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.

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Consider the Lie group {\rm Sp}(6,\mathbb{R}) defined as

{\rm Sp}(6,\mathbb{R}) = \{g\in {\rm GL}(6,\mathbb{R}): {}^tgJg=J \},

where

J=\begin{bmatrix}  &&&&&1\\  &&&&1&\\  &&&1&&\\  &&-1&&&\\  &-1&&&&\\  -1&&&&&\\  \end{bmatrix}.

Its Lie algebra sp(6,\mathbb{R}) is the set of all 6\times 6 matrices X with real entries such that

{}^tXJ+JX=0.

A general element of sp(6,\mathbb{R}) is of the form

\begin{bmatrix}   a & b & c & d & e & f\\  g & h & i & j & k & e\\  l & m & n & o & j & d\\  p & q & r & -n & -i & -c\\  s & t & q & -m & -h & -b\\  u & s & p & -l & -g & -a\\  \end{bmatrix}

and so sp(6,\mathbb{R}) is a 21-dimensional real vector space. We would like to describe its root system.

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Let C be a smooth projective variety of genus g\leq3 over \mathbb{Q}. Arithmetic information about the curve is encoded in its L-function L(C,s). The conjectures of Birch and Swinnerton-Dyer about elliptic curves over \mathbb{Q} were generalized to arbitrary abelian varieties over number fields by John Tate.

In the case of hyperelliptic curves (genus 2 curves) over \mathbb{Q}, 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

\lim_{s\rightarrow1}(s-1)^{-r}L(J,s)=\Omega\cdot{\rm Reg}\cdot\prod_{p}c_p\cdot\# Ш (J,\mathbb{Q})\cdot(\# J(\mathbb{Q})_{\rm tors})^{-2}

where L(J,s) is the L-series of J and r is its analytic rank. \Omega denotes the integral over J(\mathbb{R}) of a certain differential 2-form, Reg is the regulator of J(\mathbb{Q}), c_p = \# J(\mathbb{Q}_p)/J^0(\mathbb{Q}) is the Tamagawa number, Ш(J,\mathbb{Q}) is the Tate-Shafarevich group of J over \mathbb{Q}, and J(\mathbb{Q})_{tors} is the torsion subgroup of J(\mathbb{Q}). Here, J^0(\mathbb{Q}) is the subgroup of the Jacobian isomorphic to \mathcal{J}^0(\mathbb{Z}_p), where \mathcal{J}^0 is the open subgroup scheme of the closed fiber of the Néron model of J over \mathbb{Z}_p

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

L(C,s)=\prod_p L_p(p^{-s})^{-1}=\sum_{n=1}^\infty a_n n^{-s}.

If C has good reduction at the prime p, the factor in the Euler product at this prime is determined by a polynomial L_p(T) of degree 4. It appears in the local zeta function of the curve over the finite field \mathbb{F}_p of order p.

Z(C/\mathbb{F}_p;T)=exp\left(\sum_{k=1}^\infty N_kT^k/k\right)=\dfrac{L_p(T)}{(1-T)(1-pT)},

where N_k is the number of \mathbb{F}_{q^k}-points on C.

By a theorem of Weil, the polynomial L_p(T)=\sum_{n=0}^4 a_nT^n, can be determined by counting points on C only over \mathbb{F}_p and \mathbb{F}_{p^2} since the coefficients must satisfy a_0=1, a_3=pa_1, and a_4=p^2. 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: y^2=x^6 + 4x^5 + 4x^4 + 2x^3 + 1
Conductor: 249
Discriminant: 261095424 = 2^{20} \cdot 3 \cdot 83
Local L-factor at p=7: L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+8\cdot 7^{1-3 s}+44\cdot     7^{-2 s}+8\cdot 7^{-s}+1}

Curve: y^2=x^6 + 2x^5 + 3x^4 + 4x^3 - x^2 - 2x + 1
Conductor: 277
Discriminant: 290455552 = 2^{20} \cdot 277
Local L-factor at p=7: L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+6\cdot 7^{1-3 s}+54\cdot     7^{-2 s}+6\cdot 7^{-s}+1}

Curve: y^2=x^6 - 2x^3 - 4x^2 + 1
Conductor: 295
Discriminant: 309329920 = 2^{20} \cdot 5 \cdot 59
Local L-factor at p=7: L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+6\cdot 7^{1-3 s}+48\cdot     7^{-2 s}+6\cdot 7^{-s}+1}

Curve: y^2=x^6 - 2x^5 + 3x^4 - x^2 - 2x + 1
Conductor: 349
Discriminant: 365953024 = 2^{20} \cdot 349
Local L-factor at p=7: L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+9\cdot 7^{1-3 s}+69\cdot     7^{-2 s}+9\cdot 7^{-s}+1}

Curve: y^2=x^6 + 2x^5 + 5x^4 + 2x^3 + 2x^2 + 1
Conductor: 353
Discriminant: 370147328 = 2^{20} \cdot 353
Local L-factor at p=7: L_7(7^{-s})^{-1}=\dfrac{1}{7^{2-4 s}+7^{2-3     s}+7^{1-s}+37\cdot 7^{-2 s}+1}

If one were to attempt to find an automorphic representation \pi=\otimes\pi_p 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 \Pi({\rm St}\otimes 1), \Pi(\xi{\rm St}\otimes 1), \Pi({\rm St}\otimes{\rm St}):

\Pi({\rm St}\otimes 1) : L_p(s,\Pi_p)^{-1} = (1-p^{-s-1/2})^2(1-p^{-s+1/2}),
\Pi(\xi{\rm St}\otimes 1) : L_p(s,\Pi_p)^{-1} = (1-p^{-s-1/2})(1-p^{-s+1/2})(1+p^{-s-1/2}),
\Pi({\rm St}\otimes{\rm St}) : L_p(s,\Pi_p)^{-1} = (1-p^{-s-1/2})^2.

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 G be a connected reductive group over a nonarchimedean field k and let \Gamma<\Gamma_1 be congruences subgroup of G(k). Let \pi be an irreducible smooth admissible representation of G. Suppose \pi is supercuspidal and \pi^{\Gamma_1}\neq 0.

Then \pi|_\Gamma contains a cuspidal representation \rho of \Gamma/\Gamma_1.

Therefore \pi contains an extension of \rho to Z\Gamma, \tilde{\rho}. So {\rm Hom}_{Z\Gamma}(\tilde{\rho},\pi|_{Z\Gamma})\neq 0. By Frobenius reciprocity,

{\rm Hom}_G({\rm ind}_{Z\Gamma}^G \tilde{\rho},\pi)\cong {\rm Hom}_{Z\Gamma}(\tilde{\rho},\pi|_{Z\Gamma}).

Since {\rm ind}_{Z\Gamma}^G\tilde{\rho} is irreducible, it must be isomorphic to \pi.

Now consider the decomposition of the induced representation into irreducible components

{\rm ind}_{Z\Gamma}^G\tilde{\rho}|_{\Gamma_1}=\oplus_{x\in Z\Gamma\backslash G/\Gamma_1} {\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1} (\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1}).

where Z\Gamma^x=x^{-1}Z\Gamma x and \tilde{\rho}^x is the obvious representation of Z\Gamma^x (transport \tilde{\rho} through conjugation by x, i.e., \tilde{\rho}^x(x^{-1}hx) = \tilde{\rho}(h) for h\in Z\Gamma.)

When does {\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1} (\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1}) contain the trivial representation of \Gamma_1? Equivalent questions are

When is {\rm Hom}_{\Gamma_1}({\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1}(\tilde{\rho}^x), \textbf{1}_{\Gamma_1})\neq 0?

When is {\rm Hom}_{Z\Gamma^x\cap\Gamma_1}(\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1},\textbf{1}_{Z\Gamma^x\cap\Gamma_1})\neq 0?

One can show x\in Z\Gamma by considering {\rm ind}_{Z\Gamma}^G\tilde{\rho}|_{\Gamma_1}. If this contains the trivial representation, then {\rm ind}_{Z\Gamma}^G\tilde{\rho}|_\Gamma contains an irreducible representation, say \tau, such that \tau_{\Gamma_1}\supset\textbf{1}_{\Gamma_1}.

This implies that \tau is trivial on \Gamma_1. So \pi contains \rho and \tau. (A weaker approach is to use explicit representatives for Z\Gamma\backslash G/\Gamma_1.)

General theory implies that \rho and \tau intertwine, i.e., there exist x\in G such that

{\rm Hom}_{\Gamma^x\cap\Gamma}(\rho^x,\tau)\neq 0.

This implies x\in Z\Gamma. Using cuspidality of \rho plus reps for \Gamma\backslash G/\Gamma, we have that \rho\cong\tau.

It follows that \pi^{\Gamma_1}=\tilde{\rho}|_{\Gamma_1}. In particular,

{\rm dim}\, \pi^{\Gamma_1}={\rm dim}\, \tilde{\rho}={\rm dim}\, \rho.

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 \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.

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.

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Let G={\rm GL}(n,\mathbb{F}_q) and let N be the nilpotent radical of the standard Borel subgroup, i.e., N is the subgroup of G consisting of upper triangular matrices with 1′s on the diagonal. Given a non-trivial character \psi: \mathbb{F}_q\rightarrow \mathbb{C}^\times, we define a character \psi_N of N by

\psi_N( \begin{pmatrix}  1 & x_{12} & x_{13} & \cdots & x_{1n}\\  & 1 & x_{23} & \cdots & x_{2n}\\  & & 1 & \cdots & \vdots \\  & & & \ddots & \vdots\\  & & & & 1\\  \end{pmatrix}) = \psi(x_{12}+x_{23}+\cdots+x_{n-1, n}).

This defines a one-dimensional representation of N. One can show that the induced representation \mathcal{G}={\rm Ind}_N^G(\psi_N) is multiplicity-free, i.e., when we decompose \mathcal{G}={\rm Ind}_N^G(\psi_N) into a sum of irreducible constituents,

\mathcal{G}={\rm Ind}_N^G(\psi_N)=\oplus a_i\pi_i,

we have a_1=1 for all i. Each constituent \pi_i is called a generic irreducible representation. There are many reasons for calling them generic. One is that no matter which non-trivial character \psi 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 q and, once simplified, do not depend on the character \psi. Moreover, this Gelfand-Graev representation \mathcal{G} contains most irreducible representations of the group.

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

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I have been reading “Number Theory I: Fermat’s Dream” by Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito. In Chapter 1: Rational Points on Elliptic Curves, the authors state two propositions:

Proposition 1.1. There is no integral solution (x,y,z) to x^4 + y^4 = z^4 satisfying xyz \neq 0.

and

Proposition 1.2. The only rational solutions to y^2=x^3-x are
(x,y)=(0,0) and (\pm1,0).

They then show how 1.2 implies 1.1:

Suppose (x,y,z) is an integral solution to x^4 + y^4 = z^4 satisfying xyz \neq 0. We rewrite the equation as

x^4 = z^4 - y^4

We then multiply each side by \dfrac{z^2}{y^6} to get

\dfrac{x^4z^2}{y^6} = \dfrac{z^6}{y^6} - \dfrac{z^2}{y^2}
\left(\dfrac{x^2z}{y^3}\right)^2=\left(\dfrac{z^2}{y^2}\right)^3-\left(\dfrac{z^2}{y^2}\right).

Setting Y=\dfrac{x^2z}{y^3},\, X=\dfrac{z^2}{y^2}, we have
Y^2=X^3-X.

Note that X,Y are rational numbers. The only rational points of this elliptic curve are (0,0), (\pm1,0). In either case, 0=Y=\dfrac{x^2z}{y^3} implies either x=0 or z=0, contradicting our assumption. So there are only trivial integral solutions (x,y,z) to x^4 + y^4 = z^4.

We can extend this argument to prove that there are no non-trivial integral solutions to x^n + y^n = z^n, for n\equiv 0\, ({\rm mod}\, 4).

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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}.

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