Enumerative Arithmetic Geometry and Automorphic forms

This is part of the talk given by Tony Feng (MIT).

The talk starts with giving basic background about theta series.

• Theta series are generating functions for counting lattice vectors.

Example: $$\theta(q)=\sum_{\lambda\in Z^4}q^{{|\lambda|}^2}=\sum_n \#\{n=a^2+b^2+c^2+d^2\}q^n$$
where $\#\{n=a^2+b^2+c^2+d^2\}$ is number of elements of the given set. It can be computed using Siegel-Weil formula: $$\#\{n=a^2+b^2+c^2+d^2\}=8\sum_{d'|n \ and \ 4\nmid d' } d'$$
Consider $n=2$ then RHS is $24$, for LHS let’s denote
$a^2+b^2+c^2+d^2$ as $(a,b,c,d)$. Then for LHS we have the following possible combinations: $(1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1,0), (0,1,0,1), (0,0,1,1).$ Same thing can be repeated with all $-1's$ instead of $1$, combinations of $1$ and $-1$. Hence, we get $24$ combinations for LHS.

Theta series over function fields
Notation:

• $X$: smooth projective curve over a finite field $\mathbb {F_q}$
• $\nu$: $X'\rightarrow X$ an $\acute{e}$tale double cover with non-trivial automorphism $\sigma$.
• $\mathcal{F}$: vector bundle on $X'$
• $h$: $\mathcal{F}\xrightarrow{\sim} \mathcal{\sigma*}\mathcal{F}^{\vee}$(conjugate dual) a hermitian form.

Let Bun$_{U(n)}$ is the moduli space classifying a rank $n$ vector bundle $\mathcal{F}$ on $X'$ and hermitian structure $h$: $\mathcal{F}\xrightarrow{\sim} \mathcal{\sigma*}\mathcal{F}^{\vee}$ such that ${\sigma*}h^{\vee}=h$

And the connection between number theory and geometry comes from Weil’s Uniformization theorem.

Arithmetic theta series: generating functions of “special cycles” in ambient moduli space
So we have the following:
$$\theta(q)=\sum_n\#Z(n)q^n$$ where $Z(n)$=$\{\lambda\in \Lambda:Q(\lambda)=n\}$ and for arithmetic theta series: $$\Theta(q)=\sum_n[\mathcal{Z}(n)]q^n$$
where $\mathcal{Z}(n)\subset$ Sht$_{n}^{1}$ (moduli space of shtukas)
From this we have arithmetic Siegel-Weil:$$\int\Theta(z)=\frac{d}{ds}|_{s=0}Eis(z,s)$$
Rest of the talk is about moduli of shtuka, definition of special cycles mentioned earlier, higher theta series, and contruction of virtual fundamental class.