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Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive $M$ of Artin-Tate type to a connected reductive group $G$ over a field $k$. The motive $M$ and its $L$-function are particularly useful in computing some adèlic integrals that occur in the trace formula. Moreover, the motive contains some other encoded information about the group. For example, if $k=\mathbb{F}_q$ is a finite field, then the twisted dual motive $M^\vee(1)$ yields a formula for the order of $G(\mathbb{F}_q)$, as shown by Steinberg.

If $k$ is a local field with characteristic 0, then the $L$-function $L(M)$ is finite if and only if Serre’s Euler-Poincaré measure $\mu_G$ on $G(k)$ is non-zero. Also, there is a local functional equation relating the $L$-function of $M$ to that of its twisted dual motive.

The following is taken from notes distributed by Matthew Morrow at the Midwest Number Theory Conference for Graduate Students and Recent PhD’s 2010:

A two-dimensional local field F is a complete discrete valuation field whose residue field is a local field.

Some examples of two-dimensional local fields are

(i) For any local field $K$, consider $F=K((t))$. This is complete under the $t$-adic valuation and has residue field $K$.

(ii) Let $F$ be the $p$-adic completion of $\mathbb{Z}_p((t))$. $F$ is a characteristic zero complete DVR in which $p$ is prime, and its residue field is $\mathbb{F}_p((t))$.

Suppose that $X$ is a two-dimensional scheme of finite type and let $x\in X$ be a closed point and $y\subset X$ an irreducible curve containing $x$. Let $A=\mathcal{O}_{X,x}$ and let $\mathfrak{p}$ be the height one prime in $A$ which is the local equation of $y$ at $x$. Consider the following sequence of localisations and completions:

$A \rightsquigarrow \hat{A} \rightsquigarrow \hat{A}_{\mathfrak{p}\hat{A}} \rightsquigarrow \widehat{\hat{A}_{\mathfrak{p}\hat{A}}} \rightsquigarrow {\rm Frac}\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}=: F_{x,y}$

It follows from the excellence of $A$ that $\mathfrak{p}\hat{A}$ is a radical ideal of the completion $\hat{A}$. We then localise and complete at $\mathfrak{p}\hat{A}$ and again use excellence to deduce that 0 is a radical ideal in the resulting ring, i.e., $\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}$ is reduced. The total field of fractions $F_{x,y}$ is therefore isomorphic to a finite direct sum of fields, and each is a two-dimensional local field.

The two-dimensional adelic philosophy, originally due to A. Parshin, is that we should study $X$ via the family of two-dimensional local fields $(F_{x,y})_{x\in y\subset X}$.

A useful approach to finding dimensions of spaces of Siegel cusp forms is to investigate the representation theory of GSp(2n). We can translate results in representation theory to results on spaces of cusp forms and vice-versa. As observed by Harish-Chandra, cuspidal representations are the building blocks for the representation theory of certain groups in a way analogous to the construction of Eisenstein series from cusp forms. More precisely, a cusp form $f\in S_k(\Gamma(N))$ gives rise to a cuspidal automorphic representation $(\pi,V)$ of ${\rm GSp}(2n,\mathbb{A})$ and vice-versa. These cuspidal automorphic representations can be written in terms of local components. The local components of the automorphic representation in turn give rise to local components of the cusp form. The dimensions of these spaces tell us essentially how many choices we have for the local factors of the representation and therefore the number of choices of local cusp forms.

Let $F$ be a non-archimedean local field of characteristic zero with ring of integers $\mathfrak{o}$ and maximal ideal $\mathfrak{p}$ such that $\mathfrak{o}/\mathfrak{p}$ is isomorphic to $\mathbb{F}_q$, the finite field of order $q=p^n$ for $p$ an odd prime. We consider the group ${\rm GSp}(4,F)$ and hence Siegel modular forms of degree 2. By the properties of our field $F$ we have ${\rm GSp}(4,\mathfrak{o}/\mathfrak{p})\cong {\rm GSp}(4,\mathbb{F}_q)$. We define the congruence subgroup of level $\mathfrak{p}^n$, denoted by $\Gamma(\mathfrak{p}^n)$, by

$\Gamma(\mathfrak{p}^n)=\{g\in {\rm GSp}(4, \mathfrak{o})\, :\, g\equiv I\, ({\rm mod}\, \mathfrak{p}^n)\}$

For the maximal compact subgroup $K = {\rm GSp}(4,\mathfrak{o})$ and an admissible representation $(\pi, V)$ of ${\rm GSp}(4, F)$, $K$ acts on the space $V^{\Gamma(\mathfrak{p})}$ of vectors in $V$ fixed by the action of the congruence subgroup $\Gamma(\mathfrak{p})$. This space is finite dimensional by the admissibility of the representation. By definition, $\Gamma(\mathfrak{p})$ acts trivially on this space and so we have a more interesting action of the group $K/\Gamma(\mathfrak{p})\cong{\rm GSp}(4,\mathfrak{o}/\mathfrak{p})\cong{\rm GSp}(4,\mathbb{F}_q)$. We can then determine the dimension of $V^{\Gamma(\mathfrak{p})}$ by looking at the finite group analogue of $\pi$.

An investigation of this finite group analogue yields information that is then translated to the language of modular forms. We can then obtain results such as the dimension of a space of cusp forms. We have these results for all such local fields $F$ that satisfy the conditions above. Our local method can then be used for global results on cusp forms, such as dimension formulas for the space $S_k(\Gamma(N))$ of Siegel cusp forms on the principal congruence subgroup of odd square-free level $N$.