Automorphic Forms and Arithmetic

In the Langlands program, theta functions are used as integral kernels to construct automorphic forms through theta lifting, and the Rallis inner product formula relates the inner product of two theta lifts to special values of L-functions. In the Kudla program, arithmetic theta functions with values in Chow groups are used to construct algebraic cycles through arithmetic theta lifting, while the arithmetic inner product formula relates the Beilinson–Bloch height pairing of two arithmetic theta lifts to the central derivative of an L-function and could be used to give evidence for the Beilinson–Bloch conjecture.

In this talk, we propose a cohomological framework that could be used to give a uniform construction of arithmetic theta functions and of motivic theta functions with values in higher Chow groups. These motivic theta functions are used to define motivic theta lifting, which provides a new way to construct motivic classes beyond those arising from modular units. The resulting motivic theta lifts satisfy an identity relating their Beilinson regulators to non-critical L-values. This identity can be viewed as a Hodge–Deligne analogue of the Rallis inner product formula and provides evidence for Beilinson’s conjectures.

I will explain joint work in progress with Matt Emerton and Kari Vilonen in which we conjecture a Galois-theoretic formula for the cohomology of a congruence locally symmetric space in terms of categorical local Langlands at every place. This generalises a conjecture of Emerton and Xinwen Zhu in the case of Shimura varieties. The new ingredient is a prediction for the contribution from categorical Langlands at Archimedean places (generalising the Shimura datum), which turns out to be a small K-type for the dual group in the sense of Vogan. In the talk, I’ll explain the general shape of the conjecture, why small K-types appear, and give some evidence for the conjecture.

We discuss the construction of an Euler system for the (split) triple product, that is, the p-adic Galois representation V = V_1 x V_2 x V_3, where V_i is the Galois representation associated to a cuspform f_i of weight k_i. Applications of this construction include new cases of the Bloch—Kato conjecture: for example, in the case that f_i corresponds to an elliptic curve E_i over \Q and if \chi denotes a non-trivial Dirichlet character (satisfying some technical conditions), we prove that the non-vanishing of L(E_1 x E_2 x E_3 x \chi, s) at s=2 implies the vanishing of the relevant Selmer group for V. This complements the Bloch—Kato results of Yifeng Liu (for the twisted triple product) and Haining Wang (for the case E_1=E_2=E_3 and \chi=1). Our Euler system incorporates the cyclotomic variation not seen by the Euler system “base class” defined by Darmon—Rotger using diagonal cycles. We use the pullback method pioneered by Sangiovanni—Skinner in their construction of an Euler system for the adjoint of a modular form. The main technical step is the construction of distinguished Klingen—Eisenstein classes in the cohomology of the Siegel threefold. (This talk will not assume prior knowledge of Euler systems.)