In recent work in collaboration with A. Iovita and G. Stevens we provided an anlogue of the classical Eichler-Shimura isomorphism, defining an explicit, Hecke and Galois equivariant map from finite slope families of modular symbols to finite slope families of overconvergent modular forms. The key ingredient is the construction of a new integral structure on the invariant differentials of the universal elliptic curve (in mixed characteristic 0-p) via the theory of the canonical subgroup. Recently we showed that such structure can be refined to an integral structure on the full de Rham cohomology. We hope that this will allow to define Eichler-Shimura isomorphsims in the crystalline context as well.
The following is joint work with David Pollack. Several decades ago, Glenn Stevens introduced a way of studying the p-adic variation of geometric automorphic forms using large modules of p-adic distributions. Glenn and I showed that (in the generality of reductive Q-groups split at p) there exists a lift of any given classical form f to a p-adic form if f satisfies a numerical criterion of "non-criticality." These forms are actually homology classes. More recently, David and Robert Pollack showed how to compute explicit representatives of these lifts, in the ordinary case, for GL(3). David and I have now figured out how to do this in the non-ordinary non-critical case. In addition, we have a way of computing explicitly the germ of the eigenvariety at a given point. In practice, these computations are huge, and of course can only be carried out to a desired approximation. We have some hope that we may be able to compute the tangent vector mod p to an eigenvariety and compare it with what is happening on the Galois side, in some examples.
In this talk I'll discuss recent progresses in the theory of modular forms modulo p.
In this talk we will explain how to attach Galois representations to certain Hecke eigenforms in the coherent cohomology of the special fibers of good reduction Shimura varieties. As an application we will give a new proof of cases of a recent theorem of Scholze attaching Galois representations to certain torsion classes in the Betti cohomology of arithmetic locally symmetric spaces. The main tool will be some "generalized Hasse invariants" on the Ekedahl-Oort strata of the special fiber of the Shimura variety.
A few years ago, Howard's construction of big Heegner points in Hida families was extended by Longo and Vigni to a general quaternionic setting. In this talk, we will relate the higher weight specializations of their construction in the definite case to certain theta elements studied in recent work of Chida and Hsieh. Our proof (joint with Matteo Longo) is notably inspired by some of the ideas in the celebrated work of Greenberg--Stevens. If time permits, we will also explain the applications of the resulting two-variable p-adic L-function to the study (ongoing joint work with Matteo Longo and Chan-Ho Kim) of the variation of anticyclotomic Iwasawa invariants in Hida families.
This is a joint work with Francois Brunault. In this talk, we will discuss a result related to Beilinson's conjecture for Rankin-Selberg products of higher weight modular forms. In particular, we shall give an explicit formula for the complex regulator of an element in motivic cohomology, which is expressed by non-critical values of Rankin-Selberg L-functions. This is a partial generalization of works of Baba-Sreekantan and Bertolini-Darmon-Rotger.
This is a report on joint work in progress with Michael Spiess on the following conjecture of Gross. Let F be a totally real number field of degree n, and let L/K/F be a tower of field extensions with G=Gal(L/F) finite abelian. Let Θ denote the usual theta element encoding the values of the partial zeta functions of L/F at s=0, appropriately smoothed to lie in Z[G]. Let I denote the augmentation ideal of Z[G] relative to Z[Gal(K/F)]. Finally, let r denote the number of places of F that ramify in L but split completely in K. Then Gross conjectures that Θ lies in Ir. We describe an attack on this conjecture when K is totally complex using the Eisenstein cocycle. The Eisenstein cocycle was first defined for n=2 by Glenn Stevens, who used the cocycle to study partial zeta values of real quadratic fields. Stevens's idea that partial zeta values are cohomological in nature lies at the heart of our approach.
In this talk, we discuss a construction of certain families of modular forms on unitary groups of various signatures. This involves the construction of a p-adic family of Eisenstein series for unitary groups of signature (n,n), as well as the application of certain p-adic differential operators. We will also briefly discuss applications to the construction of p-adic L-functions and beyond.
I'll explain the construction of an Euler system of zeta elements, living in Galois cohomology sheaves over the eigencurve, which recovers Kato's Euler system at classical points. This naturally leads to a "two-variable algebraic p-adic L-function" which provably satisfies one divisibility in a "two-variable main conjecture" over the eigencurve. I'll also explain how these results imply the equality of the analytic (Pollack-Stevens/Bellaiche) and algebraic (Kato/Perrin-Riou) p-adic L-functions associated with a critical-slope refinement of a modular form. The key ingredients are a careful study of etale cohomology of modular curves with Glenn's distributions as coefficients, and the construction of Chern class maps from a limit of K-groups to the Galois cohomology of such modules.
I will describe the general structure of our construction of p-adic L-functions attached to families of ordinary holomorphic modular forms on Shimura varieties attached to unitary groups. The complex L-function is studied by means of the doubling method; its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein measure. This is work in progress with Eischen, Li, and Skinner.
Using a patching module without fixed level structure at primes dividing p (as in recent work of Caraiani, Emerton, Gee, Geraghty, Paskunas and Shin) one can construct some analogue of an eigenvariety. We show that this "patched" eigenvariety agrees (up to taking a product with some open polydisc) with a union of irreducible components of a space of trianguline representations. This has some consequences for eigenvarieties for unitary groups as well as some applications to Breuil's recent work on the locally analytic socle of some GLn representations. This is joint work with C. Breuil and B. Schraen.
This is a joint work with D. Burns and T. Sano. I plan to begin with an interpretation of Rubin-Stark elements in the refined Stark conjecture of Rubin by zeta elements whose existence is predicted by the ETNC. I talk about new properties of Rubin-Stark elements, and describe higher Fitting ideals of certain cohomology groups and class groups for general abelian extensions of number fields. With this theory I discuss several conjectures related to L-values, including a recent refined class number formula of Mazur-Rubin and Sano, and several conjectures by Gross, especially conjecture for tori, etc.
Kolyvagin's method of Euler systems, originally used to prove the BSD conjecture for certain elliptic curves of analytic rank ≤ 1, has proved to be an extremely fruitful idea in number theory. Conjectures due to Perrin-Riou and Fukaya--Kato predict the existence of Euler systems for a very wide class of Galois representations: these predicted Euler systems are collections of elements in the top wedge powers of global Galois cohomology groups.
Recently, some new constructions of Euler systems have appeared which interestingly do not seem to fit these conjectures: e.g. for the Rankin convolution of two modular forms, the conjecture predicts classes in the wedge square of a cohomology group (a "rank 2" Euler system), but recent work of Lei, Zerbes and myself -- building on earlier constructions of Beilinson, Flach, and Bertolini--Darmon--Rotger -- gives an Euler system living in the cohomology group itself and not its wedge square (a "rank 1" Euler system).
I will explain an extension of Perrin-Riou's conjecture, predicting the existence of not one but potentially several Euler systems, of different ranks, for a motivic Galois representation; and I will describe how this conjecture naturally predicts the existence of rank 1 Euler systems in several settings which seem to be accessible to explicit constructions. (This is joint work with Sarah Zerbes.)
We are going to discuss a new cohomology theory for a suitable class of Shimura varieties, conditional on a certain geometric conjecture. Its applications include, among others, a construction of new Euler systems. This is a joint work with Tony Scholl.
The goal of this talk is to reveal hidden structures on the Betti cohomology and the period integral of a smooth projective hypersurface X in terms of BV(Batalin-Vilkovisky) algebras and homotopy Lie theory. We construct an explicit BV algebra which is quasi-isomorphic to the middle-dimensional primitive cohomology of X, lift the Hodge filtration and the polarization to the BV algebra, and enhance the Griffiths period integral to a BV algebra morphism. As an application, we provide an explicit algorithm to compute the period matrix of a deformed hypersurface and the Gauss-Manin connection. This is a joint work with Jae-Suk Park.
In this talk I will describe the circle of ideas appearing in recent joint works with Massimo Bertolini, Henri Darmon and Alan Lauder, where we construct p-adic families of cohomology classes associated to Beilinson-Flach elements and diagonal cycles, and show that their images under suitable regulator maps are related to special values of the p-adic L-function attached to the convolution of two (resp. three) Hida families of modular forms. Specialisations of these classes taking values in the Tate module of an elliptic curve twisted by an Artin representation are shown to be cristalline at p if and only if the associated classical central critical value vanishes. This leads to the proof of new cases of BSD in rank 0, and provides an effective algorithm -via p-adic Coleman iterated integrals- for computing non-torsion points or canonical Selmer classes when the rank is 1 or 2.
We explain how to combine the work of Ash Stevens and the Teitelbaum Poisson kernel formula in order to compute Colmez primitives attached to modular forms on a Mumford curve in terms of p-adic integration. On the other hand, we introduce an operation of "taking the derivative of a family of measure" and express the Coleman group cohomological cycle obtained by means of the Coleman primitives in terms of this operation. It should be mentioned that the work is also based on ideas of Bertolini, Darmon and Iovita.
Let f be an eigenform on Γ1(N) ∩ Γ0(p). If f is non-critical then the work of Amice and Velu provides us with a p-adic L-function Lp(f,s) interpolating the critical values of f. Thanks to the work of Pollack-Stevens and Bellaiche we now have a natural notion of Lp(f,s) even when f has critical slope. In this talk we show how to use the Shintani modular symbol, which parameterizes the special values of L-functions of real quadratic fields, to compute Lp(f,s) when f is a critical slope (or "evil twin") Eisenstein series. Our main result is a factorization formula for Lp(f,s), similar to a formula for its ordinary twin, which was recently proved by Bellaiche and Dasgupta using different techniques. This gives a new proof of a conjecture of Pasol and Stevens.
A classical conjecture in the Langlands program asserts that coherent cohomology classes on Shimura varieties that are eigen for the action of the Hecke operators should give rise to Galois representations. For classes in H0 it is a consequence of the p-adic interpolation by multiplication of powers of a lifting of Hasse invariant. We present here a method to tackle the case of higher cohomology groups, building on the recent work of Scholze on Shimura varieties at infinite level and the new Hasse invariants defined at this level.
In a joint work with H. Hida, we study the relation between the level of the largest congruence subgroup contained in the image of the Galois representation associated to a "general" Hida family and the congruence ideals (defining the schematic intersection) of the irreducible component defining this family and other "special" irreducible components.
I will discuss the construction of p-adic distributions attached to certain periods of automorphic forms and state some conjectures and results relative to them. If times allows, I will describe expected consequences of the latter to Bloch-Kato type conjectures.
I will explain a technique for estimating average values Rankin-Selberg L-functions for GL(2) via spectral decompositions of certain shifted convolution sums. Such a technique can be used to determine the nonvanishing of certain moments of central values of interest to the Iwasawa main conjectures (along the lines of previous works of Greenberg, Rohrlich, Vatsal, and Cornut), allowing e.g. for an analytic study of the heights of CM points on Shimura curves. The technique applies rather more generally than this though. For instance, it also suggests some interesting avenues for the study of central values of automorphic L-functions of certain higher-rank groups, which I will describe if time permits.
We prove the compatibility of local and global Langlands correspondences for GLn up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne. More precisely, let rp(π) denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation π of GLn(AF). We show that the restriction of rp(π) to the decomposition group of a place v not dividing p of F corresponds up to semisimplification to rec(πv), the image of πv under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of rp(π) at v is `bounded by' the monodromy of rec(πv).
To prove the above, we use the fact that the representations rp(π) are constructed as a p-adic limit of representations for which local-global compatibility is already known. We are able to p-adically interpolate the traces of these representations (as well as their exterior powers), which allows us to establish the above results. If time permits, we will discuss how this argument may be modified to study the Galois representations constructed by Scholze at primes away from p.
I explain how one can use the Euler system of generalized Beilinson-Flach elements, introduced in recent joint work of Lei-Loeffler-Zerbes, to prove one inclusion of the Iwasawa main conjecture for a modular form over the Zp2-extension of an imaginary quadratic field. Consequences of this result include cases of the finiteness of the p-part of Tate-Shafarevich groups. This is joint work with Guido Kings and David Loeffler.