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There are two well-known L-functions attached to Siegel modular forms of degree two. These are the Spin and the standard L-function. They have been studied by Andrianov, Böcherer, Shimura, and others. Other L-functions associated to degree two modular forms include the adjoint L-function (degree 10), and two others (degree 14 and degree 16).

The Spin L-function is of particular interest in the cases of paramodular forms of level t and weight k. Examples of paramodular forms can be made using the Gritsenko lift, a generalization of the Saito-Kurokawa lifting. The Gritsenko lift is actually a map defined on Jacobi forms, but it may also be considered a lift of classical newforms $\mathcal{S}_{2k-2}^{new-}(\Gamma_0(t))$, where the space $\mathcal{S}_{2k-2}^{new-}(\Gamma_0(t))$ consists of newforms such that the sign in the functional equation of their L-function is -1. Skoruppa and Zagier defined an isomorphism from the space $J_{k,t}^{cusp, new}$ of cuspidal Jacobi newforms to $\mathcal{S}_{2k-2}^{new-}(\Gamma_0(t))$. So one takes a classical newform in the “minus space,” maps it to the space of Jacobi forms using the inverse of the Skoruppa-Zagier map, then lifts it with the Gritsenko lift to the space of paramodular forms $S_k(K(t))$. We will also call the composition of these two maps the Gritsenko lift.

Euler factors of the Spin L-function for paramodular forms of square-free level t=N were computed by Schmidt using representation theoretic methods. At the time of his paper, the local Langlands correspondence for GSp(4) was unverified, but Gan and Takeda have since proven its existence and so Schmidt’s results hold. He had shown that for $f\in \mathcal{S}_{2k-2}^{new-}(\Gamma_0(N))$, the lifting produces a cusp form $F\in S_k(K(N))$ of degree 2 whose completed Spin L-function is given by

$L(s,F, {\rm spin})=\dfrac{1}{4\pi}\left(s-\dfrac{1}{2}\right)Z\left(s+\dfrac{1}{2}\right)Z\left(s-\dfrac{1}{2}\right)L(s,f)$

where Z is the completed Riemann zeta function. Moreover, this lifting preserves the Atkin-Lehner eigenvalues, i.e., $\eta_pF=\varepsilon_pF$ for every prime p.

Schmidt also determined the local components of the automorphic representation $\pi_f$ associated to a classical newform $f\in\mathcal{S}_{2k-2}^{new-}(\Gamma_0(N))$. Moreover, he explained that the Gritsenko lift and similar lifts are predicted by Langlands functoriality. Schmidt also computed the Euler factors of the Spin L-function for the space of paramodular forms of square-free level that are lifts using representation theoretic methods since the lift is functorial. Schmidt and Roberts later determined the possible Euler factors of any paramodular representation, but precise information for the non-square-free level case remained unknown.

Jacquet and Langlands have shown how to associate classical holomorphic modular forms f to automorphic representations $\pi_f$ of ${\rm GL}(2,\mathbb{A})$. These representations can be written in terms of local components. In this case, the local components, which are infinite dimensional, are complex representations of ${\rm GL}(2,\mathbb{Q}_v)$. Also, the representation is realized in the action of ${\rm GL}(2,\mathbb{A})$ by right translation on a certain space of functions on ${\rm GL}(2,\mathbb{Q})\backslash{\rm GL}(2,\mathbb{A})$.

The classical theory of the passage of modular forms to automorphic representations suggests how it may be extended to Siegel modular forms of higher degree. Spaces of cuspidal Siegel modular forms of degree n are associated to cuspidal automorphic representations of ${\rm GSp}(2n,\mathbb{A})$. These cuspidal automorphic representations $\pi$ can be written in terms of local components $\pi_v$, where v is a place of $\mathbb{Q}$. Many cusp forms are associated to a single such representation $\pi$, but among them is a unique primitive f known as a newform.

In addition to representation theory, modular forms are also related to certain abelian varieties. The nature of this relationship is made precise by the famous Taniyama-Shimura conjecture, proven in 1999.

Taniyama-Shimura Conjecture: Let $E: y^2=x^3+ax+b$ be an elliptic curve with integral coefficients, conductor N, and L-series

$L(s,E)=\sum_{n=1}^\infty\dfrac{a_n}{n^s}.$

Then there is a cusp form of weight 2 and level N which is a Hecke eigenform with Fourier series

$f(\tau)=\sum_{n=1}^\infty a_nq^n,\quad {\rm where}\, q=e^{2\pi i\tau}$

The Langlands philosophy suggests that there should be abelian varieties associated with degree 2 Siegel modular forms. Brumer and Kramer have recently stated a conjectured extension of the Taniyama-Shimura conjecture to the degree 2 case for paramodular forms.

In the GL(2) case, cuspidal eigenforms determine Galois representations as well as automorphic representations. By a theorem of Deligne and Carayol, Galois representations and automorphic representations both have local components which determine each other. This theorem tells us that a modular elliptic curve has the same conductor as the level of the corresponding cusp form.

We would like to do the same thing for the GSp(4) case, but we need to know the corresponding abelian varieties. Unfortunately, these are unknown in general. But there is evidence supporting a conjectured correspondence for the paramodular group K(p) of GSp(4) due to Brumer and Kramer.

The Paramodular Conjecture: There is a one-to-one correspondence between isogeny classes of rational abelian surfaces A of conductor N with ${\rm End}_\mathbb{Q} A = \mathbb{Z}$ and weight 2 newforms f on K(N) with rational eigenvalues, not in the span of the Gritsenko lifts, such that L(A,s) = L(f,s). The $\ell$-adic representations associated to f should be isomorphic to those of the Tate module of A for any $\ell$ prime to N.

The paramodular conjecture is consistent with known examples from Brumer and Kramer and also in Poor and Yuen’s work on weight 2 Siegel paramodular forms. There are many open problems related to this conjecture, such as translating the different actions one can perform on abelian varieties to actions on spaces of paramodular forms.

More generally, Yoshida conjectured that for any rational abelian surface, there exists a discrete subgroup $\Gamma'$ of ${\rm Sp}(4,\mathbb{Q})$ and a degree 2 Siegel modular form of weight 2, $f\in\mathcal{M}_2(\Gamma')$, with the same L-series.

References
A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, arXiv:1004.4699v2 (2010).

C. Poor and D. Yuen, Paramodular Cusp Forms, arXiv:0912.0049v1 (2009).

H. Yoshida, Siegel modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193-248.

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 $N$.