# Closing: Akshay Venkatesh :::{.remark} Automorphic forms using TFQTs as a metaphor. We'll consider $\TQFT_4 = [(\Bord^3, \disjoint), (\Vect\slice \CC, \tensor_\CC)]$ where $\Bord^3$ is the category whose objects are 3-manifolds and morphisms $M\to N$ are 4-manifolds $W$ with $\bd W = M \disjoint N$. These are meant to extract invariants of 4-manifolds that are amenable to cut-and-paste arguments. The correspondences: - Manifolds $M \leadsto$ vector spaces $A_M$, - Bordisms $(M\to N)\leadsto$ linear maps $A_M\to A_N$, - A 4-manifold $Z$ with boundary $M \leadsto$ a vector $v_Z \in A_M$ - A 4-manifold $Z$ with empty boundary $\leadsto \lambda_Z\in \CC$ - A decomposition $X$ without boundary into $M\disjoint_Z N \leadsto$ vector $\ell\in A_M, r\in A_M\dual$, where $\inp{\ell}{r} = \lambda_Z$. ::: :::{.example title="A $\TQFT_2$ due to Dijkgraaf-Witten"} Fix $G\in\Fin\Grp$, and a correspondence - $S^1 \leadsto \CC[G]^{\mathrm{conj} }$, conjugacy-invariant functions in the group algebra, - Pants $\leadsto$ multiplication - $\Sigma_g$ a genus $g$ surface $\leadsto {1\over k!} N$ where $N$ is the number of ways to write $e\in G$ as a product of $g$ commutators. ::: :::{.remark} An informal definition of **extended TQFTs**, in particular $\TQFT_4$: - 4-manifolds $\leadsto \CC$ - 3-manifolds $\leadsto \Vect\slice \CC$ - 2-manifolds $\leadsto$ categories enriched over $\Vect\slice \CC$ This should yield - 3-manifolds without boundary $\leadsto A_S$ objects in $\cat C$ - $X = M\disjoint_Z N \leadsto \Hom(A_M, A_N)$ ::: :::{.remark} Last time we said $\spec \ZZ\invert p$ is like a 3-manifold with boundary $\QQpadic$, which is like a 2-manifold. A philosophy is to put all places on the same footing -- note that we haven't included the places at $p$ and $\infty$ here, so really we should have $\bd \spec \ZZ\invert p = \spec \RR, \spec \QQpadic$, and $\bd \spec \ZZ = \spec \RR$. So our new picture should be: (todo finish) ::: ## Automorphic Forms as $\TQFT_4$ :::{.remark} What should the automorphic form correspondence be in this analogy? Our 3-dimensional objects: - $\ZZ \leadsto \mca_\ZZ$ functions on $\dcosetl{G_\ZZ}{G_\RR}$, - $\ZZ\invert p\leadsto \mca_{\ZZ\invert p}$ functions on $\dcosetl{?}{G_\RR\cartpower{2}}$, - $X\slice {\FF_p}$ a smooth projective curve $\leadsto$ functions on $\Bun_G(X)$, - $\ZZpadic \leadsto$ something more difficult. The 2-dimensional objects: - $\QQpadic \leadsto \Rep G(\QQpadic)$, - $\RR \leadsto \Rep G(\RR)$, - $X\slice{\bar \FF_p}$ a smooth projective curve $\leadsto \Sh(\Bun_G(X))$. Really for a TQFT, we should assign something to objects in the category of its boundary -- here e.g. the vector space $\mca_{\ZZ}$ is an object in $\Rep G(\QQpadic)$. Idea: functions on $\Bun_G$ are hard to deal with, e.g. Hecke operators turn into infinite sums. Make things robust to passing to algebraic closures by passing from functions to sheaves! ::: :::{.remark} Last time: thinking of $\spec \ZZ$ as a 3-manifold, \[ \spec \ZZ = \spec \ZZpadic \glue{\spec \QQpadic} \spec \ZZ\invert p \approx M^3\glue{Z^2}N^3 .\] We want the following: \[ \Hom_{G(\QQpadic)}(\mca_{\ZZpadic}, \mca_{\ZZ\invert p}) =_? \mca_{\ZZ} .\] We regard $\mca_\ZZ$ as elements of $\mca_{\ZZ\invert p}$ which are also unramified at $p$, and by Frobenius reciprocity this should yield \[ \mca_\ZZ = \Hom_{G(\QQpadic)} \qty{\ts{ \text{Functions on } \dcosetr{ G(\QQpadic)}{G(\ZZpadic)} , \mca_{\ZZ\invert p} }} ,\] which should encode a Hecke algebra at $p$. ::: :::{.question} What is the Langlands correspondence in this language? What should an "arithmetic TQFT" be? ::: :::{.remark} Let $O$ be a category of arithmetic ring and $A_O$ be a category or vector space, and call an associated $O\to A_O$ an *arithmetic field theory*. Let $X\slice{\FF_p}$ be a smooth projective curve and $G=\GL_n$. The Langlands correspondence here (due to Drinfeld and Lafforgue) yields \[ \correspond{ \text{Cuspidal functions on $n\dash$dimensional} \\ \text{vector bundles on } X } &\mapstofrom \correspond{ \text{Functions on $n\dash$dimensional} \\ \text{irreducible Galois reps} }\\ T_x\actson x &\mapstofrom \Frob_x \actson ? \] We regard the LHS $A$ as "automorphic forms". This suggests the following viewpoint on the Langlands correspondence: we are only seeing one level, and there is a *second* arithmetic field theory $B^{(G\dual)}$ built out of Galois representations of the Langlands dual $G\dual$, so $\ZZ$ yields a vector space and $\QQpadic$ yields a category, and an equivalence of arithmetic field theories $A^{(G)} \mapstofrom B^{(G\dual)}$. Often $B$ is a category of coherent sheaves. This should package local, global, and geometric Langlands into a single theory! ::: :::{.remark} The abstract correspondence between automorphic forms and Galois reps isn't so useful; the real utility comes from matching structures and numerical invariants on both sides, e.g. Fourier coefficients, Rankin-Selberg or doubling integrals, the $\Theta$ correspondence, etc which all match with something on the Galois side (usually an \(L\dash \)function). This yields a panoply of matching invariants! E.g. for $E/\QQ$ an elliptic curve, \[ L(\Sym^2 E, 1) = \prod_p {p^2 \over (1-1/p) \size E(\FF_{p^2})} \in \pi\cdot \mathrm{Area}(E_\CC) \QQ ,\] where the area is of the fundamental parallelogram of $E$, which is hard to prove without automorphic forms. How can we interpret this in terms of TQFTs? ::: :::{.remark} Consider numerical invariants of automorphic forms and Galois reps landing in $\CC$. Let $\OO$ be a 3-dimensional ring of integers over $X$. The numerical invariants of Galois reps should be elements of $B_\OO^{(G\dual)}$, and numerical invariants of automorphic forms should come from $A_\OO^{(G)}$ where given $P$, one takes $\phi\mapsto \inp{P}{\phi}$. To find matching invariants, we want to match elements in $A_\OO$ to elements in $B_\OO$. More ambitiously, we can ask for matching *boundary conditions* in $A^{(G)}$ and $B^{(G\dual)}$. ::: :::{.definition title="Boundary conditions, informal definition"} A **boundary condition** in $\TQFT_4$ is a coherent assignment: - 3-manifolds $M \leadsto v\in A_M$ a distinguished vector - 2-manifolds $S \leadsto X_S$ a distinguished object in $A_S$ > See Kasputin's 2010 ICM address for a nice overview. ::: :::{.remark} Joint work with David Ben-Zvi, Sakellaridis, an informal summary: - A variety $G\dash$variety gives a boundary condition for both $A^{(G)}$ and $B^{(G\dual)}$, - For suitable choice of $Y$, this recovers familiar invariants of automorphic forms mentioned above, - On the Galois side this recovers \(L\dash \)functions, - There is a proposed specific class of dual pairs $(G, Y) \mapstofrom (G\dual, Y\dual)$ which give matching/dual invariants. E.g. each periodic integral should have a dual. ::: :::{.question} For the next generation of number theorists: why are there such similarities between TQFTs and automorphic forms? This is something deep that we barely understand at all. ::: :::{.remark} Extending to 1-dimensional objects: these should be 2-categories which are categorical reps of a loop group. :::