract: (joint work with Chris Hall and Emmanuel Kowalski) We show that 1parameter families of abelian varieties over a number field K have few fibers over boundeddegree extensions of K whose modp Galois representations have ”unexpectedly small image.” When the abelian variety is an elliptic curve, this result reduces to known facts about gonality of modular curves due to Abramovich and Zograf. The truth of the result is not surprising, but the method of proof is unexpected (at least to us) – the argument uses in a central way new results on expansion in Cayley graphs of linear groups over finite fields due to Helfgott, Gill, PyberSzabo, BreuillardGreenTao, GolsefidyVarju, etc., in combination with analytic results due to Li and Yau. If time permits, we will rephrase the result in terms of the "Bogomolov property" and discuss some arithmetic analogues. The paper can be found at http://arxiv.org/abs/1008.3675 
Tuesday, March 1 
3:00 p.m. Title "The TaylorWiles Method for Coherent Cohomology" Abstract: The TaylorWiles method, and its more elaborate variants due to Faltings, Fujiwara, Diamond, and Kisin, has been used in a variety of situations to prove that padic representations are attached to automorphic forms. The method was developed in the setting of elliptic modular forms, or of automorphic forms on totally definite unitary groups, in order to avoid complications arising from torsion in cohomology. A recent vanishing theorem of Lan and Suh makes it possible to apply the TaylorWiles method to coherent cohomology and padic de Rham and \'etale cohomology of certain Shimura varieties. The method does not yield new modularity results, but it does show that these cohomology groups tend to be free over Hecke algebras, after localization at a nonEisenstein prime. 4:30 p.m. Title: “Presentation of the Iwasawa algebra of Gamma_1 SL_(2, Z_p)” Abstract: It seems to have been assumed that explicit descriptions, by generators and relations, of the Iwasawa alebras of Chevalley groups over Z_p were inaccessible. We will give an explicit, and simple, presentation in the case announced in the title, and discuss applications and a related problem. 
Tuesday, April 12 
3:00 p.m. Title:"Arthur Packets and the Ramanujan Conjecture" Abstract: In this talk we show that under a part of Arthur's Apacket conjecture, locally generic cuspidal automorphic representations of a quasisplit group over a number field are of Ramanujan type, i.e., are tempered at almost all primes. The Apacket conjecture allows us to reduce the problem to a special case of a general local problem which we then solve in general. Our result also gives enough evidence to conjecture that, up to isomorphism, locally generic cuspidal representations are in fact globally generic and conversely. 4:30 p.m. Title: “The interpretation and distribution of cycle integrals of modular functions” Abstract: I will survey some results of two joint works, one with Imamoglu and Toth and one with Friedlander and Iwaniec, about cycle integrals of modular functions. In the first we interpret such integrals as Fourier coefficients of weight 1/2 harmonic Maass forms. These forms can be used to construct explicitly modular "integrals" having rational period functions. In the second we give as an application of general results about Weyl sums an asymptotic formula for such cycle integrals. 
20092010  
September 22 MIT Room 4153 
3:00 p.m. 4:30 p.m. 
October 20 
3:00 p.m. 4:30 p.m. 
November 17 
3:00 p.m. 4:30 p.m. 
3:00 p.m. 4:30 p.m. 

March 9 
3:00 p.m. 4:30 p.m. 
April 13 BC McElroy Conference Room 
3:00 p.m. 4:30 p.m. 