Hyperelliptic curves, local L-factors, and automorphic representations

February 26, 2012 in algebraic geometry, L-functions, representation theory | Tags: , | Leave a comment

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. MR 1 610977.

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

Lectures on the Langlands conjectures

February 7, 2012 in number theory, representation theory, videos | Leave a comment

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.

Irreducible smooth admissible supercuspidal representations and fixed vectors

January 18, 2012 in representation theory | Leave a comment

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.

The paramodular conjecture

January 11, 2012 in algebraic geometry, L-functions, modular forms, representation theory | Tags: | Leave a comment

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.

Motives of reductive groups

October 8, 2011 in group theory, L-functions, number theory, representation theory | Tags: , | Leave a comment

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.

If k is a number field, let \mathbb{A}:=\mathbb{A}_k be the adèles of k and let S be a finite set of places of k such that S contains all of the infinite places and all of the finite places where G over k_v is not quasi-split. We can define a measure \mu_S on the G(\mathbb{A}), a locally compact group, in terms of its local components by setting \mu_v=L_v(M^\vee(1))\cdot|\omega_{G_v}| on G(k_v) for all v\notin S, and \mu_v=\mu_{G_v}, the Euler-Poincaré measure on G(k_v) for all v\in S.

Let L_S(M) be the value at s=0 of the meromorphic continuation of the Euler product

\prod_{v\notin S} L_v(M,s).

This product converges in some right half plane and, by some results of Siegel, L_S(M) is a rational number, which is non-zero if and only if the measure \mu_S is non-zero on the adèlic points of G. In this case,

\int_{G(k)\backslash G(\mathbb{A}}\mu_S=L_S(M)\cdot \tau(G)\slash\prod_{v\in S}c(G_v)

where \tau(G) is the Tamagawa number of G and

c(G_v)=\dfrac{\# H^1(k_v,G)}{\#({\rm ker}:H^1(k_v,T)\longrightarrow H^1(k_v,G))},

where T is a maximal torus which is anisotropic over k. For finite places v, we have c(G_v)=\# H^1(k_v,G).

Let k^s denote the separable closure of k and let \Gamma={\rm Gal}(k^s\slash k). Let G denote a connected reductive group over k with center Z and derived subgroup G^{\rm der}. Let C denote the connected component of Z. C will be a torus. To define the motive of G, we first define the motive in the case of quasi-split groups.

For a quasi-split group G, let S be a maximal split torus in G and let T be the centralizer of S in G. Since G is quasi-split, T will be a maximal torus. Let W=N_{G(k^s)}(T(k^s)\slash T(k^s) denote the Weyl group of T in G over the separable closure k^s of k. Then the Galois group \Gamma acts on W. The \mathbb{Q}-vector space

E=X^\bullet(T)\otimes\mathbb{Q}={\rm Hom}_{k^s}(T,\mathbb{G}_m)\otimes\mathbb{Q}

admits an action of W\rtimes\Gamma.

By Chevalley, the algebra of W-invariants in the symmetric algebra on E is isomorphic to a symmetric algebra on a graded \mathbb{Q}-vector space V. If R={\rm Sym}^\bullet(E)^W and R_+ is the ideal of elements of degree at least 1 in R, then the graded vector space V=\oplus_{d\geq 1}V_d is given by

V=R_+\slash R_+^@

This V is isomorphic to E as a representation of \Gamma. The grading of V and the fact that each V_d is also a representation of \Gamma can be exploited to yield some useful information about G and its Weyl group. For example, one has

{\rm dim}\, G=\sum_{d\geq 1}(2d-1){\rm dim}\, V_d

and

\# W=\prod_{d\geq 1}d^{{\rm dim}\, V_d}

Let \mathbb{Q}(1)=H_1(\mathbb{G}_m) be the Tate motive of rank 1 and weight -2 over k. If N is an Artin motive over k, given by a rational representation of \Gamma, then N(n)=N\otimes \mathbb{Q}(1)^{\otimes n} is an Artin-Tate motive of weight -2n

The motive M of G is defined as

M=M_G=\oplus_{d\geq 1} V_d(1-d)

The rank of M is the rank of G over k^s. Each representation V_d is self-dual and so the twisted dual of M is the motive

M^\vee(1)=\oplus_{d\geq 1}V_d(d)

The weights of the motive M are all \geq 0 and the weights of its twisted dual M^\vee(1) are all \leq -2.

In the case of G split over k, then T=S, each V_d is the trivial representation of \Gamma, and the motive M is a Tate motive.

Finally, for any connected reductive group G, we define its motive as the motive of the quasi-split inner form of G.

Examples:

If G={\rm GL}_n, then
M=\mathbb{Q}+\mathbb{Q}(-1)+\mathbb{Q}(-2)+\dots+\mathbb{Q}(1-n).

If G={\rm Sp}_{2n} or G={\rm SO}_{2n+1}, then
M=\mathbb{Q}(-1)+\mathbb{Q}(-3)+\dots+\mathbb{Q}(1-2n).

If k=\mathbb{F}_q is a finite field with q elements, then \Gamma is generated by the geometric Frobenius element F, which has eigenvalue q^{-1} on the Tate motive \mathbb{Q}(1).

Steinberg found the twisted dual M^\vee(1) of M and the following formula:
\# G(\mathbb{F}_q)/q^{{\rm dim}\, G} = {\rm det}(1-F|M^\vee(1))=\prod_{d\geq 1} {\rm det}(1-F|V_d(d))=\prod_{d\geq 1}{\rm det}(1-Fq^{-d}|V_d).

In the case that k is a local non-Archimedean field of characteristic 0. Let \mathcal{O} be its ring of integers with uniformizer \varpi and let q be the cardinality of the residue field \mathcal{O}\slash\varpi\mathcal{O}. Let k^{\rm unr} be the maximal unramified extension of k contained in the separable closure k^s of k and let I be the inertia subgroup which fixes k^{\rm unr}. We define the Artin-Tate motive
M^I=\oplus_{d\geq 1}V_d^I(1-d)
over \mathcal{O}\slash\varpi\mathcal{O}.

Let G be a connected reductive group over k with motive M. We have the L-functions
L(M)={\rm det}(1-F|M^I)^{-1}
and
L(M^\vee(1))={\rm det}(1-F|M^\vee(1)^I)^{-1}
The L-function of M can be infinite, positive, or negative.

Now, let |\omega_G| be the Haar measure on G(k) which corresponds to a differential \omega_G with good reduction mod \varpi and also to the canonical absolute value |\cdot|_v:k^\times\rightarrow\mathbb{R}_+^* with |\varpi|_v=q^{-1}.

Let H be the quasi-split inner form of G over k. Then, following Kottwitz, we attach the sign
e(G)=(-1)^{{\rm rank}(G/k)-{\rm rank}(H/k)}
to G.

The local function equation relating the L-functions of the motive M with its twisted dual is given by Gross:

Theorem (Gross) Assume that the connected center C of G is anisotropic. Then

L(M)\cdot\mu_G\cdot e(G)\cdot\#H^1(k,G)=L(M^\vee(1))\cdot|\omega_G|

in the space of invariant measures on G(k).

References:

B. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287–313.

R. Kottwitz, Tamagawa numbers, Ann. of Math. 127 (1988), 629–646.

G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. IHES 69 (1989), 91–114.

R. Steinberg, Endomorphisms of linear algebraic groups, Memoir AMS 80 (1968).

J. Tate, Les conjectures de Stark sur les fonctions L d’Artin en s=0, Birkhäuser Progress in Math. 47 (1984).

Gelfand-Graev representations

August 7, 2011 in representation theory | Tags: , | Leave a comment

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.

J. A. Green wrote a paper about the irreducible characters of {\rm GL}(n,\mathbb{F}_q) in which he gives a generating function for the number c(n,q) of classes of {\rm GL}(n,\mathbb{F}_q):

Let p_n be the number of partitions of n. We define the partition function

p(x)=\sum_{n=0}^\infty p_n x^n = 1/(1-x)(1-x^2)\cdots.

The generating function for the number c(n,q) of classes of {\rm GL}(n,\mathbb{F}_q) is

\sum_{n=0}^\infty c(n,q)x^n = \prod_{d=1}^\infty p(x^d)^{w(d,q)}

where

w(d,q)=\frac{1}{d}\sum_{k|d}\mu(k)q^{d/k}

if the number of irreducible polynomials f(t) of degree d over \mathbb{F}_q.

Comparing c(n,q) with the character inner product (\mathcal{G},\mathcal{G}), one can see that most irreducible representations of G are generic.

General symplectic groups

We have a similar story for G={\rm GSp}(2n,\mathbb{F}_q). Again, let N be 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})

and induce to G. The induced representation is multiplicity-free and contains most of the irreducible representations of G. The case n=2 was made explicit in my thesis.

Conjugacy classes of {\rm GSp}(4,\mathbb{F}_q)

The conjugacy classes can be found by using a paper by Wall to compute the classes of {\rm SO}(5, \mathbb{F}_q)\cong {\rm PGSp}(4,\mathbb{F}_q):={\rm GSp}(4,\mathbb{F}_q)/Z, where Z denotes the center of G. The list of conjugacy classes of {\rm PGSp}(4,\mathbb{F}_q) is then used to determine the classes of {\rm GSp}(4,\mathbb{F}_q). Let’s investigate how the class representatives of {\rm PGSp}(4,\mathbb{F}_q) lead to representatives for the classes of {\rm GSp}(4,\mathbb{F}_q).

Consider the natural projection map from {\rm GSp}(4,\mathbb{F}_q) to {\rm PGSp}(4,\mathbb{F}_q) given by

{\rm GSp}(4,\mathbb{F}_q) \longrightarrow {\rm PGSp}(4, \mathbb{F}_q), \text{ } g \mapsto \overline{g}.

Let g,h\in {\rm GSp}(4, \mathbb{F}_q). If g=xhx^{-1}, then, by taking multipliers on each side, it is clear that the multiplier of g is equal to the multiplier of h. Moreover, under the projection map, we have

\overline{g}=\overline{xhx^{-1}}=\overline{x}\cdot\overline{h}\cdot \overline{x^{-1}}.

So if two elements are conjugate in {\rm GSp}(4,\mathbb{F}_q), they must be conjugate in {\rm PGSp}(4,\mathbb{F}_q). The list of class representatives in {\rm PGSp}(4,\mathbb{F}_q), when pulled back to {\rm GSp}(4,\mathbb{F}_q), hit class representatives of all the conjugacy classes of {\rm GSp}(4,\mathbb{F}_q). Suppose now that two elements g,h\in {\rm GSp}(4, \mathbb{F}_q) are conjugate in {\rm PGSp}(4,\mathbb{F}_q), i.e. \overline{g}=\overline{x}\cdot\overline{h}\cdot\overline{x^{-1}}, for some \overline{x}\in {\rm PGSp}(4, \mathbb{F}_q). Then, for some \gamma^i\in \mathbb{F}_q^\times,

g = \begin{pmatrix} \gamma^i & & & \\ & \gamma^i & & \\ & & \gamma^i & \\ & & & \gamma^i \\ \end{pmatrix}x h x^{-1}.

Taking multipliers on both sides of the equation above, we have \lambda(g)=\gamma^{2i}\lambda(h). So if the multiplier of g is a square, then the multiplier of h is a square and if the multiplier of g is a non-square, then the multiplier of h is a non-square. Write g and h in the following way

g = \begin{pmatrix} 1 & & & \\ & 1 & & \\ & & \gamma^{i_g} & \\ & & & \gamma^{i_g} \\ \end{pmatrix}\cdot\begin{pmatrix} \gamma^{j_g} & & & \\ & \gamma^{j_g} & & \\ & & \gamma^{j_g} & \\ & & & \gamma^{j_g} \\ \end{pmatrix}\cdot g',
h = \begin{pmatrix} 1 & & & \\ & 1 & & \\ & & \gamma^{i_h} & \\ & & & \gamma^{i_h} \\ \end{pmatrix}\cdot\begin{pmatrix} \gamma^{j_h} & & & \\ & \gamma^{j_h} & & \\ & & \gamma^{j_h} & \\ & & & \gamma^{j_h} \\ \end{pmatrix}\cdot h',

with g',h'\in {\rm Sp}(4, F_q), i_g,i_h\in\{0,1\}, and j_g,j_h\in T_3. So \lambda(g)=\gamma^{i_g+2j_g}, \lambda(h)=\gamma^{i_h+2j_h}. If g and h are conjugate, then i_g=i_h. Then \gamma^{2j_h} = \gamma^{2j_h}, or ({\gamma^{j_g}})^2 = ({\gamma^{j_h}})^2. So \lambda(h)=\pm\lambda(g), i.e., the multipliers can only differ by a minus sign.

It is possible that an element g of {\rm GSp}(4,\mathbb{F}_q) is conjugate to -g. An example is

\begin{pmatrix} 1 & & & \\ & 1 & & \\ & & -1 & \\ & & & -1 \\ \end{pmatrix}.\begin{pmatrix} & & 1 & \\ & & & 1 \\ 1 & & & \\ & 1 & & \\ \end{pmatrix}.\begin{pmatrix} 1 & & & \\ & 1 & & \\ & & -1 & \\ & & & -1 \\ \end{pmatrix}
=\begin{pmatrix} & & -1 & \\ & & & -1 \\ -1 & & & \\ & -1 & & \\ \end{pmatrix}.

The centralizers are somewhat affected when we pull back our representatives in {\rm PGSp}(4,\mathbb{F}_q) to {\rm GSp}(4,\mathbb{F}_q). There are two types of pullbacks. The first type consists of elements g such that g\neq x(-g)x^{-1} for any x\in {\rm GSp}(4, \mathbb{F}_q). The second type consists of elements g such that g=x(-g)x^{-1} for some x\in {\rm GSp}(4, \mathbb{F}_q). Let {\rm Cent_{PGSp}}(\overline{g}) denote the centralizer of \overline{g} in {\rm PGSp}(4,\mathbb{F}_q) and let {\rm Cent_{GSp}}(g) denote the centralizer of g in {\rm GSp}(4,\mathbb{F}_q).

Type 1

Let g\in {\rm GSp}(4,\mathbb{F}_q) be of the first type, i.e., g is not conjugate to -g. Let \overline{h}\in {\rm Cent_{PGSp}}(\overline{g}). Then \overline{g} = \overline{h}\cdot \overline{g}\cdot \overline{h^{-1}}. When pulled back to {\rm GSp}(4,\mathbb{F}_q), g = z_0hgh^{-1}, with z_0=\pm I. z_0\neq -I since g is not conjugate to -g. So z_0=I, g=hgh^{-1}, and h\in {\rm Cent_{GSp}}(g). We get a short exact sequence

1 \longrightarrow Z \longrightarrow {\rm Cent_{GSp}}(g) \longrightarrow {\rm Cent_{PGSp}}(\overline{g}) \longrightarrow 1.

Therefore \# {\rm Cent_{GSp}}(g) = (q-1)\cdot \# {\rm Cent_{PGSp}}(\overline{g}).

Type 2

Let g\in {\rm GSp}(4,\mathbb{F}_q) be of the second type, i.e., g is conjugate to -g. Define the set S_g=\{h\in {\rm GSp}(4, F_q) : hgh^{-1}=-g\}. Fix s_0\in S_g. S_g is not a group, but there is a bijection of sets S_g \longrightarrow {\rm Cent_{GSp}}(g), given by the map h\mapsto s_0h. Given \overline{h}\in {\rm Cent_{PGSp}}(\overline{g}), either h\in {\rm Cent_{GSp}}(g) or h\in S_g. S_g\sqcup {\rm Cent_{GSp}}(g) maps onto {\rm Cent_{PGSp}}(\overline{g}) via the projection map. Moreover, {\rm Cent'_{GSp}}(g) := S_g\sqcup {\rm Cent_{GSp}}(g) is a group with respect to matrix multiplication and the projection map is a group homomorphism. {\rm Cent_{GSp}}(g) is a subgroup of {\rm Cent'_{GSp}}(g) of index 2. We get a short exact sequence

1 \longrightarrow Z \longrightarrow {\rm Cent'_{\rm GSp}}(g) \longrightarrow {\rm Cent_{\rm PGSp}}(\overline{g}) \longrightarrow 1

Then \# {\rm Cent'_{GSp}}(g) = (q-1) \cdot \# {\rm Cent_{\rm PGSp}}(\overline{g}). Also, 2\cdot \# {\rm Cent_{\rm GSp}}(g) = \# {\rm Cent'_{\rm GSp}}(g). So {\rm Cent_{\rm GSp}}(g) = \dfrac{q-1}{2} \cdot \# {\rm Cent_{\rm PGSp}}(\overline{g}).

Thus, given a class representative \overline{g}\in {\rm PGSp}(4, \mathbb{F}_q), we pull it back to {\rm GSp}(4,\mathbb{F}_q). Then, we determine if g is of Type 1 or Type 2. If it is of Type 1, then there are q-1 conjugacy classes zg, for z\in Z, each of order \dfrac{\# {\rm GSp}(4,\mathbb{F}_q)}{(q-1)\cdot\# {\rm Cent}_{\rm PGSp}(g)}. If the pullback is of Type 2, then there are \dfrac{q-1}{2} conjugacy classes z_i g, with

z_i = \begin{pmatrix} \gamma^i & & & \\ & \gamma^i & & \\ & & \gamma^i & \\ & & & \gamma^i \\ \end{pmatrix},\, i\in T_2

Each of these classes is of order

\dfrac{\# {\rm GSp}(4, \mathbb{F}_q)}{\frac{(q-1)}{2}\cdot \# {\rm Cent_{\rm PGSp}}}(g).

Some of the conjugacy classes of {\rm GSp}(4,\mathbb{F}_q) are given in the following table. Let T_3=\{1,2,...,q-1\}.

Notation Class representative      Number of classes Order of centralizer
A_1(k),
k\in T_3		 \begin{pmatrix} \gamma^k & & & \\ & \gamma^k & & \\ & & \gamma^k & \\ & & & \gamma^k \\ \end{pmatrix}      q-1 \# {\rm GSp}(4, \mathbb{F}_q)
A_2(k),
k\in T_3		 \begin{pmatrix} \gamma^k & & & \\ & \gamma^k & \gamma^k & \\ & & \gamma^k & \\ & & & \gamma^k \\ \end{pmatrix}      q-1 q^4(q^2-1)(q-1)
A_{31}(k),
k\in T_3		 \begin{pmatrix} \gamma^k & & & \gamma^k \\ & \gamma^k & -\gamma^k & \\ & & \gamma^k & \\ & & & \gamma^k \\ \end{pmatrix}      q-1 2q^3(q-1)^2
A_{32}(k),
k\in T_3		 \begin{pmatrix} \gamma^k & & & \gamma^{k+1} \\ & \gamma^k & -\gamma^k & \\ & & \gamma^k & \\ & & & \gamma^k \\ \end{pmatrix}      q-1 2q^3(q^2-1)
A_5(k),
k\in T_3		 \begin{pmatrix} \gamma^k & \gamma^k & -\gamma^k & \\ & \gamma^k & -\gamma^k & \\ & & \gamma^k & -\gamma^k \\ & & & \gamma^k \\ \end{pmatrix}      q-1 q^2(q-1)

We find that there are (q^2+2q+4)(q-1) conjugacy classes.

The conjugacy classes of N

Every element g\in N_{\rm GSp(4)} can be written uniquely in the form

g = \begin{pmatrix} 1 & & & \\ & 1 & x & \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}. \begin{pmatrix} 1 & \lambda & \mu & \kappa \\ & 1 & & \mu \\ & & 1 & -\lambda \\ & & & 1 \\ \end{pmatrix},

with x, \kappa, \lambda, \mu \in \mathbb{F}_q. The order of N is therefore q^4. The multiplier of the matrix g given above is \lambda(g)=1. The conjugacy classes of N = N_{\rm GSp(4)} are listed in the following table.

Notation Class representative      Number of classes Order of centralizer
NA_1 \begin{pmatrix} 1 & & & \\ & 1 & & \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}      1 q^4
NA_2^1(k),
k\in T_3		 \begin{pmatrix} 1 & & & \\ & 1 & \gamma^k & \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}      q-1 q^3
NA_2^2(k),
k\in T_3		 \begin{pmatrix} 1 & & & \gamma^k \\ & 1 & & \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}      q-1 q^4
NA_{31}^1(i ,j , \kappa),
i,j\in T_3
\gamma^{2i}-\gamma^j\kappa = \gamma^{2n}
for some n\in T_3		 \begin{pmatrix} 1 & & \gamma^i & \kappa \\ & 1 & \gamma^j & \gamma^i \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}      \dfrac{(q-1)^2}{2} q^3
NA_{31}^2(k),
k\in T_3		 \begin{pmatrix} 1 & \gamma^k & & \\ & 1 & & \\ & & 1 & -\gamma^k \\ & & & 1 \\ \end{pmatrix}      q-1 q^2
NA_{31}^3(K),
k\in T_3		 \begin{pmatrix} 1 & & \gamma^k & \\ & 1 & & \gamma^k \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}      q-1 q^3
NA_{32}(i ,j , \kappa),
i,j\in T_3
\gamma^{2i}-\gamma^j\kappa = \gamma^{2n+1}
for some n\in T_3		 \begin{pmatrix} 1 & & \gamma^i & \kappa \\ & 1 & \gamma^j & \gamma^i \\ & & 1 & \\ & & & 1 \\ \end{pmatrix}      \dfrac{(q-1)^2}{2} q^3
NA_5(i,j),
i,j\in T_3		 \begin{pmatrix} 1 & \gamma^i & & \\ & 1 & \gamma^j & -\gamma^{i+j} \\ & & 1 & -\gamma^i \\ & & & 1 \\ \end{pmatrix}      (q-1)^2 q^2

Generic representations of {\rm GSp}(4,\mathbb{F}_q)

The character table of \mathcal{G} is

Conjugacy class         \mathcal{G} character value
A_1(q-1) (q^4-1)(q^2-1)(q-1)
A_2(q-1) -(q^2-1)(q-1)
A_{31}(q-1) -(q^2-1)(q-1)
A_{32}(q-1) -(q^2-1)(q-1)
A_5(q-1) q-1

We compute that there are q^2(q-1) irreducible generic representation, which is the majority of the (q^2+2q+4)(q-1) total irreducible representations.

References

J. Breeding II, Irreducible non-cuspidal characters of {\rm GSp}(4,\mathbb{F}_q), Ph.D. thesis, University of Oklahoma, Norman, OK, 2011.

D. Bump, Lie Groups, Springer, 2004.

W. Fulton and J. Harris, Representation Theory, A First Course, Springer, 2004.

J.A. Green, The Characters of the Finite General Linear Groups, Transactions of the American Mathematical Society, 80, 2 (1955), 402–447.

G.E. Wall, On the Conjugacy Classes in the Unitary, Symplectic and Orthogonal Groups, Journal of the Australian Mathematical Society, 3 (1963), 1–62.

Elliptic curves and Fermat’s Last Theorem for n=4k

February 1, 2011 in algebraic geometry | Tags: , | Leave a comment

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).

Write n=4k.

x^n + y^n = z^n
x^{4k} + y^{4k} = z^{4k}
x^{4k} = z^{4k} - y^{4k}

We multiply each side by \dfrac{z^{2k}}{y^{6k}} to get
\dfrac{x^{4k}z^{2k}}{y^{6k}} = \dfrac{z^{6k}}{y^{6k}} - \dfrac{z^{2k}}{y^{2k}}
\left(\dfrac{x^{2k}z^k}{y^{3k}}\right)^2=\left(\dfrac{z^{2k}}{y^{2k}}\right)^3-\left(\dfrac{z^{2k}}{y^{2k}}\right).

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

So we find that there are only trivial integral solutions (x,y,z) to x^n + y^n = z^n for n\equiv 0\, ({\rm mod}\, 4).

Two-dimensional adelic analysis

December 8, 2010 in number theory | Tags: , | Leave a comment

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}.

As in dimension one, we can wrap together the local data using a ring of adeles.

\mathbb{A}_X = \prod_{x\in y\subset X} ' F_{x,y},

where the restricted product condition is more complicated. In fact, there are several different candidates for the rind of adeles, some better for geometry, others for arithmetic.

K. Kato, S. Saito, S. Bloch, and others contributed to the development of two-dimensional class field theory for X, describing the abelian extensions of K(X) in terms of idele groups constructed from the Milnor K-theory of the fields F_{x,y}.

For X a variety, properties of its zeta function follow from the theory of étale cohomology, but current techniques for treating arithmetic varieties usually only work in certain special cases. However, when X is (the spectrum of) the ring of integers of a number field K then we recover the Dedekind zeta function of K, which is known to have meromorphic continuation and a functional equations, with the most insightful proof being that of Tate and Iwasawa.

With that in mind, the possibility of generalising Tate and Iwasawa’s techniques to higher dimensions has been dreamt of by several mathematicians over the years. The problem is that higher dimensional local fields are not locally compact in any reasonable topology, and therefore there is no existing theory of integration and harmonic analysis.

We can never hope to have a reasonable real-valued integration theory: two-dimensional local fields are just too large. A way around this problem is to add an extra degree of freedom and let our measure take values in \mathbb{R}(T).

Siegel cusp forms of degree 2

November 28, 2010 in modular forms, representation theory | Tags: , , | Leave a comment

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.

First!

October 30, 2010 in videos | Tags: | Leave a comment

I thought the following video would be a good start to this blog. It is of the announcement of John Tate receiving the 2010 Abel Prize at the Norwegian Academy of Science and Letters on March 24, 2010

John Tate wins the Abel Prize 2010

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