Zeta Functions for \mathop{\rm GL}\nolimits(n)

Zeta Functions for the Adjoint Action of GL(n) and density of residues of Dedekind zeta functions

Abstract.

We define zeta functions for the adjoint action of on its Lie algebra and study their analytic properties. For we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms for . Our construction naturally yields a regularisation, which is necessary for the improvement of the properties of these zeta function, in particular for the analytic continuation if .

We further obtain upper and lower bounds on the mean value as , where runs over totally real cubic number fields whose second successive minimum of the trace form on its ring of integers is bounded by . To prove the upper bound we use our new zeta function for . These asymptotic bounds are a first step towards a generalisation of density results obtained by Datskovsky in case of quadratic field extensions.

1. Introduction

The first purpose of this paper is to provide another point of view for the construction of the Shintani zeta function for the binary quadratic forms and to generalise this approach to the action of on the Lie algebra . To improve its properties, has to be “adjusted” (cf. [Yuk92, Dat96]), and the advantage of our approach is that a suitable modification (for as well as the higher dimensional case) naturally emerges. The second purpose of this paper is to make a first step towards the generalisation of a result from [Dat96]: We prove upper and lower bounds on the density of residues of Dedekind zeta functions for totally real cubic number fields. For the upper bound we use our new zeta function for .

There has been a long interest in zeta functions attached to group actions, in particular in the Shintani zeta functions attached to prehomogeneous vector spaces, cf. [SS74, Shi75, Yuk92, Kim03]. One basic example of a prehomogeneous vector space is the space of binary quadratic forms on which acts by multiplication by scalars and by changing basis. There are two natural generalisation of this space to higher dimensions corresponding to different viewpoints: From the point of view of quadratic forms, the obvious generalisation is to consider acting on quadratic forms in variables. This is again a prehomogeneous vector space studied, e.g., in [Shi75, Suz79].

On the other hand, we can equally well identify the space of binary quadratic forms with the Lie algebra of so that the action of becomes the adjoint representation on . From this point of view, it seems more natural to generalise to higher dimensions by considering the action of on by letting act by multiplication of scalars and by the adjoint action. This is the point of view we take in this paper. The problem is that this is not a prehomogeneous vector space if so that the general theory of Shintani zeta functions does not apply.

Shintani zeta functions often turned out to be useful to obtain information on certain arithmetic quantities encoded in these zeta functions, cf. [Shi75, WY92, Dat96]. In particular, the Shintani zeta function for the binary quadratic forms can be used to deduce density theorems for class numbers of binary quadratic forms as well as for residues of Dedekind zeta functions for quadratic field extensions, cf. [Shi75, Dat96]. We will later find that in our zeta function for , the residues of the Dedekind zeta functions for cubic number fields are encoded. For general , one could find the respective objects for number fields of degree .

The paper consists of two main parts. The second part is independent of the techniques of the first one, we only use results from the first part.

To describe our results in more detail, let , or , and let accordingly or be the Lie algebra of . Put . Then acts on by the adjoint action . Let denote the set of regular elliptic elements in , i.e. matrices having an irreducible characteristic polynomial over , and let denote the set of orbits of regular elliptic elements under .

Part I

We generalise the zeta function to higher dimensions by defining the “main” (or unregularised) zeta function for by

for , , and a Schwartz-Bruhat function. We will see (cf. Theorem 1.1 below) that this defines a holomorphic function for . For the function basically coincides with the (unmodified) Shintani zeta function from [Shi75, Yuk92, Dat96] (cf. §6 and [Mat11]).

To study one needs to regularise it in a suitable way. For a regularisation is needed to obtain a “nice” functional equation (cf. [Yuk92, Dat96]), but for higher dimensions, the regularisation appears to be even more essential: Already for , it seems that can not be continued to all of , cf. [Mat11, IV.iii]. Our method of regularisation is different from the one used for so far: In [Yuk92, Dat96] smoothed Eisenstein series were used to cut off diverging integrals. In contrast to this we use a more geometric truncation process that is analogous to the one employed by Arthur for his trace formula; cf. also [Lev99] for a similar truncation for the Shintani zeta function for the binary quartic forms. For this we use Chaudouard’s trace formula for (= truncated summation formula) from [Cha02]: Let denote the set of equivalence classes on . This set corresponds bijectively to orbits of semisimple elements, cf. §2.4. Let be the nilpotent variety in . One can attach to every and to every truncation parameter in the coroot space of a distribution on the space of Schwartz-Bruhat functions , cf. §2.7. They are defined similar to Arthur’s distributions on the space of test functions on a reductive algebraic group appearing in Arthur’s trace formula for the group. We now define the regularised zeta function as follows: If , set . Then

 (1) ΞT(s,Φ):=∫∞0λ√D(s+√D−12)∑o∈O∖{n}JTo(Φλ)d×λ

provided this integral converges. We need to extend this definition to non-smooth test functions for sufficiently large, where denotes the space of functions , which are of rapid decay, but only differentiable up to order , cf. §2.5. This extension to non-smooth functions is important for later applications in Part II. In the definition of , the function corresponds to the partial sum over such which are attached to orbits of regular elliptic elements. The function is also closely connected to Arthur’s trace formula for , cf. §6. Our first result is the following:

Theorem 1.1.

[cf. Theorem 3.3] Let . There exists depending only on such that for every the following holds:

1. If is sufficiently regular, the integral defining converges absolutely and locally uniformly for . In particular, is holomorphic in this half plane.

2. is a polynomial in of degree at most and can be defined for every . Then for every the function is holomorphic in .

In this way we get a well-defined family of zeta functions indexed by the parameter and varying continuously with . By the nature of our construction this family depends on an initial choice of minimal parabolic subgroup in . We can, however, choose a zeta function in this family which is independent of this choice: Taking , the function does not depend on the fixed minimal parabolic subgroup anymore (cf. [Art81, Lemma 1.1]) so that can be viewed as ”the“ zeta function associated with acting on .

One of the standard methods to get the meromorphic continuation and functional equation of zeta functions is to use the Poisson summation formula. In our context, Chaudouard’s trace formula takes the place of the Poisson summation formula, and the main obstruction to obtain the meromorphic continuation and the functional equation for is to understand the nilpotent contribution . Restricting to , we are able to analyse the nilpotent distribution completely (see §4 and §5), obtaining our main result of Part I:

Theorem 1.2.

[cf. Theorems 5.6] Let or with , and let be given. Then there exists such that for every and the following holds.

1. has a meromorphic continuation to all with , and satisfies for such the functional equation

 ΞT(s,Φ)=ΞT(1−s,^Φ).
2. The poles of in are parametrised by the nilpotent orbits . More precisely, its poles occur exactly at the points

 s−N=1−√D2+dimN2√D    and     s+N=1+√D2−dimN2√D

and are of order at most . In particular, the furthermost right and furthermost left pole in this region are both simple, correspond to , and are located at the points and , respectively. The residues at these poles are given by

 ress=s−0ΞT(s,Φ) ress=s+0ΞT(s,Φ)

Note that if , then is a Schwartz-Bruhat function and can be meromorphically continued to all of .

Chaudouard’s trace formula is valid for any reductive group. In principle, it is possible to define the zeta function as in (1) for an arbitrary reductive group acting on its Lie algebra. At least Theorem 1.1 should stay true (in fact, our proof should go through as it is without major difficulties; we restricted to and mainly to make it not more technical as it already is). One can of course also conjecture that the analogue of Theorem 1.2 holds, and the main difficulty then lies in the analysis of the nilpotent contribution . One could take this approach even further, by considering a general rational representation of the group instead of its adjoint representation. In [Lev01] equivalence classes and corresponding distributions are defined for such a representation, and also a kind of trace ”formula” is proved for this situation. For the Shintani zeta function of binary quartic forms such an approach has already been used in [Lev99].

For and , we can show that is indeed the main part of in the following sense:

Proposition 1.3.

[cf. Corollaries 7.3 and 7.5] If or , then continues holomorphically at least to . In particular, the furthermost right pole of and coincide and have the same residue.

This result will become important in Part II, where we will use the analytic properties of to apply a Tauberian theorem.

The organisation of Part I is as follows: In §3 we will define and prove Theorem 1.1. In §4 and §5 we study the nilpotent distribution for and conclude the proof of Theorem 1.2. In §6 we describe the connection of our construction of the zeta function to the Arthur-Selberg trace formula for , and of to the classical Shintani zeta function for . Finally, restricting to , , we prove Proposition 1.3 in §7.

Part Ii

The Shintani zeta function for the space of binary cubic forms was used by Shintani to establish mean values for the class numbers of binary quadratic forms, cf. [Shi75]. From our point of view, another closely related density result obtained from is more important: Datskovsky (cf. [Dat96]) proved that if is a finite set of prime places of including the archimedean place, and is a fixed signature for quadratic number fields, then as one has

 (2) ∑L: DL≤Xress=1ζL(s)=α(rS)X,

where runs over all quadratic fields of signature and absolute discriminant bounded by , and is a suitable non-zero constant. As a first step towards generalising this, we prove upper and lower bounds for the densities of residues of Dedekind zeta functions of totally real cubic number fields.

Suppose is a totally real number field of degree with ring of integers . We denote by the positive definite quadratic form for , where denotes the field trace of . We denote the successive minima of on by . If , then for every quadratic field so that the sum in (2) runs over all quadratic fields with . Our main result of Part II is the following:

Theorem 1.4.

[cf. Theorem 10.1] We have

 (3) limsupX→∞X−52∑E: m1(E)≤Xress=1ζE(s)<∞

where the sum extends over all totally real cubic number fields for which the first successive minimum is bounded by . Here denotes the Dedekind zeta function attached to .

We complement the above upper bound (3) with the following result:

Proposition 1.5.

[cf. Proposition 10.2] For every , we have

 liminfX→∞X−52+ε∑E: m1(E)≤Xress=1ζE(s)=∞,

where the sum extends over totally real cubic number fields .

This is a first step towards a generalisation of (2) to the cubic case and the signature of totally real cubic number fields. As in the quadratic case, one expects that in fact the limit of the left hand side in (3) exists and is non-zero:

Conjecture 1.6.

There exists a constant such that as

 ∑E: m1(E)≤Xress=1ζE(s)∼α3X52,

where the sum extends over all totally real cubic number fields for which the first successive minimum is bounded by .

Let us make a few remarks on the (quite different) strategies to prove Theorem 1.4 and Proposition 1.5: First we use a suitable sequence of test functions and apply a Tauberian Theorem to to obtain an asymptotic for the density of certain orbital integrals in Proposition 9.2. These orbital integrals are basically products of and a quantity , , obtained from the non-archimedean part of the test function. For an appropriate we have for every relevant so that Theorem 1.4 is a direct consequence of Proposition 9.2. To prove Proposition 1.5, on the other hand, we go a completely different way (independent of our results for ): We basically show that there are sufficiently many irreducible cubic polynomials.

In fact, we would like to deduce all of the conjectured asymptotic from Proposition 9.2. In Appendix B we give a sequence of test functions for which . However, a certain uniformity of the convergence with respect to is needed to prove Conjecture 1.6, which we were not able to show so far.

Our methods can at least heuristically be applied to for every . In particular, the first pole of for is expected to be at . This suggests:

Conjecture 1.7.

For every there exists such that as

 ∑E: m1(E)≤Xress=1ζE(s)∼αnXn(n+1)−24,

where the sum extends over all totally -dimensional number fields for which the first successive minimum is bounded by .

Ordering fields with respect to the first successive minimum of (in contrast to the discriminant) is also related to a conjecture of Ellenberg-Venkatesh, cf. [EV06, Remark 3.3]: Basically they conjecture that has a non-zero limit as where runs over -dimensional number fields. As remarked in [EV06], it is possible to show a “weak form” of this asymptotic under a strong hypothesis on the existence of sufficiently many squarefree polynomials. If one can prove an -dimensional analogue of Proposition 9.2 and make the passage from to work (e.g., with a sequence of test function as ), this should lead to another approach to (a slightly weaker form of) the conjecture of Ellenberg-Venkatesh.

This second part of the paper is organised as follows: In §8 we first recall and prove some properties of orbital integrals, before stating and proving an asymptotic for the mean value of certain orbital integrals in §9, cf. Proposition 9.2. Our main result Theorem 1.4 in §10 will then be an easy consequence of Proposition 9.2 together with results in §8. Finally, we will prove Proposition 1.5 at the end of §10.

Acknowledgments

The work on Part I was started while the author stayed at the MPI in Bonn during August/September 2011, and continued during the year 2011/2012 when the author was supported by grant #964-107.6/2007 from the German-Israeli Foundation for Scientific Research and Development and by a Golda Meir Postdoctoral Fellowship. Part II was essentially contained in the author’s doctoral dissertation [Mat11] which was written under the direction of T. Finis and supported by grant #964-107.6/2007 from the German-Israeli Foundation for Scientific Research and Development. The author would like to thank T. Finis and E. Lapid for communicating their idea for the proof of Proposition 5.2 carried out in Appendix A.

2. Notation and general conventions

2.1. General notation

We fix notation following [Cha02, Art05]:

• denotes the ring of adeles of . If is a place of , denotes the completion of at , and if is non-archimedean, is the usual -adic norm on , i.e. if is the prime corresponding to , then is normalised by . Then denotes the norm on given by the product of the ’s.

• is an integer, and denotes or as a group defined over with Lie algebra or . We put ( or ). denotes the identity element.

• is the minimal parabolic subgroup of upper triangular matrices with the torus of diagonal elements and its unipotent radical of unipotent upper triangular matrices. If is a -defined parabolic subgroup with Levi component , then denotes the set of (-defined) parabolic subgroups containing , and the subset of parabolic subgroups with Levi component . For with Levi decomposition , we denote by the corresponding decomposition of the Lie algebra. For with , put and for the opposite parabolic subgroup. denotes the identity component of the split component of the center in .

• is called standard if and we write for the set of standard parabolic subgroups.

• is the root space, i.e. the -vector space spanned by all rational characters , and is the coroot-space. denotes the set of reduced roots of the pair

We denote by the set of simple roots and by the set of all positive roots of the action of on . If , then denotes the corresponding coroot. Similarly, is the set of simple weights, and if , then denotes the corresponding coweight. If , we denote by the weight such that for all (here is the Kronecker ).

• If and , write . For , let

 AP2P1=A21={a∈AP1∣∀α∈ΔP2: α(a)=1}≃AP1/AP2,

and . For let be the intersection of the kernels of all rational characters . Let be the positive chamber in with respect to our fixed minimal parabolic subgroup. Similarly, we define . Denote by the unique element in such that the modulus function satisfies for all and write and .

• Let be the map characterised by and for all .

• We denote by the set of weights of with respect to so that . Then we have a direct sum decomposition for the eigenspace of in . We take the usual vector norm on obtained by identifying with via the matrix coordinates. Then if , with , then .

• If , we write , , , etc., and further put and .

2.2. Characteristic functions

Let be parabolic subgroups with . We define the following functions (cf. [Art78]):

• is the characteristic function of the set

 {H∈a0∣∀ϖ∈^Δ21: ϖ(H)>0}.

If , we also write .

• is the characteristic function of the set

 {H∈a0∣∀α∈Δ21: α(H)>0}.

If , we also write .

• is the characteristic function of the set

 {H∈a0∣∀α∈Δ21: α(H)>0; ∀α∈Δ1∖Δ21: α(H)≤0; ∀ϖ∈^Δ2: ϖ(H)>0}.
Remark 2.1.

The function is related to and by .

• is called sufficiently regular if is sufficiently large, i.e., if is sufficiently far away from the walls of the positive Weyl chamber (cf. [Art78]). We fix a small number such that the set of sufficiently regular satisfying is a non-empty open cone in .

• For sufficiently regular the function is defined as the characteristic function of all , , satisfying

 ϖ(H0(μm)−T)≤0

for all and . If , we sometimes write .

• If is sufficiently regular, [Art78, Lemma 6.4] gives for every the identity

 ∑R: P0⊆R⊆P∑δ∈R(Q)∖P(Q)FR(δx,T)τPR(H0(δx)−T)=1.

2.3. Measures

We fix the following maximal compact subgroups: If is a non-archimedean place, then , and at the archimedean place, . Globally, we take . Up to normalisation there exists a unique Haar measure on , and we normalise it by for all , and then take the product measure on . We further choose measures as follows:

• and , : normalized by .

• , , , , : usual Lebesgue measures.

• , : twice the usual Lebesgue measure.

• and : product measures.

• : measure induced by the exact sequence .

• finite dimensional -vector space with fixed basis: take the measures induced from (resp. ) on (resp. ) via this basis. This in particular defines measures on and if we take the canonical bases corresponding to the root coordinates.

• and : measures induced from and via the diagonal coordinates.

• and : compatible with the Iwasawa decomposition (resp. ) such that for every integrable function on we have

 ∫G(A)f(g)dg=∫T0(A)∫U0(A)∫Kf(tuk)dk du dt=∫T0(A)∫U0(A)∫Kδ0(t)−1f(utk)dk du dt

(similarly for the local case).

• : measure induced by the exact sequence .

• Levi and parabolic subgroups: compatible with previous cases.

2.4. Equivalence classes

Let (resp. ) denote the set of semisimple elements in (resp. ). We define an equivalence relation on as follows: Let and write , for the Jordan decomposition with , semisimple and , nilpotent, where is the centraliser of in . We call and equivalent if and only if there exists such that . We denote the set of equivalence classes in by .

Let denote the set of nilpotent elements. Then constitutes exactly one equivalence class (corresponding to the orbit of ) and decomposes into finitely many nilpotent orbits under the adjoint action of . On the other hand, if corresponds to the orbit of a regular semisimple element (i.e., the eigenvalues of (in an algebraic closure of ) are pairwise different), then is in fact equal to the orbit of .

2.5. Test functions

Let denote the Lie algebra of one of the standard parabolic subgroups of , of one of their unipotent radicals or of one of their Levi components. We fix the standard vector norm on by identifying via the usual matrix coordinates. Let denote the universal enveloping algebra of the complexification . For every we fix a basis of the finite dimensional -vector space of elements in of degree . For a real number and a non-negative integer we define seminorms on the spaces , , by setting for

 ∥f∥a,b:=supx∈b(R)((1+∥x∥)a∑X∈Bb∣∣(Xf)(x)∣∣)

with . We put

 Sν(b(R)):={f∈Cν(b(R))∣∀a<∞, b<ν: ∥f∥a,b<∞}.

Then is the usual space of Schwartz functions on . If , then operates on by acting on the archimedean part of the function. We then define seminorms on and on the spaces and similar as before. In particular, is the usual space of Schwartz-Bruhat functions on . Dualy to we put

 Sν(b(A)):={f∈C∞(b(A))∣∀a<ν, b<∞: ∥f∥a,b<∞}

so that .

If is a finite prime, we let denote the set of compactly supported smooth functions , and define analogously.

The topology induced by the set of seminorms , , (resp. , ) makes (resp. ) into a Frechet space. The words “seminorm” and “continuous seminorm” on one of these spaces will be used synonymously.

We fix a non-degenerate invariant bilinear form by setting for . Let be the non-trivial character constructed in [Lan94, XIV, §1]. We define the Fourier transform

 ˆ:Sν(g(A))⟶Sν(g(A)),   ˆΦ(Y):=∫g(A)Φ(X)ψ(⟨X,Y⟩)dX

with respect to this bilinear form.

2.6. Siegel sets

If , let denote the set of all with for all . Reduction theory tells us that there exists such that

 G(A)1=G(Q)P0(A)1AG0(T1)K.

We fix such a from now on and write

 ST1={g=pk∈P0(A)K∣∀α∈Δ0: α(H0(a)−T1)>0}.

If then is measurable, we have

 (4)

2.7. Distributions associated with equivalence classes

For and sufficiently regular define for (cf. [Cha02])

and, if is integrable, we set

 JTo(Φ)=∫AGG(Q)∖G(A)kTo(x,Φ)dx

provided the sum-integrals converge.

Part I The zeta function

3. The trace formula for Lie algebras and convergence of distributions

Let us recall some of the main results from [Cha02].

Theorem 3.1 ([Cha02], Théorem̀e 3.1, Théorem̀e 4.5).

For all and sufficiently regular we have

 (5)

and

 (6) ∑o∈OJTo(Φ)=∑o∈OJTo(^Φ).

The distributions and are polynomials in of degree at most .

The Poisson summation like identity (6), is what we refer to as Chaudouard’s trace formula for the Lie algebra .

Remark 3.2.
1. Since the distributions in the theorem are polynomials in for varying in a non-empty open cone of , they can be defined at any point , with (6) then being valid for all .

2. The results in [Cha02] hold for arbitrary reductive groups .

3. (5) holds for every