Counting local systems via the trace formula

and Yuval Z. Flicker
  • Corresponding author
  • University of Ariel, Ariel, 4070000, Israel, and The Ohio State University, Columbus, Ohio, 43210, USA
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

Let 𝔽q be a finite field with q elements, 𝔽=𝔽¯q an algebraic closure of 𝔽q, X1 an absolutely irreducible projective smooth curve over 𝔽q, and S1 a finite set of closed points of X1. Let N1 be the cardinality of S1. Erasing the index 1 indicates extension of scalars to 𝔽. Replacing it by m indicates extension of scalars to 𝔽qm⊂𝔽. Let Fr be the Frobenius endomorphism of the curve X over 𝔽, deduced from the 𝔽q-form X1 of X. For ℓ a prime not dividing q, the pullback by Fr is an autoequivalence of the category of smooth ℓ-adic sheaves (=ℚ¯ℓ-local systems) on X-S. It induces a permutation Fr∗ of the set of isomorphism classes of irreducible smooth ℓ-adic sheaves of rank n on X-S, whose local monodromy at each s∈S is unipotent with a single Jordan block (= principal unipotent). Let T(X1,S1,n) be the number of fixed points of Fr∗ acting on this set. We will also consider the number of fixed points T(X1,S1,n,m) of the iterate Fr∗,m of Fr∗. As Fr∗,m : X→X is the Frobenius, for X1/𝔽q replaced by Xm/𝔽qm, one has T(X1,S1,n,m) =T(Xm,Sm,n).

The number T(X1,S1,n) is computed when N1≥ 2 in [4] where it is shown that the T(X1,S1,n,m) are given by a formula reminiscent of a Lefschetz fixed point formula. The automorphic-Galois reciprocity is used there to reduce the computation of T(X1,S1,n) to counting automorphic representations on GL(n), and the assumption N1≥ 2 to move the counting to a division algebra, where the trace formula is easier to use. The trace formula used in Section 4 of [4] is based on a new, categorical approach to computing the trace formula in the very special case of counting that is needed there. It is not yet known if it extends to the use of Hecke correspondences and in the non-compact quotient case. Our aim here is to give an alternative proof of Section 4 of [4] using the usual trace formula in the compact quotient case. As stated in [4], after Lemma 4.9, the equivalence between the two approaches seems to us interesting.

1 Introduction, and the result

Let X1 be a smooth projective absolutely irreducible curve over 𝔽q of genus g. Let S1 be a finite set of closed points of X1 of cardinality N1≥2. Fix ℤ∋n≥2. If 2|n and n≥4, suppose 2|N1. The case of odd n>2 or N1>2 is reduced in [4, Section 5] to this case. To simplify the notations, instead of X1, S1, N1 (as in [4]), we write X, S, N. Fix a prime ℓ with the property that (ℓ,q)=1. Let F be the function field of X over 𝔽q and FS the maximal extension of F unramified at each closed point of X-S, inside a fixed separable closure F¯ of F. Fix an algebraic closure 𝔽 of 𝔽q in FS⊂F¯. Denote the geometric fundamental group of the affine curve X-S by π1⁢((X-S)⊗𝔽q𝔽)=Gal⁡(FS/F⊗𝔽q𝔽). It is the kernel of the restriction map from the arithmetic fundamental group π1⁢(X-S)=Gal⁡(FS/F) to Gal⁡(F⊗𝔽q𝔽/F)=Gal⁡(𝔽/𝔽q). Then Gal⁡(𝔽/𝔽q) acts on the set of the isomorphism classes of the representations of π1⁢((X-S)⊗𝔽q𝔽).

The work [4] computes, and attempts to interpret, the number T of isomorphism classes of the n-dimensional irreducible ℓ-adic representations ρ¯ of the geometric fundamental group π1⁢((X-S)⊗𝔽q𝔽) of the affine curve X-S, invariant under the Frobenius, with a single Jordan block (of rank n) unipotent monodromy at each point of S (equivalently: the number of isomorphism classes of rank n irreducible local ℚ¯ℓ-systems, namely smooth ℚ¯ℓ-sheaves on (X-S)⊗𝔽q𝔽, with principal unipotent local monodromy at each v∈S and fixed by the Frobenius, or equivalently: the number of rank n irreducible local ℚ¯ℓ-systems (smooth ℚ¯ℓ-sheaves) on X-S, with principal unipotent local monodromy at each v∈S, up to isomorphism and twisting by a character of Gal⁡(𝔽/𝔽q)).

To compute T, using the automorphic-Galois reciprocity law in the function field case ([13]) we relate T to the number of cuspidal representations of GL⁡(n) over the function field F of X over 𝔽q which are unramified except for at S, where the component is Steinberg twisted by an unramified character.

This number is related – using the correspondence from GL⁡(n) to its inner forms – to that of the unramified cuspidal representations on the multiplicative group of a division algebra over F which remains a division algebra at the places of S, and is split elsewhere. In this compact quotient case the trace formula can be used to carry out the computation.

The trace formula used in [4, Section 4] is based on a completely new, categorical approach, based on the notion of the mass of a category. But so far it has been developed only to count the cardinality of interest in [4, Section 4]. It is not yet clear if it can be extended to the non-compact quotient case or to deal with Hecke correspondences.

The aim of the present work is to compute the same number using the usual trace formula in the compact quotient case. As stated in [4] following Lemma 4.9, the equivalence between these two approaches seems to us interesting.

To understand T, [4, Proposition 6.28] shows that there exists a virtual motive on the moduli stack ℳg;[N′,N′]′ over ℤ of triples (X,S′,S′)′ consisting of a smooth projective geometrically irreducible curve X and disjoint sets S′, S′′ of cardinalities N′, N′≥′1 of points on X, such that T is the trace of the Frobenius on this virtual motive.

This leads to an understanding of how T varies with the base field 𝔽q.

The fact that the cardinality N of S is at least 2 permits us to use a simple – compact quotient – form of the trace formula. When n=2, thus for GL⁡(2), we deal with all N in [9]. The remaining cases being N=1 and N=0. These cases require using the full trace formula and computing each term in it explicitly. The explicit form of the trace formula for GL⁡(2) over a function field is developed for this purpose in [8]. The case of N=0, thus counting the objects in question which are nowhere ramified in the (n=) two-dimensional case, was done by Drinfeld [5] in 1981. His pioneering work is the foundation to [4], [8] and [9].

The case of even n>2 and odd N is reduced in [4, Section 5] to the case dealt with here, which is the case of [4, Section 4].

The contents of this paper are as follows. We state our counting results in this section. In Section 2 we recall what is meant by monodromy and examine the behavior by extensions 𝔽qj/𝔽q (fixed points by power of Frobenius). The number of Theorem 1.1 appears to take the form ∑iαij-∑kβkj. This suggests it can be expressed as the trace of the Frobenius on a compatible system of smooth ℓ-adic sheaves (local systems) on some moduli stack over ℤ. This is done in [4, Section 6]. In Section 3 we translate the question to one about counting automorphic representations of a suitable adèle group. In Sections 4–7 we develop and evaluate a trace formula.

The analytic part of Sections 3–7 can be read independently, it is just the motor behind the complete machine. For the full picture one needs the arithmetic-geometric avatar of Section 2. See [4, Sections 2 and 6].

To state our results, denote by Frq the pullback Fr∗ by Fr introduced in the abstract, and put

fm(t)=∏1≤i≤2⁢g(1-tαim)=det[1-tFrqm|H1(X⊗𝔽q𝔽,ℚ¯ℓ)].

The fixed algebraic closure 𝔽 of 𝔽q is the union ⋃m≥1𝔽qm of its subfields 𝔽qm with qm elements. Put

hm=#⁢Pic0⁡(X)⁢(𝔽qm).

Thus hm=fm⁢(1). Let Xm=X⊗𝔽q𝔽qm be the curve over 𝔽qm obtained from X/𝔽q. Let ζ⁢(Xm,t) be the rational function of t such that ζ⁢(Xm,q-m⁢s) is the zeta function of Xm. Then

fm⁢(t)=ζ⁢(Xm,t)⁢(1-t)⁢(1-qm⁢t)∈ℤ⁢[t].

Write |X| for the set of closed points of X. Let F be the function field of X over 𝔽q. Let deg⁡(v) be the residual exponent of v (the residue field of Fv is 𝔽qv, qv=qdeg⁡(v)) for each v∈|X|. Write (n/S) for the greatest quotient of n prime to deg⁡(v) for each v∈S. If k≥1, then put μ⁢(k)=(-1)r if k is the product of r distinct primes, μ⁢(k)=0 if not. For m≥1 put cm=∑k|mμ⁢(k)⁢(qm/k-1). It is the number of x∈𝔽qn× generating 𝔽qm over 𝔽q.

Theorem 1.1.

The number of the isomorphism classes of the irreducible rank n local Q¯ℓ-systems (= smooth Q¯ℓ-sheaves) on (X-S)⊗FqF, invariant under the Frobenius, with a single Jordan block (of rank n) unipotent monodromy at each point of S, or equivalently the number of rank n irreducible local Q¯ℓ-systems on X-S, with principal unipotent local monodromy at each v∈S, up to isomorphism and twisting by a character of Gal⁡(F/Fq), or equivalently the number of n-dimensional irreducible (continuous) ℓ-adic representations ρ¯ of the geometric fundamental group π1⁢((X-S)⊗FqF) of the affine curve X-S, invariant under the Frobenius, with a single Jordan block (of rank n) unipotent monodromy at each point of S, is

T:=T⁢(q)=(-1)N⁢(n-1)⁢h1⁢((∑m|(n/S)cm⁢Tm)-1),

where

Tm=Tm⁢(n,S,q)=hmh1⁢mN-2qn-1⁢∏jfm⁢(qm⁢j)∏j(1-qm⁢j)2⁢∏v∈S∏j(1-qvm⁢j),1≤j<nm.

For example, suppose X is ℙ1 (thus we have g=0 and fm⁢(t)=1). If n=2, N≥2 and ∑v∈Sdeg⁡(v)=4, then T=q. Indeed, c1=q-1 and c2=q2-q. If S consists of N=4 points of degree 1, then (n/S)=2, T1=q-1q+1, T2=4q2-1, and

T=c1⁢T1+c2⁢T2-1=q.

If S consists of one point of degree 2 and two points of degree 1, then N=3, (n/S)=1, T1=-1, so

T=-[-1⁢(q-1)-1]=q.

If S consists of two points of degree 2 each, then (n/S)=1, N=2, T1=q+1q-1,

T=q.

If S consists of two points, of degrees 1 and 3, then (n/S)=2, N=2, T1=q2+q+1q2-1, and T2=1q2-1, so

T=q.

When g=0, n=2, ∑v∈Sdeg⁡(v)=3, we have

T=0.

If g=0, and S consists of two points of degree 1 (any n≥2), then we have (n/S)=n, ∑m|ncm=qn-1 (see [14, Chapter V, Example 21]), Tm=(qn-1)-1, and

T=0.

Let 𝔸 denote the ring of adèles of F. Let O𝔸=∏v∈|X|Ov be the ring of integers in 𝔸. Put O𝔸S=∏v∈|X-S|Ov. There is a degree homomorphism

deg:𝔸×→ℤ,(av:v∈|X|)↦-∑vdeg(v)valv(av).

Fix an idèle α∈𝔸× of degree 1 whose components are 1 outside a finite set Tα of places of F. Put αℤ={αn:n∈ℤ}.

A representation ρ¯ of the geometric fundamental group

π¯1=π1⁢((X-S)⊗𝔽q𝔽)=Gal⁡(FS/F⊗𝔽q𝔽)

is invariant under the Frobenius precisely when it extends to a representation, say ρ, of the arithmetic fundamental group π1⁢(X-S). Here FS denotes the maximal extension of F in a fixed algebraic closure F¯ which is unramified outside S. For any α in

π1W⁢(X-S)ab=W⁢(FS/F)ab≃𝔸×/F×⁢O𝔸S×

which corresponds by CFT to a fixed idèle of degree 1 in 𝔸×/F×⁢O𝔸S×, ρ can be chosen to have det⁡ρ⁢(α)=1. If ρ is irreducible, so is ρ¯. Conversely, if ρ¯ is irreducible, ρ is too, by the condition that we put at the v in S in Theorem 1.1, as long as S is not empty. Any other representation which restricts to ρ¯ is of the form ρ⊗χ, χ a character of Gal⁡(𝔽/𝔽q), and ρ⊗χ≃ρ if and only if χ=1.

The number of ρ¯ stated in Theorem 1.1 is the number of equivalence classes of irreducible representations ρ of π1⁢(X-S), with det⁡ρ⁢(α)=1 and of dimension n, with principal unipotent monodromy at each v∈S, up to twisting by a character χ of Gal⁡(𝔽/𝔽q)=π1⁢(X-S)/π¯1. If det⁡ρ⁢(α)=1, and

1=det⁡(ρ⊗χ)⁢(α)=χ⁢(α)n⁢det⁡ρ⁢(α)=χ⁢(α)n,

there are n possibilities for the character χ. The representations ρ⊗χ are inequivalent; see [4, Theorem 1.9 (ii)], which uses the fact that the restriction to the geometric fundamental group is irreducible. The number of ρ not up to twisting is n times that of Theorem 1.1.

Note that the automorphic counterpart, [4, Theorem 1.11 (ii)], asserts that if π is a cuspidal automorphic ℚ¯ℓ-representation of GL⁡(n,𝔸), unramified outside the finite set S of closed points of X, or places of F, such that for each s∈S the local component πs is of the form St⊗χs⁢(det), where St stands for Steinberg, for χs an unramified character of Fs× in ℚ¯ℓ, then all twists π⊗χ~ by χ~:GL⁡(n,𝔸)→ℚ¯ℓ× defined by χ~⁢(g)=χ⁢(deg∘det⁡(g)), for some χ:ℤ→ℚ¯ℓ×, are distinct. This is clear, from the automorphic point of view, since πs=πs⊗η with a character η≠1, implies in particular that πs (s∈S) is not a twist of a Steinberg (see, e.g., [10], or just consider the induced containing St, that is, the cuspidal support of St).

We shall now rewrite the analytic expression T of Theorem 1.1 in a geometric form. First we note that if ηv runs through the dv=deg⁡(v) roots of 1, and ζ=ζm through the mth roots of 1, we have that

∏ζm≠1,ηv(1-ζm⁢ηv)=∏ζm≠1(1-ζmdv)

is 0 if (m,dv)≠1, and it is m if (m,dv)=1, thus ∏v∈S∏ζm≠1,ηv(1-ζm⁢ηv) is 0 if there is v∈S with (m,dv)≠1, and it is mN if (m,dv)=1 for all v∈S. Replacing S by the multiset S′ consisting of all ηv, v∈S, with 1 removed twice (this makes sense since N≥2), we see that ∏η∈S′,ζm≠1(1-ζ⁢η) is 0 or mN-2.

Further, write B for the multiset consisting of the eigenvalues αi (1≤i≤2⁢g) of the Frobenius Frq on H1⁢(X⊗𝔽q𝔽,ℚ¯ℓ), of the deg⁡(v)th roots of 1 for each v∈S, minus twice 1. Then B is the multiset of eigenvalues of the Frobenius Frq on Hc1⁢((X-S)⊗𝔽q𝔽,ℚ¯ℓ)-ℚ¯ℓ, where the last ℚ¯ℓ signifies the trivial one-dimensional space. It is shown in [4, Corollary 6.4] that

Corollary 1.2.

We have

T=(-1)N⁢(n-1)⁢h1⁢(∑m|ncm⁢Tm′-1),

where

Tm′=1qn-1⁢∏β∈B(∏ζm≠1(1-ζ⁢β)⋅∏j,ζm(1-ζ⁢β⁢qj)).

Indeed, the first product over β is ∏i,ζm≠1(1-ζ⁢αi)⋅∏η∈S′,ζm≠1(1-ζ⁢ηv), which is mN-2⁢hm/h1 times 1 if m|(n/S) and 0 if m does not divide (n/S), while the second product is the product over j of ∏i,ζm(1-ζ⁢αi⁢qj)=fm⁢(qm⁢j) and

∏η∈S′,ζm(1-ζm⁢η⁢qj)=(∏v∈S(1-qvm⁢j))⁢(1-qm⁢j)-2.

Proposition 6.28 of [4] concludes from this the following.

Theorem 1.3.

The cardinality T is equal to the trace of the Frobenius Frq on a virtual motive on the moduli stack Mg;[N′,N′]′ over Z of triples (X,S′,S′)′ consisting of a smooth projective absolutely irreducible curve X and two disjoint sets S′, S′′ of geometric points on X of cardinalities N′, N′≥′1.

2 Arithmetic perspective

To recall what we mean by monodromy, prepare to relating the arithmetic expression of Theorem 1.1 to analysis, and review the study of variation by base change in [4, Section 6], we leisurely review the objects under consideration, in relatively classical language.

The Galois group Gal⁡(𝔽/𝔽q) of the finite field 𝔽q of q elements is isomorphic to the projective limit

ℤ^=lim←⁡ℤ/n.

The arithmetic Frobenius automorphism Frq:x↦xq is a topological generator of Gal⁡(𝔽/𝔽q). We identify Gal⁡(𝔽/𝔽q) with ℤ^ by sending Frq to -1. The geometric Frobenius automorphism x↦x1/q is mapped to 1∈ℤ^.

Let Fv be a local non-Archimedean field of positive characteristic, 𝔽v its residue field, its cardinality is qv, F¯v a separable algebraic closure of Fv, and Gal⁡(F¯v/Fv) the Galois group. The natural homomorphism jv:Gal⁡(F¯v/Fv)→Gal⁡(𝔽¯v/𝔽v)=ℤ^ is onto. Here 𝔽¯v is the algebraic closure of 𝔽v in F¯v, also denoted by 𝔽. The kernel, Iv, is called the inertia subgroup. The pullback of ℤ by jv is the Weil groupWv=WFv of Fv. It is endowed with the topology induced by the embedding of Wv in Gal⁡(F¯v/Fv)×ℤ, where ℤ is discrete. Thus we have an exact sequence

1⟶Iv⟶Wv⁢⟶jv⁢ℤ⟶1,

of topological groups, where Wv is locally compact. Let Gab signify the quotient of a topological group G by the closure of its commutator subgroup. Local class field theory creates a canonical isomorphism between Wvab and Fv×. There are two such isomorphisms. We choose the one whose composition with the valuation valv:Fv×→ℤ, which takes a uniformizer 𝝅v to 1, is jv|Wvab; note that jv:Wv→ℤ factorizes via Wvab. Denote it by τv:Fv×→Wvab.

Let F be a global field of characteristic p>0 and field of constants 𝔽q. The natural homomorphism Gal⁡(F¯/F)→Gal⁡(𝔽/𝔽q)=ℤ^ is onto. The pullback of ℤ is called the Weil group of F and is denoted by W, as we fixed F and its separable closure F¯. The topology of W is defined as in the local case. Denote by 𝔸 the ring of adèles of F, by Fv the completion of F at a place v. There is a natural class of embeddings of Wv=WFv in W, differing from each other by inner automorphisms of W. Thus there is a natural homomorphism Wvab→Wab. Global class field theory asserts that there is a unique isomorphism τ:𝔸×/F×→Wab such that for every v,

Fv×⟶𝔸×/F×⁢⟶𝜏⁢Wab

coincides with

Fv×⁢⟶τv⁢Wvab⟶Wab.

Denote by X the smooth projective model of the field F. Thus X is a smooth projective absolutely irreducible curve over 𝔽q. The set of places of F coincides with the set |X| of closed points of X. Let S be a finite set of closed points of X. Let FS be the maximal extension of F in F¯, the fixed separable closure of F, which is unramified at each v in |X-S|. Let ξ denote the geometric point of X-S corresponding to F↪F¯. Denote π1⁢(X-S,ξ)=Gal⁡(FS/F) by π1⁢(X-S). At each v in |X-S| we have a (unique up to conjugacy) geometric Frobenius element Frv∈π1⁢(X-S). The composition Wv→Gal⁡(F¯v/Fv)→π1⁢(X-S) is trivial on the inertia subgroup Iv (for v∉S). It then defines a homomorphism ℤ→π1⁢(X-S), and Frv is the image of 1.

For a geometric definition of Frv, view the closed point v of X as the scheme Spec⁡𝔽v, with geometric point ξv:𝔽v→𝔽¯v, where 𝔽¯v is the algebraic closure of 𝔽v in F¯. There is a natural homomorphism π1⁢(v,ξv)→π1⁢(X-S,ξ). Note that changing the base point the Frobenius changes by conjugation in π1⁢(X-S). Indeed, the fundamental group is independent of the choice of a base point, namely there is a natural class of isomorphisms π1⁢(X-S,ξ)≃π1⁢(X-S) which differ from each other by inner automorphisms of π1⁢(X-S). Now π1⁢(v,ξv) is the absolute Galois group of the residue field 𝔽v at v, i.e. it is ℤ^. Then Frv is the image of 1 under ℤ^=π1⁢(v,ξv)→π1⁢(X-S).

Denote by π1W⁢(X-S) the inverse image of ℤ under the homomorphism π1⁢(X-S)→ℤ^. The topology is defined as on W and Wv, and Frv lies in π1W⁢(X-S) if v∉S. The Picard group Pic⁡X consists of the isomorphism classes of invertible sheaves on X, or classes of divisors on X, or the idèle classes 𝔸×/F×⋅∏vOv×, where Ov× denotes the units in the ring Ov of integers in Fv. The class of a divisor D is denoted D¯, thus for any closed point v in |X|, v¯ is the class of the divisor which has a single point v with multiplicity one. Unramified Class Field Theory (CFT) asserts that there is a unique isomorphism π1W⁢(X)ab≃Pic⁡X with Frv↦v¯ for each closed point v in |X|, while CFT establishes a unique isomorphism

π1W⁢(X-S)ab≃Pic⁡X-S=𝔸×/F×⁢∏v∉SOv×

with Frv↦v¯ for all v∉S.

An ℓ-adic representation of a topological group G is a representation of G in a finite-dimensional vector space V over ℚ¯ℓ such that for one, equivalently any, basis of V, the homomorphism ρ:G→GL⁡(n,ℚ¯ℓ) is continuous and Im⁡ρ⊂GL⁡(n,E), where E is a finite extension of ℚℓ.

Class Field Theory establishes a natural bijection between one-dimensional representations of π1W⁢(X-S) and homomorphisms Pic⁡X-S=𝔸×/F×⁢∏v∈X-SOv×→ℚ¯ℓ× for all prime ℓ≠p.

The automorphic-Galois global reciprocity theorem, proven by Lafforgue [13], asserts, for all prime ℓ≠p, that there is a bijection between the set of equivalence classes of irreducible n-dimensional ℓ-adic representations ρ of π1W⁢(X-S), and the set of cuspidal (irreducible, automorphic) representations π=⊗πv of GL⁡(n,𝔸) over ℚ¯ℓ which are unramified outside S, such that the n-tuple of the roots of det⁡(I-qv-1/2⁢ρ⁢(Frv)) (we choose a square root of qv in ℚ¯ℓ) equals the n-tuple of Satake parameters of the unramified component πv of π for each v∈X-S. By the latter we mean the values at the uniformizer 𝝅v of the unramified characters on the torus Fv×n in the upper triangular subgroup of GL⁡(n,Fv) from which πv is induced. Moreover, the restriction ρv of ρ to the subgroup Wv corresponds to the component πv of π at each v in S. We do not elaborate on this local correspondence, but we now review the structure of the ℓ-adic representations of Wv.

There exists an epimorphism tℓ:Iv→ℤℓ whose kernel is the inverse limit of finite groups whose orders are prime to ℓ. This tℓ is determined uniquely up to multiplication by an element of ℤℓ×. Define ν:Wv→ℚℓ× by ν⁢(x)=qv-jv⁢(x). Here jv is the map Wv→ℤ. One has tℓ⁢(h⁢x⁢h-1)=ν⁢(h)⁢tℓ⁢(x) for h∈Wv, x∈Iv.

Grothendieck showed that for an ℓ-adic representation ρ:Wv→Aut⁡V there exists a unique endomorphism N:V→V such that for all x∈Iv close enough to the identity, ρ⁢(x)=exp⁡(tℓ⁢(x)⁢N). Moreover, N is nilpotent and ρ⁢(h)⁢N⁢ρ⁢(h)-1=ν⁢(h)⁢N for all h∈Wv. To see this, note that Aut⁡V contains an open subgroup U which is a pro-ℓ-group. Then the restriction of ρ to U~=ρ-1⁢(U)∩Iv factors through tℓ⁢(U~), an open subgroup of ℤℓ. Thus there is ψ:tℓ⁢(U~)→U with ρ=ψ∘tℓ on U~. By the ℓ-adic Lie theory, the restriction of any ℓ-adic representation of tℓ⁢(U~) to some open subgroup is given by y↦exp⁡(y⁢N), for an endomorphism N:V→V. From ψ⁢(tℓ⁢(h⁢x⁢h-1))=ψ⁢(tℓ⁢(x))ν⁢(h) we get

ρ⁢(h)⁢exp⁡(tℓ⁢(x)⁢N)⁢ρ⁢(h)-1=exp⁡(ν⁢(h)⁢tℓ⁢(x)⁢N),

thus ρ⁢(h)⁢N⁢ρ⁢(h)-1=ν⁢(h)⁢N. So the spectrum of N is invariant under multiplication by qv, so it reduces to zero, thus N is nilpotent. We name N the logarithm of the unipotent part of the monodromy of ρ.

If ρ is irreducible, then its kernel is open. Indeed, from ρ⁢(h)⁢N=ν⁢(h)⁢N⁢ρ⁢(h) we see that ker⁡N is invariant with respect to the action of ρ. Thus ker⁡N is 0 or N is 0. But N is nilpotent, hence N=0. Since ρ⁢(x)=exp⁡(tℓ⁢(x)⁢N) for x∈Iv near the identity, ker⁡ρ is open in Iv, hence in Wv, by definition of the topology on Wv.

A Weil–Deligne representation (WD-representation) is a triple (V,ρ′,N), where V is a finite-dimensional vector space over ℚ¯ℓ, ρ′:Wv→Aut⁡V a homomorphism with open kernel, N:V→V an endomorphism with ρ′⁢(h)⁢N⁢ρ′⁢(h)-1=ν⁢(h)⁢N for all h∈Wv. This N is nilpotent and called the monodromy of ρ. Weil–Deligne representations have the advantage of being defined algebraically, over any field, not equipped with topology.

To associate a WD-representation to an ℓ-adic representation ρ:Wv→Aut⁡V, choose Frv∈Wv whose image in ℤ is 1. Any element of Wv has a unique presentation Frvm⁡x with m∈ℤ and x∈Iv. Define ρ′:Wv→Aut⁡V by ρ′⁢(Frm⁡x)=ρ⁢(Frm⁡x)⁢exp⁡(-tℓ⁢(x)⁢N), where N is the monodromy of ρ. The exp term makes sense since N is nilpotent. Then (V,ρ′,N) is a WD-representation and the map ρ↦(V,ρ′,N) is one-to-one. The induced map from the set of isomorphism classes of ℓ-adic representations of Wv to those of WD-representations over ℚ¯ℓ is independent of the choice of Frv and tℓ ([3, Lemme 8.4.3]).

A WD-representation (ρ′,N) is said to be Frv-semisimple if ρ′ is semisimple, and an ℓ-adic representation of Wv is called Frv-semisimple if the associated (ρ′,N) is.

The example which interests us here is ρ′:Wv→GL⁡(n,ℚ¯ℓ) with

ρ′⁢(h)=diag⁡(1,ν⁢(h)-1,ν⁢(h)-2,…,ν⁢(h)-(n-1))

and N=(ai⁢j), ai⁢j=0 unless j=i+1 (1≤i<n), where ai,i+1=1. Then (ρ′,N) is a semisimple WD-representation. The associated ℓ-adic representation ρ of Wv is given by ρ⁢(Frvm⁡x) being diag⁡(1,qvm,qv2⁢m,…,qv(n-1)⁢m) times exp⁡(tℓ⁢(x)⁢N). This ρ is denoted by Stv below, since the admissible representation of GL⁡(n,Fv) corresponding by the local reciprocity law to this ρ is the Steinberg representation, which is square integrable mod center.

We express our computation in terms of the rational function ζ⁢(X,t) such that ζ⁢(X,q-s) is the zeta function of X:

ζ(X,t)=∏0≤i≤2det(1-tFrq|Hi(X⊗𝔽q𝔽,ℚ¯ℓ))(-1)i+1=f1⁢(t)(1-t)⁢(1-q⁢t),

where

f1⁢(t)=∑0≤k≤2⁢gak⁢tk∈ℤ⁢[t]

(see [16, Chapter VII, Section 6, Theorem 4]). It satisfies the functional equation

ζ⁢(X,t)=qg-1⁢t2⁢g-2⁢ζ⁢(X,1/q⁢t),

where g=dim𝔽q⁡H1⁢(X,𝒪X) is the genus of X. This implies the relation a2⁢g-k=ak⁢qg-k. Further, a0=1, a2⁢g=qg. Consider ζ⁢(Xm,t), Xm=X⊗𝔽q𝔽qm. From

fm(t)=det(1-tFrqm|H1(X⊗𝔽q𝔽,ℚ¯ℓ))=∏1≤i≤2⁢g(1-tαim)=∑0≤k≤2⁢gak(m)tk,

since |αi|=q1/2, we see that the ak(m) depend on q, are bounded by qm⁢k/2, and are dominated by a2⁢g(m)=qg⁢m. Define hm to be

#Pic0(X)(𝔽qm)=det(1-Frqm|H1)=∑k=02⁢gak(m)=fm(1).

It is dominated by a2⁢g(m)=qg⁢m.

We would like to study the behavior of the number T⁢(qj) of Theorem 1.1 (where j=1) as the base field 𝔽qj, namely Frqj, varies. The Tm and q suggest the appearance of eigenvalues of the Frobenius on a compatible system of ℓ-adic sheaves on some moduli space over ℤ. However the numbers deg⁡(v), v∈S, hence also the number (n/S), the greatest quotient of n prime to ∏v∈Sdeg⁡(v), of Theorem 1.1, do not remain unchanged as j increases. Consider each point of S as a Gal⁡(𝔽/𝔽q)-orbit of a geometric point of X. Then S can be viewed as a set of M=∑v∈Sdeg⁡(v) distinct geometric (over 𝔽) points of X, M being independent of the base field 𝔽q. This motivates Corollary 1.2 above, in which ∑m|(n/S)cm⁢Tm is expressed as ∑m|ncm⁢Tm′, a similar expression which behaves well with respect to base change. In [4, Corollaries 6.8, 6.9, 6.15] it is shown:

Corollary 2.1.

Unless g=0 and N=2, or g=0 and N=3 and n=2, there is r≥1 and r-tuples (m1,…,mr) in Zr and (ν1,…,νr) in Cr such that the number T⁢(qj) of equivalence classes of n-dimensional ℓ-adic representations ρ of π1⁢((X-S)⊗FqF) invariant under Frqj with a single Jordan block (of rank n) monodromy at each point of S, is

(-1)N⁢(n-1)⁢qj⁢[(n2-1)⁢(g-1)+n⁢(n-1)2⁢M]+∑1≤i≤rmi⁢νij.

For each i, |νi|=qbi, 2⁢bi∈Z, 0≤bi<(n2-1)⁢(g-1)+n⁢(n-1)2⁢M, and mi∈Z.

3 Analytic perspective

The strategy of proof is to transform the arithmetic question to an analytic one, concerning automorphic representations. Indeed, by the automorphic-Galois global reciprocity law ([13]) we need to count the cuspidal representations π=⊗πv of GL⁡(n,𝔸) which are unramified at each place v∉S of F, with Steinberg (twisted by an unramified character) component at each v∈S, with central character ωπ whose value at α is 1, plus the cardinality of 𝔸×/F×⁢O𝔸×⋅αn⁢ℤ.

Let D be a central division algebra over F of rank n which is split at each v∉S, and such that Dv is a division algebra at each v∈S. When n is even, such a D exists provided S has even cardinality.

The Deligne–Kazhdan (or inner-forms) correspondence (for the main ideas, in the number fields case, see [1, Section 25] (for the trace formula) and [6, Section III] (for the correspondence), and [7] when there is a cuspidal component; the case n=2 is due to Jacquet–Langlands; see also [4, Appendix]) is a bijection from the set of automorphic representations πD of D𝔸× to the set of automorphic representations π of GL⁡(n,𝔸) which are one-dimensional or have square-integrable components at each place v∈S where Dv is a division algebra. It is determined by πvD≃πv at each v∉S, π is one-dimensional if and only if πD is, and when dim⁡π>1, πv is Steinberg twisted by an unramified character, if and only if dim⁡πvD=1.

The underlying F-group G is defined by the multiplicative group G⁢(F)=D× of D. Put Gv for Dv×, Dv=D⊗FFv. Put G⁢(𝔸) for the adèle group D𝔸×, D𝔸=D⊗F𝔸.

Let 𝒟v be a maximal order in Dv, thus 𝒟v× is a maximal compact subgroup in Dv×. The group 𝒟v× is isomorphic to GL⁡(n,Ov), and Gv is isomorphic to GL⁡(n,Fv) for all v∈X-S. For a general division algebra D over F the Gv is GL⁡(mv,Dv′), where Dv′ is a division algebra of rank dv=n/mv, and 𝒟v×=GL⁡(mv,𝒟v′), where 𝒟v′ is the maximal order of the x in Dv′ whose reduced norms are in Ov. In the case that we consider mv=1 for all v∈S, thus Dv is a division algebra, 𝒟v× is the group of x∈Dv whose reduced norm is in Ov×. Then

K=𝒟𝔸×=∏v∈|X|𝒟v×

is a maximal compact open subgroup of G⁢(𝔸).

Consider the space L2⁢(G⁢(F)⋅αℤ\G⁢(𝔸)) of automorphic forms on G⁢(𝔸) which are invariant under the central element α. The group G⁢(𝔸) acts on this space by right translation:

(r⁢(h)⁢ϕ)⁢(g)=ϕ⁢(g⁢h).

Since the group G is anisotropic (mod center), the quotient G⁢(F)⋅αℤ\G⁢(𝔸) is compact. Hence the space L2 decomposes as a direct sum with finite multiplicities of irreducible submodules, called automorphic representations. The multiplicities are known to be one by the Deligne–Kazhdan correspondence of these automorphic representations from this group to GL⁡(n), since GL⁡(n) has multiplicity one. Since G has no proper F-parabolic subgroup, these G⁢(𝔸)-modules are all cuspidal.

An irreducible representation π of G⁢(𝔸) decomposes as a product ⊗vπv of irreducible admissible representations πv of Gv, where v ranges over |X|. For almost all v – that is, except for finitely many – πv is unramified, i.e. has a nonzero vector fixed by the maximal compact subgroup Kv=𝒟v×, which is GL⁡(n,Ov) for v∉S.

Our aim is then to compute the number of everywhere unramified automorphic representations of G⁢(𝔸) which are trivial on αℤ.

Theorem 3.1.

Let N be the number of places in S. Suppose 2|N if n is even and n>3. The number of nowhere ramified infinite-dimensional irreducible automorphic representations π of DA× whose central character takes the value 1 on αZ is n⁢T⁢(q), with

T⁢(q)=(-1)N⁢(n-1)⁢h1⁢((∑m|(n/S)cm⁢Tm)-1)

as described in Theorem 1.1.

The number T=T⁢(q) of Theorem 1.1 is the number of orbits of π under twisting by a character χ(|Nrd(x)|), where Nrd denotes the reduced norm from D𝔸× to 𝔸×, |⋅|:𝔸×→qℤ and χ is a character of qℤ≃ℤ of order n. This is the number of ρ¯ described in Theorem 1.1, or the number of orbits of the irreducible n-dimensional ρ of π1⁢(X-S) with det⁡ρ⁢(α)=1 under twisting with a character of Gal⁡(𝔽/𝔽q).

The unramified one-dimensional automorphic representations π of D𝔸× with central character ωπ with ωπ⁢(α)=1 is the number of characters of 𝔸×/F×⁢O𝔸×⁢αn⁢ℤ, namely n⁢h1.

Had we fixed a central character ω, the number of all π, one-dimensional or not, without taking orbits under twisting, is n⁢(T+h1)/h1. This is the number of Theorem 1.1, plus h1, times n, and divided by h1. Here h1 is the cardinality of Pic0⁡(X)⁢(𝔽q)=𝔸×/F×⁢O𝔸×⁢αℤ. This is the number of ω with ω⁢(α)=1.

The number of one-dimensional unramified automorphic representations of D𝔸× with central character ω is zero unless ω is trivial on the n-torsion subgroup Picn0 of Pic0⁡(X)⁢(𝔽q), in which case it is n⁢|Picn0|. Subtracting this number we get the number of infinite-dimensional representations: n⁢(T+h1)/h1-n⁢{0⁢ or ⁢|Picn0|}. Dividing by n we get (T+h1)/h1-{0⁢ or ⁢|Picn0|}, the number of orbits under twisting by χ⁢(|Nrd|), as explained 3–4 paragraphs before Corollary 1.2.

Our proof of Theorem 3.1 is relatively elementary. It is independent of the automorphic-Galois reciprocity law, and of the correspondence from GL⁡(n) to inner-forms.

Note that each local component of the representations of the theorem, viewed as representations of GL⁡(n) by the inner-forms correspondence, lies in the Bernstein component [2] associated with the trivial character of the Borel subgroup.

Up to a scalar multiple, there is one nonzero K-fixed vector in an unramified automorphic representation. So the number of unramified automorphic representations whose central character is 1 at α, is the cardinality of D×\D⁢(𝔸)×/K⋅αℤ. This set is in bijection with the set of principal D×-bundles on X, a set with a structure of an Artin stack. Our technique relies on an evaluation of the trace formula, but all we do is simply count the indicated cardinality. However, the technique of the trace formula extends to deal with the cases N=0 and N=1, see [9] when n=2.

4 The trace formula

Our proof of Theorem 3.1 is based on an application of the trace formula. In fact, we simply need to compute the cardinality of the finite set K\D𝔸×/D×⋅αℤ. Our trace formula here is simply computing this number. However, the trace formula can be developed to treat the cases where N=0 and N=1, where G⁢(𝔸)/G⁢(F)⋅αℤ is not compact. See [9] when n=2. This is why we use the trace formula formalism. This is done in [4, Section 4] using a new, categorical approach, using masses of objects in the category. As stated in [4] after Lemma 4.9, the equivalence between the categorical and analytic approaches seems to us interesting.

So, choose a Haar measure dy=⊗vdyv on G⁢(𝔸), and let f=⊗vfv be a test function on G⁢(𝔸), thus fv∈Cc∞⁢(Gv) for each v and fv=fv0, the characteristic function of Kv divided by vol⁡(Kv), for almost all v. Thus f⁢d⁢y is a smooth compactly supported measure on G⁢(𝔸), in the convolution algebra of such measures. The convolution operator

(r⁢(f⁢d⁢y)⁢ϕ)⁢(x)=∫G⁢(𝔸)f⁢(y)⁢ϕ⁢(x⁢y)⁢𝑑y

is of finite rank hence has a trace, which is equal to the sum over all irreducible π in L2⁢(αℤ⋅D×\D𝔸×) of tr⁡π⁢(f). This last sum is finite. The last integral can be written as

∫αℤ⋅G⁢(F)\G⁢(𝔸)[∑γ∈αℤ⋅G⁢(F)f⁢(x-1⁢γ⁢y)]⁢ϕ⁢(y)⁢𝑑y.

Hence the trace is equal to the integral of the kernel over the diagonal, thus to

∫G⁢(𝔸)/αℤ⋅G⁢(F)[∑γ∈αℤ⋅G⁢(F)f⁢(x⁢γ⁢x-1)]⁢𝑑x.

Since G is anisotropic (mod center), the elements of G⁢(F) are either central, thus we have γ∈αℤ⋅Z⁢(F), where Z⁢(F)≃F× is the center of G⁢(F), or elliptic non-central, which generate a field extension F⁢(γ) of F of degree d>1 dividing n. The centralizer Zγ⁢(F) of γ in G⁢(F) is the multiplicative group of a central division algebra over F⁢(γ) of degree m with m⁢d=n. When γ is regular, thus d=n, this division algebra is F⁢(γ) itself, and the centralizer in D× is an F-torus T⁢(F)≃F⁢(γ)×. We consider γ only up to conjugacy, ∼, in G⁢(F). The geometric side of the trace formula then takes the form

vol(D𝔸×/αℤ⋅D×)∑γ∈F×⋅αℤf(γ) (recall: D𝔸×=G(𝔸),D×=G(F))

plus

∑γ∈αℤ⁢(G⁢(F)-F×)⁣/∼∫D𝔸×/Zγ⁢(F)⋅αℤf⁢(x⁢γ⁢x-1)⁢𝑑x.

This is equal to the spectral side of the trace formula, which is ∑πtr⁡π⁢(f). If we choose fv to be the characteristic function of Kv divided by the volume |Kv| of Kv, then tr⁡πv⁢(fv) is 1 if πv has a nonzero Kv-fixed vector, and 0 otherwise. We make this choice at every place v. For the resulting function f=⊗vfv the number of everywhere unramified automorphic representations of G⁢(𝔸) is then given by the value of the geometric side of the trace formula at our f. We now evaluate it.

5 Scalars

Recall that 𝒟𝔸=∏v𝒟v, where 𝒟v is a maximal order in Dv for all v∈|X|, and 𝒟v×=Kv is a maximal compact subgroup in Gv=Dv×. A differential form ω of highest degree on the group G=D× determines a Haar measure μv on Dv× for every v∈|X|. The isomorphism of G with GL(n) (over a separable algebraic closure F¯ of F) defines a differential form, also denoted ω, on GL(n), which defines a Haar measure, also denoted μv, on GL⁡(n,Fv), for each v. The volumes of 𝒟v×=Kv and GL⁡(n,Ov) are the same for v∈|X-S|, where G and GL(n) are isomorphic over F. Thus we have

vol⁡(𝒟𝔸×)=vol⁡(GL⁡(n,O𝔸))⁢∏v∈Sξ⁢(Dv×),

where ξ⁢(Dv×)=vol⁡SL⁡(n,Ov)/vol⁡SL⁡(mv,𝒟v′) is computed in Proposition 6.1.

In this section we prove:

Proposition 5.1.

The value of the contribution from the part of the geometric side of the trace formula corresponding to the central elements is

n⁢h1⁢ζ⁢(X,q)⁢ζ⁢(X,q2)⁢⋯⁢ζ⁢(X,qn-1)⁢∏v∈Sξ⁢(Dv×).

Proof.

Consider the sum over γ in αℤ⋅F×. If f⁢(γ)≠0, then the scalar γ lies in both O𝔸× and F×⁢αℤ, thus in 𝔽q×, and so this sum equals (q-1)/vol⁡(𝒟𝔸×), as 𝒟𝔸=∏v𝒟v, 𝒟v×=Kv. By the remarks before the proposition, this can be expressed as

[(q-1)/vol⁡(GL⁡(n,O𝔸))]⁢∏v∈Sξ⁢(Dv×).

To compute the volume of D𝔸×/αℤ⋅D×, denote by D𝔸0 the kernel of the reduced norm D𝔸×↠𝔸×. See [16, Chapter XI, Section 3, Proposition 3] for the surjectivity. Then

vol⁡(D𝔸×/αℤ⋅D×)=vol⁡(D𝔸0/D0)⋅vol⁡(𝔸×/F×⋅αn⁢ℤ).

The second factor here is equal to n⋅vol⁡(𝔸×/F×⋅αℤ). Using the exact sequence

1→𝔽q×→O𝔸×→𝔸×/F×⋅αℤ→(Pic⁡X)⁢(𝔽q)/αℤ(≃(Pic0⁡(X))⁢(𝔽q))→1

we conclude that

vol⁡(𝔸×/F×⋅αℤ)=h1⋅vol⁡(O𝔸×)/(q-1),

where h1, the cardinality of (Pic0⁡(X))⁢(𝔽q), is given by

det[1-Frq|H1(X⊗𝔽q𝔽,ℚℓ)]=∑0≤k≤2⁢gak.

Note that vol⁡GL⁡(n,O𝔸)/vol⁡O𝔸× equals vol⁡SL⁡(n,O𝔸), where the measure on SL⁡(n) is obtained from the restriction of the differential form ω on GL⁡(n), and that on GL⁡(1) is the quotient measure.

The F-group D0 is an inner form of SL⁡(n,F), hence, for our choice of compatible measures on inner forms, the volume of D𝔸0/D0 is equal to that of SL⁡(n,𝔸)/SL⁡(n,F) ([12]). We shall compute below the volume of SL⁡(n,O𝔸). In general, let G be an algebraic group over F of dimension m. Fix an invariant m-form ω on G. For each closed point v in |X| fix a Haar measure d⁢xv on Fv such that (vol(𝔸/F)=)∫𝔸/F∏vdxv=1. For each v in |X| the form ω and the measure d⁢xv determine a Haar measure μv on G⁢(Fv). The group G is the generic fiber of a group scheme G′ over X-S′, where S′ is a finite subset of X. Thus for almost all v in |X| there exists a Gv′=G′⁢(Ov) in G⁢(Fv)=Gv. Replacing G′ by another group scheme with generic fiber G changes only a finite number of Gv′. Assuming ∏v∫Gv′μv is finite, the Tamagawa measure μT=∏vμv is well defined (see [17]).

Recall that the Tamagawa number of SL⁡(n,𝔸)/SL⁡(n,F) (and of D𝔸0/D0) is one. We need to compute the volume volT⁡SL⁡(n,O𝔸) of SL⁡(n,O𝔸) by the Tamagawa measure. We shall use the exact sequence

0→𝔽q→O𝔸→𝔸/F→H1⁢(X,𝒪X)→0.

Here H1⁢(X,𝒪X) is a vector space over 𝔽q, whose dimension, by definition, is the genus g of X. Hence vol⁡(𝔸/F)/vol⁡(O𝔸)=qg-1. We may and do choose the form ω on SL⁡(n) to be defined over 𝔽q⊂F. Put bv=∫Ov𝑑xv. Then

vol⁡SL⁡(n,Ov)=(bvqv)n2-1⁢|SL⁡(n,𝔽qv)|=(bvqv)n2-1⁢[∏0≤j<n(qvn-qvj)]⁢(qv-1)-1=bvn2-1⁢∏2≤j≤n(1-qv-j).

Hence

volT⁡SL⁡(n,O𝔸)=∏v[bvn2-1⁢∏2≤j≤n(1-qv-j)].

Since ∏vbv=vol⁡(O𝔸)/vol⁡(𝔸/F)=q1-g, and ζ⁢(X,t)=∏v(1-tdeg⁡v)-1, thus

ζ⁢(X,q-j)=∏v(1-qv-j)-1,

we obtain

volT⁡SL⁡(n,O𝔸)=q(n2-1)⁢(1-g)⁢ζ⁢(X,q-2)-1⁢ζ⁢(X,q-3)-1⁢⋯⁢ζ⁢(X,q-n)-1.

Next we use the functional equation

ζ⁢(X,t)=qg-1⁢t2⁢g-2⁢ζ⁢(X,1/q⁢t)

for the ζ-function, at the values t=q-j, thus

ζ⁢(X,q-j)=qg-1⁢q-j⁢(2⁢g-2)⁢ζ⁢(X,qj-1).

As (n2-1)⁢(1-g)-(n-1)⁢(g-1)+(2⁢g-2)⁢(2+3+…+n)=0, we have

volT⁡SL⁡(n,O𝔸)=∏1≤j<nζ⁢(X,qj)-1,

and the value of the contribution from the part of the geometric side of the trace formula corresponding to the central elements is as asserted. ∎

6 Volumes

We still need to compute the factors ξ⁢(Dv×), in fact only when mv=1.

Proposition 6.1.

Suppose Dv×=GL⁡(mv,Dv′) and Dv′ is a maximal order in the division algebra Dv central of rank dv=n/mv over Fv. Then the quotient

ξ⁢(Dv×)=vol⁡SL⁡(n,Ov)/vol⁡SL⁡(mv,𝒟v′)

is equal to ξ⁢(dv,mv,qv). Here ξ⁢(d,m,q) is the positive integer

ξ⁢(d,m,q)=(∏1≤j<n(qj-1))⁢(∏1≤j<m(qd⁢j-1))-1.

Here Fv is a non-Archimedean local field with ring Ov of integers and generator 𝝅v of the maximal ideal in the local ring Ov. The residue field kv=kFv=Ov/𝝅v has cardinality qv. A central division algebra Dv over Fv is cyclic, and can be explicitly described using the unramified extension Ev of degree dv=degFv⁡Dv. Denote the ring of integers of Ev by OEv, and the residue field OEv/𝝅v by kEv. The cardinality of kEv is qvdv. Let σ denote a generator of Gal⁡(Ev/Fv). To simplify the notations, in the rest of this section we delete the subscript v. Let M⁢(d,E) denote the algebra of d×d matrices with entries in E.

The cyclic algebra D can be realized as the algebra of matrices (ai⁢j)∈M⁢(d,E) fixed by σ which acts by σ⁢((ai⁢j))=ρ-1⁢(σ⁢(ai⁢j))⁢ρ, where

ρ=(010…0001…0000…1𝝅00…0),

thus

(ai⁢j)=(a1a2a3…ad-1ad𝝅⁢σ⁢adσ⁢a1σ⁢a2…σ⁢ad-2σ⁢ad-1𝝅⁢σ2⁢ad-1𝝅⁢σ2⁢adσ2⁢a1…σ2⁢ad-3σ2⁢ad-2⋮⋮⋮⋮⋮⋮𝝅⁢σd-2⁢a3𝝅⁢σd-2⁢a4……σd-2⁢a1σd-2⁢a2𝝅⁢σd-1⁢a2𝝅⁢σd-1⁢a3……𝝅⁢σd-1⁢adσd-1⁢a1).

Then D⁢(O) is the order with ai∈OE, and reduction D¯ mod 𝝅 is well-defined:

D¯=(RkE/k⁢𝔾a)d,

thus

D¯⁢(k)=kEd.

Superscript (1) will mean reduced norm 1. By SL⁡(1,D¯⁢(k)) we mean D¯⁢(k)(1).

Lemma 6.2.

We have

#⁢SL⁡(m,D¯⁢(k))=[∏0≤j<m(qm⁢d-qd⁢j)]⁢qd⁢(d-1)⁢m2q-1.

Proof.

To see this, we count the number of entries in the m×m-matrix in the first row which are of the form (a~11,a~12,…,a~1⁢m), where a~1⁢i – the ith entry in the first row – is diag⁡(a1⁢i,σ⁢a1⁢i,σ2⁢a1⁢i,…,σd-1⁢a1⁢i). The a1⁢i range over kE, but not all a1⁢i are 0. We get the factor qm⁢d-1.

Then we count the number of second rows (a~21,a~22,…,a~2⁢m),

a~2⁢i=diag⁡(a2⁢i,σ⁢a2⁢i,σ2⁢a2⁢i,…,σd-1⁢a2⁢i),

which are not a multiple of the first row by a∈kE. We get a factor qm⁢d-qd. We continue through the rows to get the product over i.

The entries a2,…,ad∈kE in each of the m×m entries of SL⁡(m,D¯⁢(k)) are arbitrary. We get qd⁢(d-1)⁢m2.

We need the reduced norm, which is in kF×, to be 1. So we divide by q-1, and obtain the asserted cardinality of SL⁡(m,D¯⁢(k)). ∎

This SL⁡(m,D¯⁢(k)) has a Borel subgroup B¯=T¯⁢U¯, where the torus T¯⁢(k) consists of diag⁡(a~11,a~21,…,a~m⁢1), ai⁢1∈kE×. Then T¯⁢(k) has (qd-1)m/(q-1) elements, but we do not use this fact.

Proof of Proposition 6.1.

To prove the proposition, we need to know the cardinality of U¯⁢(k). As it is a p-Sylow subgroup of SL⁡(m,D¯⁢(k)), we can find its cardinality at once from that of SL⁡(m,D¯⁢(k)). That U¯⁢(k) is p-Sylow can be seen by arguing that any p-subgroup consists of unipotents (if xpr=1, then (x-1)pr=0 and x is unipotent (=1+nilpotent), hence – by Kolchin’s theorem – lies in U¯⁢(k) up to conjugation. As U¯ is an extension of affine lines, U¯⁢(k) is a p-group, hence maximal. Alternatively, by Bruhat decomposition G=⋃w∈WU⁢w⁢B, G/B=⋃w∈WU/U∩w⁢U⁢w-1, where #⁢U/U∩w⁢U⁢w-1 is 1 if w=1 and a positive power of q otherwise. Hence #⁢G/T⁢U is 1 mod q. But #⁢T is prime to q, so #⁢U is the p-part of #⁢G. Here for brevity we write #⁢G etc. for the cardinality of G¯⁢(k) etc.

To compute #⁢U¯⁢(k) directly, we note that along the diagonal it has entries from D¯⁢(k) with a1=1, so we get qd⁢(d-1)⁢m possibilities. The entries (ai⁢j) (j>i) of SL⁡(m,D¯⁢(k)), ai⁢j∈kE, give a factor of qd2⁢m⁢(m-1)/2. The m⁢(m-1) under diagonal entries whose diagonal entries σi⁢a1 are equal to 0, are qd⁢(d-1)⁢m⁢(m-1)/2 in cardinality. We conclude that

[SL(m,D¯(k)):U¯(k)]=[∏1≤j≤m(qd⁢j-1)](q-1)-1.

We can now use a theorem of Gopal Prasad ([15, Theorem 2.3]). It asserts that the quotient of vol⁡SL⁡(n,O) by vol⁡SL⁡(m,D⁢(O)) equals the quotient of [SL⁡(n,k):U⁢(k)] (here U is the unipotent upper triangular subgroup of SL(n)) by [SL⁡(m,D¯⁢(k)):U¯⁢(k)]. The proposition follows. ∎

7 Elliptic terms

The nonscalar elements are elliptic since D is a division algebra. The contribution from these elements is the sum over the conjugacy classes in D× of elements γ∈αℤ⋅(D×-F×), of ∫D𝔸×/Zγ⁢(F)⋅αℤf⁢(x⁢γ⁢x-1)⁢𝑑x. The centralizer Zγ⁢(F) of a γ in D× is the multiplicative group of a division algebra Dd,Fm central over Fm=F⁢(γ) of degree d=n/m, where F⁢(γ) is a field extension of F of degree m dividing n. Fix a subfield 𝔽qn in D.

Lemma 7.1.

If f⁢(x⁢γ⁢x-1)≠0, for γ in αZ⋅(D×-F×) and x∈DA×, then a conjugate of γ, which we denote again by γ, lies in Fqn-Fq and Fq⁢(γ)=Fqm, m|n. Further, L=F⁢(γ) is F⊗FqFqm, and L/F≃Fqm/Fq is Galois. Finally, m divides (n/S), the largest quotient of n which is prime to all deg⁡(v), v∈S.

Proof.

If f⁢(x⁢γ⁢x-1)≠0, the characteristic polynomial pγ⁢(t) of γ lies in O𝔸⁢[t], it is monic of degree n and det⁡γ∈O𝔸×. Write γ=αj⁢γ′ with j∈ℤ and γ′∈D×-F×, to conclude from det⁡γ∈O𝔸× that j=0, thus γ∈D×-F×. The coefficients of pγ are then in O𝔸∩F=𝔽q, and det⁡γ∈O𝔸×∩F×=𝔽q×. The first two claims of the lemma follow.

This γ lies in D if and only if L⊂D, if and only if Fv⊗FL=Fv⊗𝔽q𝔽qm lies in Dv=Fv⊗FD for each v∈S, if and only if m is prime to the residual exponent deg⁡(v) of Fv (the residual field of Fv is then 𝔽qdeg⁡(v)). The last claim of the lemma follows. ∎

Two γ1,γ2∈𝔽qn are conjugate in D× or D𝔸× if and only if they are Gal⁡(𝔽qn/𝔽q)-conjugate. See [4, Section 3.3].

The sum then ranges over all m dividing (n/S) and over the elements γ∈𝔽qn-𝔽q with 𝔽q⁢(γ)=𝔽qm, a set of cm=∑k|mμ⁢(k)⁢(qm/k-1) elements, up to conjugacy, namely a set of cm/m elements. Here μ is the function on the positive integers taking the value 1 at 1, (-1)r at a k which is a product of r distinct primes, and 0 elsewhere (see [14, Chapter V, Example 22], where the word “monic” is missing in the definition of ψ⁢(d)). The case of m=1 is dealt with in Section 5.

We now compute the integrals

∫D𝔸×/Zγ⁢(F)⋅αℤf⁢(x⁢γ⁢x-1)⁢𝑑x=|Zγ⁢(𝔸)/Zγ⁢(F)⋅αℤ|⁢∏v∫Dv×/Zγ⁢(Fv)fv⁢(x⁢γ⁢x-1)⁢𝑑x.

Lemma 7.2.

When v∉S, the group Gv is GL⁡(n,Fv). For our γ∈Fqn× the local integral equals |Kv|-1⋅|Kv|/|Kv∩Zγ,v|. Here Zγ,v=Zγ⁢(Fv) is GL⁡(d,Fm,v). Their product over all v∉S is |KS∩Dd⁢(AmS)×|-1, where AmS signifies the ring of adèles of Fm without S-components, and KS=∏v∉SKv.

Proof.

We have x⁢γ⁢x-1∈Kv if and only if x∈Kv⁢Zγ,v/Zγ,v=Kv/Kv∩Zγ,v; see [11, Proposition 7.1]. ∎

Lemma 7.3.

For v∈S, we have

∫Dv×/Zγ⁢(Fv)fv⁢(x⁢γ⁢x-1)⁢𝑑x=m⁢|Dd,Fm⁢(Ov)×|-1.

Proof.

For v∈S, Dv is a division algebra. Denote by Tv a maximal torus in Dv. We may assume Dv is the cyclic algebra of Section 6, and Tv is its diagonal torus. Then Dv×/Tv is the disjoint union over 0≤j<n of ρj⁢D⁢(Ov)×/T⁢(Ov), with ρ as in Section 6. We have x⁢γ⁢x-1∈Kv for all x∈Dv× (conjugation by ρ induces Galois conjugation). If γ is regular, that is, its centralizer in Dv× is Tv, the value of the integral ∫Dv×/Zγ⁢(Fv)fv⁢(x⁢γ⁢x-1)⁢𝑑x is n⁢|T⁢(Ov)|-1.

For general γ in D×, the centralizer Zγ of γ in D× is the multiplicative group of a division algebra Dd,Fm over the Galois field extension Fm=F⊗𝔽q𝔽qm of degree m over F. Note that Dd,Fm⁢(A)=Dd⁢(A⊗𝔽q𝔽qm), thus Dd,Fm⁢(𝔸)=Dd⁢(𝔸m). Since γ∈D×⊂Dv×, we have (m,deg⁡(v))=1, and Dd,Fm,v is a division algebra. In particular, Dd,Fm,v×/Tv is the disjoint union over 0≤j<d of ρmj⁢Dd,Fm⁢(Ov)×/T⁢(Ov), where ρm=ρm. Moreover, Dv×/Dd,Fm,v× is the disjoint union over 0≤j<m of ρj⁢D⁢(Ov)×/Dd,Fm⁢(Ov)×. We have x⁢γ⁢x-1∈Kv for all x∈Dv×, so the lemma follows. ∎

Corollary 7.4.

If there are N places v∈X where Dv is a division algebra, each of the summands is then |Dd⁢(Am)×/Dd⁢(Fm)×⋅αZ|⋅|Dd,Fm⁢(OA)|-1⋅mN.

The volume |Dd⁢(𝔸m)×/Dd⁢(Fm)×⋅αℤ| is the product of |Dd⁢(𝔸m)0/Dd⁢(Fm)0|, where superscript 0 means here determinant 1, and of |𝔸m×/Fm×⋅αd⁢ℤ|.

The last factor is d⁢|𝔸m×/Fm×⋅αℤ|. Note that the restriction to 𝔸× of the degree function degFm on 𝔸m is the degree function deg=degF on 𝔸.

The factor |𝔸m×/Fm×⋅αℤ| was computed in Section 5 when m=1. Replacing q by qm there, we get that this factor is equal to hm⁢|O𝔸m×|/(qm-1), hm=#⁢Pic0⁡(X)⁢(𝔽qm).

As for |K∩Dd⁢(𝔸m)×/O𝔸m×|=|Dd⁢(O𝔸m)0|, this again was computed in Section 5 when m=1. Thus

volTSL(d,O𝔸m)-1=∏1≤j<dζ(Xm,qm⁢j).

This multiplied by (qm-1)-1 is

=1(qm-1)⁢∏1≤j<dfm⁢(qm⁢j)(1-qm⁢j)⁢(1-qm⁢(j+1))=1(qn-1)⁢∏1≤j<dfm⁢(qm⁢j)∏1≤j<d(1-qm⁢j)2.

At the places v∈S, Dd,Fm,v is a division algebra of degree d over Fm,v=Fv⊗𝔽q𝔽qm, a field extension of degree m over Fv as (deg⁡(v),m)=1. So we get from Section 6 the factor

∏v∈S∏1≤j<d(qvm⁢j-1)=∏v∈S∏1≤j<d(1-qvm⁢j).

For the last equality note that either n is even, and so is N, or n is odd, so is then d, and d-1 is even.

Acknowledgements

This work is based on preliminary notes to [4] that were not included in the published version of [4]. It was written up after the author gave a series of lectures on the subject at the University of Tokyo, at the kind invitation of Takayuki Oda. Warm thanks also to David Kazhdan, Dipendra Prasad, Eric Opdan, Elmar Große-Klönne, Thomas Zink for invitations to discuss this work at the Hebrew University, TIFR, University of Amsterdam, Humboldt-Universität zu Berlin, Universität Bielefeld, to Sasha Beilinson, Pierre Deligne, Vladimir Drinfeld, Maxim Kontsevich, Gopal Prasad for useful conversations and correspondence, and to the referee for careful reading.

References

  • [1]↑

    J. Arthur, An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4, American Mathematical Society, Providence (2005), 1–263.

  • [2]↑

    J. Bernstein (rédigè par P. Deligne), Le centre de Bernstein, Représentations des groupes réductifs sur un corps local, Hermann, Paris (1984), 1–32.

  • [3]↑

    P. Deligne, Les constantes des equations fonctionnelles des fonctions L, Modular functions of one variable II, Lecture Notes in Math. 349, Springer, Berlin (1972), 501–597.

  • [4]↑

    P. Deligne and Y. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), 921–982.

  • [5]↑

    V. Drinfeld, Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field, Funct. Anal. Appl. 15 (1981), 294–295.

  • [6]↑

    Y. Flicker, Rigidity for automorphic forms, J. Anal. Math. 49 (1987), 135–202.

  • [7]↑

    Y. Flicker, Transfer of orbital integrals and division algebras, J. Ramanujan Math. Soc. 5 (1990), 107–122.

  • [8]↑

    Y. Flicker, The trace formula for GL ⁡ ( 2 ) {\operatorname{GL}(2)} over a function field, Doc. Math. 19 (2014), 1–62.

  • [9]↑

    Y. Flicker, Counting Galois representations with at most one, unipotent, monodromy, Amer. J. Math. 137 (2015), 1–25.

  • [10]↑

    D. Kazhdan, On lifting, Lie group representations II (College Park 1982/1983), Lecture Notes in Math. 1041, Springer, Berlin (1984), 209–249.

  • [11]↑

    R. Kottwitz, Stable trace formula: Elliptic singular terms, Math. Ann. 275 (1986), 365–399.

  • [12]↑

    R. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629–646.

  • [13]↑

    L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), 1–241.

  • [14]↑

    S. Lang, Algebra, 3rd ed., Addison Wesley, Reading 1993.

  • [15]↑

    G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989), 91–117.

  • [16]↑

    A. Weil, Basic number theory, 3rd ed., Grundlehren Math. Wiss. 144, Springer, Berlin 1974.

  • [17]↑

    A. Weil, Adeles and algebraic groups, Birkhäuser, Boston 1982.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    J. Arthur, An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4, American Mathematical Society, Providence (2005), 1–263.

  • [2]

    J. Bernstein (rédigè par P. Deligne), Le centre de Bernstein, Représentations des groupes réductifs sur un corps local, Hermann, Paris (1984), 1–32.

  • [3]

    P. Deligne, Les constantes des equations fonctionnelles des fonctions L, Modular functions of one variable II, Lecture Notes in Math. 349, Springer, Berlin (1972), 501–597.

  • [4]

    P. Deligne and Y. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), 921–982.

  • [5]

    V. Drinfeld, Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field, Funct. Anal. Appl. 15 (1981), 294–295.

  • [6]

    Y. Flicker, Rigidity for automorphic forms, J. Anal. Math. 49 (1987), 135–202.

  • [7]

    Y. Flicker, Transfer of orbital integrals and division algebras, J. Ramanujan Math. Soc. 5 (1990), 107–122.

  • [8]

    Y. Flicker, The trace formula for GL ⁡ ( 2 ) {\operatorname{GL}(2)} over a function field, Doc. Math. 19 (2014), 1–62.

  • [9]

    Y. Flicker, Counting Galois representations with at most one, unipotent, monodromy, Amer. J. Math. 137 (2015), 1–25.

  • [10]

    D. Kazhdan, On lifting, Lie group representations II (College Park 1982/1983), Lecture Notes in Math. 1041, Springer, Berlin (1984), 209–249.

  • [11]

    R. Kottwitz, Stable trace formula: Elliptic singular terms, Math. Ann. 275 (1986), 365–399.

  • [12]

    R. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), 629–646.

  • [13]

    L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), 1–241.

  • [14]

    S. Lang, Algebra, 3rd ed., Addison Wesley, Reading 1993.

  • [15]

    G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989), 91–117.

  • [16]

    A. Weil, Basic number theory, 3rd ed., Grundlehren Math. Wiss. 144, Springer, Berlin 1974.

  • [17]

    A. Weil, Adeles and algebraic groups, Birkhäuser, Boston 1982.

OPEN ACCESS

Journal + Issues

The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

Search