# Harvard Number Theory Seminar, Fall 2018

### Wednesdays 3-4pm at Science Center 507

Date Speaker Title
Sep 12 Daniel Kriz (MIT) A new p-adic Maass-Shimura operator and supersingular Rankin-Selberg p-adic L-functions
Sep 19 Ben Howard (Boston College) Moduli spaces of shtukas and a higher derivative Gross-Kohnen-Zagier formula
Sep 26 Frank Thorne (Univ. of South Carolina/Tufts) Error Terms in Arithmetic Statistics
Oct 3 Lynnelle Ye (Harvard) Geometry of eigenvarieties for definite unitary groups over the boundary of weight space
Oct 10 Max Menzies (Harvard) The p-curvature and Bost's Conjecture for the Gauss-Manin connection on non-abelian cohomology
Oct 17 Sean Howe (Stanford) A unipotent circle action on p-adic modular forms
Oct 24 Wei Ho (UMichigan) Integral points on elliptic curves
Oct 31 Chris Skinner (Princeton) Some recent results on Euler systems
Nov 7 Florian Herzig (UToronto) Ordinary representations and locally analytic vectors for GL_n(Q_p)
Nov 14 Eric Urban (Columbia) Eisenstein congruences and Euler systems
Nov 21 No seminar (Thanksgiving)
Nov 28 John Bergall (Bryn Mawr College) Upper bounds for constant slope p-adic families of modular forms
Dec 5 Sol Friedberg (Boston College) Langlands functoriality, the converse theorem, and integral representations of L-functions

### Organizers

Chi-Yun Hsu and Alison Miller

### Abstracts

Speaker: Daniel Kriz (MIT)
Abstract: We introduce a new p-adic Maass-Shimura operator acting on a space of "generalized p-adic modular forms" (extending Katz's notion of p-adic modular forms), defined on the p-adic (preperfectoid) universal cover of a Shimura curve. Using this operator, we construct new p-adic L-functions in the style of Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang for Rankin-Selberg families over imaginary quadratic fields K, in the case where p is inert or ramified in K. We also establish new p-adic Waldspurger formulas, relating p-adic logarithms of elliptic units and Heegner points to special values of these p-adic L-functions.

Speaker: Ben Howard (Boston College)
Title: Moduli spaces of shtukas and a higher derivative Gross-Kohnen-Zagier formula
Abstract: The Gross-Zagier formula expresses the Neron-Tate height of a Heegner point as the derivative of an L-function, while the Gross-Kohnen-Zagier formula expresses the Neron-Tate pairing of two different Heegner points as the product of the derivative of an L-function and a period integral. Yun and Zhang have proved a higher derivative version of the Gross-Zagier theorem, but with Heeger points on modular curves replaced by Heegner-Drinfeld cycles on moduli spaces of PGL(2)-Shtukas. I will describe a higher derivative version of Gross-Kohnen-Zagier for these Heegner-Drinfeld cycles. This is joint work with A. Shnidman.

Speaker: Frank Thorne (Univ. of South Carolina/Tufts)
Title: Error Terms in Arithmetic Statistics
Abstract: Recent years have seen tremendous advances in arithmetic statistics: counting results for number field discriminants, class group torsion, and other arithmetic objects. In many cases, these can be expressed in terms of lattice point counting problems, often in prehomogeneous or coregular vector spaces. I will give an overview of some work on the following question: how to obtain the best possible error terms, especially when "local conditions" are imposed? Although sieve methods lead to some interesting applications, in this talk I will concentrate on the algebraic side of the story. Aiming to prove optimal error terms has revealed some interesting structure in prehomogeneous vector spaces, and it has posed several questions which I will discuss. This is joint work with Takashi Taniguchi.

Speaker: Lynnelle Ye (Harvard)
Title: Geometry of eigenvarieties for definite unitary groups over the boundary of weight space
Abstract: The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid space whose points are in bijection with $p$-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will state a structure theorem about Chenevier's eigenvarieties for definite unitary groups which generalizes slope bounds of Liu-Wan-Xiao for dimension $2$ to all dimensions, and discuss, time permitting, the ideas of the proof and a geometric consequence.

Speaker: Max Menzies (Harvard)
Title: The p-curvature and Bost's Conjecture for the Gauss-Manin connection on non-abelian cohomology
Abstract: I'll begin with Bost's generalization of the p-curvature conjecture, and describe the classical geometric concepts at play such as the horizontal subbundle corresponding to a connection and parallel transport. This naturally motivates the discovery of the Gauss-Manin connection on algebraic de Rham cohomology, and its non-abelian analogue due to Simpson. I'll state Katz's theorem that the p-curvature conjecture (equivalently Bost) holds for the abelian Gauss-Manin connection, and the ingredients to even make that statement, such as the Hodge filtration, conjugate filtration, and Kodaira-Spencer map. I'll then define non-abelian analogues of these objects, and state a theorem which suitably equates them. This is the non-abelian analogue of part 1 of Katz's theorem, and is progress towards proving Bost for the non-abelian Gauss-Manin connection.

Speaker: Sean Howe (Stanford)
Title: A unipotent circle action on p-adic modular forms
Abstract: The horizontal translation action of the real line on the complex upper half plane descends to an action of the circle group S^1 on the "unstable locus", or image of Im \tau > 1, in the complex modular curve. In this talk, we explain an analogous action of the p-adic formal multiplicative group on the Katz moduli problem over the p-adic ordinary locus, whose ring of functions is the ring of p-adic modular forms. The analogy is richer than one might expect, and leads to new perspectives on classical notions in the p-adic theory of modular curves and modular forms such as Dwork's equation \tau=\log q and Hida's space of ordinary p-adic modular forms.

Speaker: Wei Ho (UMichigan)
Title: Integral points on elliptic curves
Abstract: We show that the second moment for the number of integral points on elliptic curves over Q is bounded. The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The results on moments then follow from Holder’s inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families. This is joint work with Levent Alpoge.

Speaker: Chris Skinner (Princeton)
Title: Some recent results on Euler systems
Abstract: Euler systems remain one of the best tools for relating Selmer groups to L-values and especially for making progress on special value formulas, at least whenever a related Euler system exists. In this talk we describe some recent work on Euler systems, which includes some new results in the anticyclotomic setting as well as some new constructions of Euler systems in the cyclotomic setting.

Speaker: Florian Herzig (UToronto)
Title: Ordinary representations and locally analytic vectors for GL_n(Q_p)
Abstract: Suppose that rho is an irreducible automorphic n-dimensional global p-adic Galois representation that is upper-triangular locally at p. In previous work with Breuil we constructed a unitary representation of GL_n(Q_p) on a p-adic Banach space (depending only on rho locally at p) that is an extension of finitely many principal series, and we conjectured that this representation occurs globally in a space of p-adic automorphic forms cut out by rho. Together with C. Breuil we prove many new cases of this conjecture, assuming that rho is moreover crystalline. If time permits, I'll also discuss how we can construct certain amalgams inside the locally analytic vectors cut out by rho (even when rho is not ordinary).

Speaker: Eric Urban (Columbia)
Title: Eisenstein congruences and Euler systems
Abstract: It is now well-known since the work of Mazur-Wiles that one can find lower bound for the size of Selmer groups by studying ordinary Eisenstein congruences. On the other hand, upper bound have been mainly obtained thanks to the use of Euler systems constructed from special units (cyclotomic, elliptic or Siegel units) or Heegner points. In this talk, I will present a general and new construction of Euler systems using Eisenstein congruences. I will stress on the key aspects of the construction and as an example, I will show that this method recovers the Euler system of cyclotomic units.

Speaker: John Bergdall (Bryn Mawr College)
Title: Upper bounds for constant slope p-adic families of modular forms
Abstract: This talk is concerned with the radius of convergence of p-adic families of modular forms --- q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the nineties predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa--Mazur prediction was false. It has since remained open question how to salvage it. Here we will present some recent theoretical results towards such a salvage, backed up by numerical data.

Speaker: Sol Friedberg (Boston College)
Title: Langlands functoriality, the converse theorem, and integral representations of L-functions
Abstract: Langlands functoriality predicts maps between automorphic forms on different groups, dictated by a map of L-groups. One important class of such maps are endoscopic liftings, established by Arthur using the trace formula. In this talk I describe an approach to endoscopic lifting that does not use the trace formula. Instead it relies on the converse theorem of Cogdell and Piatetski-Shapiro and on new integral representations of L-functions of Cai, Friedberg, Ginzburg and Kaplan.