Abstract: The function c(x) defined on rationals x = h/k with (h, k) = 1 and k > 0 by

satisfies a remarkable reciprocity formula. This function arises in conjunction with the Nyman-Beurling approach to the Riemann Hypothesis.

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 zeta-function.

Steven D. Miller (Rutgers)
Title: "Fourier Coefficients of Automorphic Forms on Exceptional Groups"

Abstract: Fourier coefficients of automorphic forms on classical groups are used in a wide variety of contexts, e.g., integral representations of L-functions and multiplicity one theorems. I'll discuss some recent work with Wilfried Schmid on decay estimates for these unipotent periods, with applications to L-functions. 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)

Dihua Jiang (Minnesota)
Title:  "Constructions of Cuspidal Automorphic Forms for Classical Groups"

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 Rallis-Soudry and the classical theta correspondences. It is work in progress with Ginzburg and Soudry.

Wenzhi Luo (Ohio State)
Title: "Asymptotic Variance for the Linnik Distribution"

Abstract: It is well-known 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 half-integral 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 L-function. 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 Jacquet-Langlands correspondence and Yoshida lift. This talk is partly based on my joint work with Z. Rudnick and P. Sarnak.

3:00 - 4:00 p.m.
Kannan Soundararajan (Stanford)
Title: Moments and the distribution of values of L-functions

Abstract: I will discuss work on the value distribution and moments of families of L-functions. 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 L-values here have a different flavor with Selberg's theorem on log-normality being a representative result. I will discuss here work on upper and lower bounds for moments of L-functions, 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 pm
Samit Dasgupta (UC Santa Cruz)
Title: On the p-adic L-functions of totally real fields.

Abstract: We present a new construction of the p-adic L-functions associated to ray class characters of totally real fields. We define a certain measure-valued 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 p-adic L-function. As a corollary of our construction and a result of Spiess, we prove that the order of vanishing at s=0 of the p-adic L-function 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.

Wen-Ching Winnie Li (Penn State)
Title: Recent progress on noncongruence modular forms

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 Swinnerton-Dyer in 1971, who proposed 3-term p-adic congruence relations replacing the usual 3-term recursive relations satisfied by eigenforms of the Hecke operator at p.In 1985 Scholl attached to noncongruence forms a family of Galois representations, and establishedp-adic 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.

Alex Kontorovich (Yale)
Title: On Zaremba's Conjecture

Abstract: It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year 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 "low-lying" 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.

Tuesday, September 21
MIT - Room 56-114
Map of MIT

3:00 p.m.
Kai-Wen Lan - Princeton/IAS

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.
Michael Rapoport - Bonn

Title: "The Langlands-Kottwitz method for the zeta function, beyond the parahoric case"

Abstract: The LK-method 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
BC - 9 Lake Street, Room 035 (north of Commonwealth Avenue near the "B" line)
Note room change 
Directions

3:00 p.m.
Eyal Goren - McGill

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.
Pierre Colmez - Jussieu

Title: “On the p-adic local Langlands correspondence for GL2”

Tuesday, November 16
MIT - Room 56-114
Map of MIT

3:00 p.m.
Brian Smithling - Toronto

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 p-component 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 Rapoport-Zink 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.
Matt Baker - Georgia Tech

Title: “Tropical and Berkovich analytic curves”

Abstract: We will discuss the relationship between a Berkovich analytic curve over a complete and algebraically closed non-Archimedean 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
BC - Campion 009
Note change of room
Directions

3:00 p.m.
Amanda Folsom - Yale

Title: "ell-adic 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 ell-adically to 0.  Little is known about the ell-adic behavior of these sequences for primes ell > 11.  We show that modulo powers of primes ell at least 5, these sequences of generating functions ell-adically converge to linear combinations of a(n explicitly given) finite number of special q-series.  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.
Jordan Ellenberg - Wisconsin

Title: "Expander graphs, gonality, and Galois representations"

Abstract: (joint work with Chris Hall and Emmanuel Kowalski) We show that 1-parameter families of abelian varieties over a number field K have few fibers over bounded-degree extensions of K whose mod-p 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, Pyber-Szabo, Breuillard-Green-Tao, Golsefidy-Varju, 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
MIT - Room 4-159
Map of MIT

3:00 p.m.
Michael Harris - Jussieu

Title "The Taylor-Wiles Method for Coherent Cohomology"

Abstract: The Taylor-Wiles method, and its more elaborate variants due to Faltings, Fujiwara, Diamond, and Kisin, has been used in a variety of situations to prove that p-adic 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 Taylor-Wiles method to coherent cohomology and p-adic 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 non-Eisenstein prime.

4:30 p.m.
Laurent Clozel (Orsay)

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
BC - Campion 009
Note change of room
Directions

3:00 p.m.
Freydoon Shahidi - Purdue

Title:"Arthur Packets and the Ramanujan Conjecture"

Abstract:  In this talk we show that under a part of Arthur's A-packet 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 A-packet 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.
William Duke - UCLA

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.

3:00 p.m.
Yiannis Sakellaridis - University of Toronto
"A 'relative' Langlands program and periods of automorphic forms"

4:30 p.m.
Matthew Emerton - Northwestern University
"p-adically completed cohomology and the p-adic Langlands program" 

October 20
BC
McElroy Conference Room

3:00 p.m.
Ze'ev Rudnick - Tel-Aviv University and IAS
"Statistics of the zeros of zeta functions over a function field" 

4:30 p.m.
Haruzo Hida - UCLA
"Characterization of abelian components of the 'big' Hecke algebra" 

November 17
MIT
Room 4-153

3:00 p.m.
Akshay Venkatesh - Stanford University
"Torsion in the homology of arithmetic groups"

4:30 p.m.
Ken Ono - University of Wisconsin
"p-adic coupling of harmonic Maass forms" 

February 9
BC
McGuinn Hall 334

3:00 p.m.
Gautam Chinta - CUNY
"Orthogonal periods of Eisenstein series" 

4:30 p.m.
Mihran Papikian - Pennsylvania State University
"On the arithmetic of modular varieties of D-elliptic sheaves" 

March 9
MIT
Room 4-145

3:00 p.m.
Elena Mantovan - Caltech
"l-adic etale cohomology of PEL Shimura varieties with non-trivial coefficients" 

4:30 p.m.
Karl Rubin - UC Irvine
"Selmer ranks of elliptic curves in families of quadratic twists" 

3:00 p.m.
Shou-Wu Zhang - Columbia University
"Calabi-Yau theorem and algebraic dynamics" 

4:30 p.m.
Ching-Li Chai - University of Pennsylvania
"CM lifting of abelian varieties" 

September 23
MIT
3-4 p.m.
Room 4-163

4-6 p.m.
Room 4-149

3:00 p.m.
Wee Teck Gan - UC San Diego
"Towards a Gross-Prasad Conjecture for A-Packets"

4:30 p.m.
Daniel Bump - Stanford
"Metaplectic Whittaker Functions and Crystal Bases

October 28
BC
3-6 p.m.
McElroy Conference Room

3:00 p.m.
Steve Kudla - Toronto
"Arithmetic Cycles for Unitary Groups" 

4:30 p.m.
Chris Skinner - Princeton
"Some Remarks on p-adic Galois Representations for
GL (2) and Other Groups" 

November 18
BC
3-6 p.m.
McElroy Conference Room

3:00 p.m.
Henri Darmon - McGill
"On the Gross-Stark Conjecture" 

4:30 p.m.
Peter Sarnak - Princeton
"Recent Progress on the QUE Conjecture" 

February 17
MIT
3-6 p.m.
Room 4-270

3:00 p.m.
Brooke Feigon - Toronto
"Unitary periods" 

4:30 p.m.
Kartik Prasanna - Maryland
"Heegner cycles, p-adic L-functions and rational points"

March 17
BC
3-6 p.m.
McElroy Conference Room

3:00 p.m.
Dorian Goldfeld - Columbia
"Symmetry types of higher rank Rankin-Selberg L-functions" 

4:30 p.m.
Brian Conrad - Stanford
"Pseudo-reductive groups" 

3:00 p.m.
Matt Papanikolas - Texas A&M
"Periods and logarithms of Drinfeld modules and algebraic independence"

4:30 p.m.
Dinakar Ramakrishnan - Caltech
"Hyperbolic 3-manifolds of arithmetic type, S^1 fibrations, and modular forms on quaternion algebras"