Automorphic Forms and Arithmetic

abstract : In his thesis, M. Bhargava proved parameterizations and identified local conditions which he used to give asymptotic counts for $S_4$ quartic and quintic number fields, ordered by discriminant. This talk will discuss results in an ongoing project to add detail to Bhargava's work by considering in addition to the field discriminant, the lattice shape of the ring of integers in the canonical embedding, and by giving strong rates with lower order terms in the asymptotics. These results build on earlier work of Taniguchi-Thorne, Bhargava-Shankar-Tsimerman and Bhargava Harron.

abstract : in a joint work in progress with Eric Urban, given a cohomological cuspidal representation $\pi$ on $GL(N)$ over a CM field, we construct a map from the fundamental group of the simplicial deformation ring of $\overline{\rho}_\pi$ to the dual adjoint Selmer group of $\rho_\pi$. In the case of Hida families, these objects still make sense but we conjecture that the corresponding simplicial deformation ring has no higher homotopy groups, so that our map vanish. We investigate evidences and consequences of this conjecture.

abstract: I will discuss a theory of endoscopy for certain symmetric spaces associated to unitary groups, the main result being the fundamental lemma for the "Lie algebra". This is motivated by the study of certain periods of automorphic forms on unitary groups with applications to arithmetic. After explaining where the fundamental lemma fits into this broader picture, I will describe its proof.

abstract: Tate and Voloch conjectured that, in an abelian variety over a p-adic field, torsion points outside a closed subvariety cannot be p-adically too close to it. There is a well-known analogy between torsion points in abelian varieties and CM points in Shimura varieties. In this talk, I will prove the analog of the Tate-Voloch conjecture for ordinary CM points in a product of Siegel spaces. I will focus on how the theory of perfectoid spaces is applied.

abstract: Every smooth proper algebraic variety over a p-adic field is expected to have semistable model after passing to a finite extension. This conjecture is open in general, but its analogue for Galois representations, the p-adic monodromy theorem, is known. In this talk, we will explain a generalization of this theorem to etale local systems on a smooth rigid analytic variety.

Abstract: In this talk, we will first summarize the final outcome we (= Liu-Tian-Xiao-Zhang-Zhu) obtained on bounding Selmer groups of Rankin-Selberg motives. Then we will explain how Galois deformation can help to give an explicit computation of certain local Galois cohomology, which is a key step toward the Selmer groups. We will also explain a related question, which has been partially answered by us, on automatic minimality of deformations of automorphic Galois representations.

Abstract: In his work on modularity theorems, Wiles proved a numericalv criterion for a map of rings R->T to be an isomorphism of complete intersections. In addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function. In this talk I will consider the case of a Hecke algebra acting on the cohomology a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of rings R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles's numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect") at a newform f which gives rise to an augmentation T -> Z_p. The defect turns out to be determined entirely by local information at the primes q dividing the discriminant of the quaternion algebra at which the mod p representationarising from f is ``trivial''. (For instance if f corresponds to a semistable elliptic curve, then the local defect at q is related to the ``tame regulator'' of the Tate period of the elliptic curve at q.) This is joint work with Gebhard Boeckle and Jeffrey Manning.

Abstract: We will first give an introduction to the formalism of (p,q)-forms and currents on non-archimedean spaces, as defined by Chambert-Loir--Ducros and Gubler--Künnemann. We explain how a notion of Greens current can be used to reinterpret intersection numbers on p-adic formal schemes in their generic fiber; at least in some cases. We finally apply these considerations to an intersection problem of quadratic CM-cycles on Lubin-Tate space. Here, all expressions in question can be evaluated neatly at infinite level and we recover an intersection number formula from Qirui Li.

Abstract: Skinner and Urban have been able to constuct nontrivial elements in the Selmer group for the Galois representation attached to a certain type of cuspidal automorphic representation of a unitary group, under the hypothesis that the L-function of this representation vanishes at its central critical value. This is predicted by the Bloch--Kato conjecture. We will explain an analogous construction for the symmetric cube of the Galois representation attached to a cuspidal eigenform, assuming the same kind of vanishing hypothesis. The method will involve studying Eisenstein series on the exceptional group G_2 and, in the absense of a G_2 Shimura variety, making a careful analysis of the representation theory of this group at the archimedean place.