Automorphic Forms and Arithmetic

Let $C$ be a close rigid annulus of radii $r < r' \in \mathbb{Q}_{\geq 0}$ over a complete discrete valuation field with algebraically closed residue field of positive characteristic.
The Swan conductor function ${\rm sw}_{\mathcal{F}}: [r, r'] \to \mathbb{Q}$ of an étale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on $C$, ramified at most at a finite set of rigid points of $C$, is continuous, convex and piece-wise linear, outside the radii of the ramification points of $\mathcal{F}$, and has finitely many slopes, all integers. Moreover, the change of slopes between radii $t$ and $t'$ is the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t'$.
I will present the construction of ${\rm sw}_{\mathcal{F}}$ as well as projectivity and rationality properties of some nearby cycles that play an important part in establishing the aforementioned properties of ${\rm sw}_{\mathcal{F}}$, with an emphasis on convexity.

In joint works with Gehrmann, Guitart and Masdeu we proposed and numerically computed plectic generalizations of Stark-Heegner (or Darmon) points. Moreover, inspired by Nekovar and Scholl's conjectures, we predicted the algebraicity of plectic points and their significance for Mordell-Weil groups of elliptic curves of large rank.
In this talk I will report on joint work with Lennart Gehrmann where we establish the algebraicity and the arithmetic significance of plectic points of "rank two" over certain biquadratic extensions.

This is a report on two related projects, the first with Gan and Sawin, the second with Ciubotaru, on the local properties of the Langlands parametrization constructed by V. Lafforgue. Let $F = k((t))$ where $k$ is the field of $q$ elements, of characteristic $p$, and let $G$ be a connected reductive group over $F$. To any irreducible admissible representation $\pi$ of $G(F)$, Genestier and Lafforgue assign a semisimple Weil group parameter $\phi_\pi$ with coefficients in the $L$-group of $G$. We say that $\pi$ is pure if the Frobenius eigenvalues of $\phi_\pi$ (composed with any linear representation of the $L$-group) are Weil $q$ numbers of the same weight. Suppose $G$ is a split reductive group. For $p$ outside a small list of exceptional primes, depending only on $G$, Gan, Sawin, and I prove the following results, among others:
1. If $\pi$ is a pure supercuspidal representation of $G$ then $\phi_\pi$ is a ramified parameter.
2. If $\pi$ is any discrete series representation of $G$ then the semisimple parameter $\phi_\pi$ can be completed to a Weil-Deligne parameter that has integral Frobenius weights and satisfies purity of the monodromy weight filtration.
Point 2 depends in general on a lemma of Beuzart-Plessis that has not yet been written down. Using this lemma, Ciubotaru and I have proved the following result. Let now $G$ be a semisimple group over a global function field $K$, and let $\Pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $w$ and $v$ (satisfying a mild restriction on the residue field cardinality) such that $\Pi_w$ is discrete series and $\pi_v$ is unramified and generic. Then $\Pi$ is tempered at all unramified places.

The motivic action conjecture of Venkatesh posits that for a Hecke eigensystem $h$, there is a natural action of the motivic cohomology of the adjoint motive associated to $h$ on the $h$-isotypic part of the rational cohomology of locally symmetric spaces. This explains and relates the arithmetic of various appearances of the same Hecke eigensystem in different degrees of cohomology. Although such multiple occurrences mostly happen when there can be no discrete series, they sometimes happen for groups with discrete series; a prototypical example is a weight one modular form appearing in both $H^{0}$ and $H^{1}$ of the modular curve. We formulate the analogous conjecture for coherent cohomology of Shimura varieties and explain how archimedean $L$-packets enter into the story. This suggests that there should be a hidden archimedean symmetry of coherent cohomology of Shimura varieties.

Let p be a prime. A special case of the Fontaine-Mazur conjecture predicts that 2-dimensional absolutely irreducible geometric Galois representations appearing in the completed cohomology of the modular curves come from eigenforms (up to twist). This was proved by Emerton in the regular case using the p-adic local Langlands correspondence for GL2(Qp). In this talk, I plan to explain a different proof of Emerton's result. Our approach is purely geometric. One main ingredient is a construction of some differential operators on the modular curves with infinite level at p which essentially agree with the nilpotent operator in Fontaine's classification of almost de Rham representations.

We will interpolate a version of the cycle appearing in the arithmetic Gan-Gross-Prasad conjecture in Hida families, which constructs a generalization of Howard's big Heegner point. We will then indicate how this can be used to derive rank 0 and 1 results in Iwasawa theory.

Kodaira-Spencer isomorphisms on classical modular curves relate differentials on the curve to those on the universal elliptic curve over it. I'll present a generalization that describes dualizing sheaves of integral models of Hilbert modular varieties with Iwahori level at p. I'll also describe properties of the associated degeneracy maps to prime-to-p level, including a relative cohomological vanishing result, with applications to the construction and properties of Hecke operators at p, mod p Galois representations, and integral models of Hilbert modular varieties.

Given a family of K3 surfaces over a curve, I will explain several results pertaining to the density (Zariski, analytic...) of the locus where the geometric Picard rank jumps. These results are formulated in the general framework of GSpin Shimura varieties and have avatars for abelian surfaces, cubic fourfolds... I will also discuss applications to the Hecke orbit conjecture and the construction of rational curves on K3's.
The results in this talk are joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.