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# Algebraic Geometry and Representation Theory Seminar

For a Nash manifold X and a Nash vector bundle E on X, one can form the topological vector space of Schwartz sections of E, i.e. the smooth sections which decay fast along with all derivatives. It was shown by Aizenbud and Gourevitch, and independently by Luca Prelli, that for a Nash manifold X, th complex of Schwartz sections of the de Rham complex of X has cohomologies isomorphic to the compactly supported cohomologies of X.

In my talk I will present a work in progress, joint with Avraham Aizenbud, to generalize this result to the relative case, replacing the Nash manifold M with a Nash submersion f:M-->N. Using infinity categorical methods, I will define the notion of a Schwartz section of a Nash bundle E over a complex of sheaves with constructible cohomologies, generalizing the notion of Schwartz section on an open semialgebraic set. I will then relate the Schwartz sections of the relative de Rham complex of a Nash submersion f:M-->N with the Schwartz functions on N over the derived push-forward with proper support of the constant sheaf on M. Finally, I will coclude with some applications to the relation between the Schwartz sections of the relative de Rham complex and the topology of the fibers of f.

Zoom meeting: https://weizmann.zoom.us/j/98304397425

Consider the function field $F$ of a smooth curve over $\FF_q$, with $q\neq 2$.

L-functions of automorphic representations of $\GL(2)$ over $F$ are important objects for studying the arithmetic properties of the field $F$. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.

I will present a conceptual proof that the two families coincide, by categorifying the question. This correspondence will necessitate comparing two very different sets of data, which will have significant implications for the representation theory of $\GL(2)$. In particular, we will obtain an exotic symmetric monoidal structure on the category of representations of $\GL(2)$

It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

Zoom meeting: https://weizmann.zoom.us/j/98304397425