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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 , where the space 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 of cuspidal Jacobi newforms to . 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 . 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 , the lifting produces a cusp form of degree 2 whose completed Spin *L*-function is given by

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

Schmidt also determined the local components of the automorphic representation associated to a classical newform . 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.

Consider the Lie group defined as

where

Its Lie algebra is the set of all matrices with real entries such that

A general element of is of the form

and so is a 21-dimensional real vector space. We would like to describe its root system.

Let be a smooth projective variety of genus over . Arithmetic information about the curve is encoded in its *L*-function . The conjectures of Birch and Swinnerton-Dyer about elliptic curves over were generalized to arbitrary abelian varieties over number fields by John Tate.

In the case of hyperelliptic curves (genus 2 curves) over , the first conjecture is that the order of vanishing of the *L*-function of the Jacobian at *s=1* (the analytic rank) is equal to the Mordell-Weil rank of the Jacobian. The second is

Ш

where is the *L*-series of *J* and *r* is its analytic rank. denotes the integral over of a certain differential 2-form, Reg is the regulator of , is the Tamagawa number, Ш is the Tate-Shafarevich group of *J* over , and is the torsion subgroup of . Here, is the subgroup of the Jacobian isomorphic to , where is the open subgroup scheme of the closed fiber of the Néron model of *J* over

The *L*-series of the curve *C* is given as both an Euler product and a Dirichlet series.

If *C* has good reduction at the prime *p*, the factor in the Euler product at this prime is determined by a polynomial of degree 4. It appears in the local zeta function of the curve over the finite field of order *p*.

where is the number of -points on *C*.

By a theorem of Weil, the polynomial , can be determined by counting points on *C* only over and since the coefficients must satisfy and . We consider five special hyperelliptic curves (those associated to paramodular forms taken from Brumer and Kramer’s paper) and compute their discriminant and local *L*-factor at *p*=7.

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

Curve:

Conductor:

Discriminant:

Local *L*-factor at *p*=7:

If one were to attempt to find an automorphic representation that could be associated to these varieties, one could use this information about the local L-factors to rule out possible local components. For example, Schmidt determined the local *L*-factors of the representations , , :

:

:

:

Comparing these with the *L*-factors of the curves we found at p=7, we can say that the local component at p=7 of an associated automorphic representation is not one of these.

**References**

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

E. Flynn, F. Leprévost, et. al., *Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves*, Math. Comp. **70** (2001), no. 236, 1675-1697

K. Kedlaya and A. Sutherland, *Computing L-series of hyperelliptic curves*, Algorithmic number theory, 312–326, *Lecture Notes in Comput. Sci.*, **5011**, Springer, Berlin, 2008.

R. Schmidt, *On classical Saito-Kurokawa liftings*, J. Reine Angew. Math. 604 (2007), 211-236.

J. Tate, *On the conjectures of Birch and Swinnerton-Dyer and a geometric analog*, Séminaire Bourbaki, **306** 1965/1966.

A. Weil, *Number of solutions of equations in finite fields*, Bull. Amer. Math. Soc. **55**, (1949), 497-508.

One of the motivations of studying the Langlands Conjectures is the question

What are the finite extensions of the rational numbers?

The local Langlands Conjectures give a dictionary between the Galois Theory of local fields and the representation theory of locally compact reductive groups.

The following videos are of the first lecture by Dick Gross in a series of lectures given at Columbia University in Fall 2011 about the Langlands Conjectures where he explains this in more detail.

Let be a connected reductive group over a nonarchimedean field and let be congruences subgroup of . Let be an irreducible smooth admissible representation of . Suppose is supercuspidal and .

Then contains a cuspidal representation of .

Therefore contains an extension of to , . So . By Frobenius reciprocity,

Since is irreducible, it must be isomorphic to .

Now consider the decomposition of the induced representation into irreducible components

where and is the obvious representation of (transport through conjugation by , *i.e.*, for .)

When does contain the trivial representation of ? Equivalent questions are

When is ?

When is ?

One can show by considering . If this contains the trivial representation, then contains an irreducible representation, say , such that .

This implies that is trivial on . So contains and . (A weaker approach is to use explicit representatives for .)

General theory implies that and intertwine, i.e., there exist such that

This implies . Using cuspidality of plus reps for , we have that .

It follows that . In particular,

**References**

L. Morris, *Tamely ramified supercuspidal representations*, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 5, 639–667.

A. Moy and G. Prasad, *Unrefined minimal K-types for p-adic groups*, Invent. Math. 116 (1994), no. 1-3, 393–408.

Jacquet and Langlands have shown how to associate classical holomorphic modular forms *f* to automorphic representations of . These representations can be written in terms of local components. In this case, the local components, which are infinite dimensional, are complex representations of . Also, the representation is realized in the action of by right translation on a certain space of functions on .

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 . These cuspidal automorphic representations can be written in terms of local components , where *v* is a place of . Many cusp forms are associated to a single such representation , 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 be an elliptic curve with integral coefficients, conductor *N*, and *L*-series

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

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 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 -adic representations associated to *f* should be isomorphic to those of the Tate module of *A* for any 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 of and a degree 2 Siegel modular form of weight 2, , 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.

Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive of Artin-Tate type to a connected reductive group over a field . The motive and its -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 is a finite field, then the twisted dual motive yields a formula for the order of , as shown by Steinberg.

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

Let and let be the nilpotent radical of the standard Borel subgroup, i.e., is the subgroup of consisting of upper triangular matrices with 1’s on the diagonal. Given a non-trivial character , we define a character of by

This defines a one-dimensional representation of . One can show that the induced representation is multiplicity-free, i.e., when we decompose into a sum of irreducible constituents,

we have for all . Each constituent is called a *generic* irreducible representation. There are many reasons for calling them *generic*. One is that no matter which non-trivial character we choose, we obtain the same induced representation up to equivalence. One can see this by looking at the induced character values on each conjugacy class. These values will be polynomials in and, once simplified, do not depend on the character . Moreover, this *Gelfand-Graev representation* contains *most* irreducible representations of the group.

To see this, we can compare the number of irreducible constituents of with the number of irreducible representations of . By a standard result in the representation theory of finite groups, the number of irreducible representations of is equal to its number of conjugacy classes.

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 gives rise to a cuspidal automorphic representation of 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 be a non-archimedean local field of characteristic zero with ring of integers and maximal ideal such that is isomorphic to , the finite field of order for an odd prime. We consider the group and hence Siegel modular forms of degree 2. By the properties of our field we have . We define the congruence subgroup of level , denoted by , by

For the maximal compact subgroup and an admissible representation of , acts on the space of vectors in fixed by the action of the congruence subgroup . This space is finite dimensional by the admissibility of the representation. By definition, acts trivially on this space and so we have a more interesting action of the group . We can then determine the dimension of by looking at the finite group analogue of .

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 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 of Siegel cusp forms on the principal congruence subgroup of odd square-free level .