DENNIS GAITSGORY AND JACOB LURIE
Abstract. LetXbe an algebraic curve defined over a finite fieldFqand letGbe a smooth affine group scheme over X with connected fibers whose generic fiber is semisimple and simply connected. In this paper, we affirm a conjecture of Weil by establishing that the Tamagawa number ofGis equal to 1.
Contents
1. Overview 3
Acknowledgements 3
1.1. The Mass Formula and Weil’s Conjecture 3
1.2. Weil’s Conjecture for Function Fields 10
1.3. Cohomological Formulation 16
1.4. Analyzing the Homotopy Type of BunG(X) 19
1.5. Summary of this Paper 24
2. Generalities on`-adic Homology and Cohomology 26
2.1. Higher Category Theory 27
2.2. `-adic Cohomology of Algebraic Varieties 34
2.3. `-adic Cohomology of Prestacks 40
2.4. Acyclicity of the Ran Space 49
2.5. Universal Homology Equivalences 52
3. Nonabelian Poincare Duality 59
3.1. Motivation: Poincare Duality in Topology 59
3.2. Statement of the Theorem 66
3.3. Outline of Proof 69
3.4. Proof of Theorem 3.3.2 72
3.5. Equivariant Sections 77
3.6. Sections of Vector Bundles 82
3.7. Existence of Rational Trivializations 85
3.8. Digression: Maps of Large Degree 91
3.9. Existence of Borel Reductions 92
4. The Formalism of`-adic Sheaves 98
4.1. Etale Sheaves´ 99
4.2. Constructible Sheaves 105
4.3. `-adic Sheaves 111
4.4. The t-Structure on`-Adic Sheaves 121
4.5. Base Change Theorems and Dualizing Sheaves 128
4.6. K¨unneth Formulae and the !-Tensor Product 138
5. The Product Formula 145
5.1. The Cohomology Sheaf of a Morphism 148
5.2. !-Sheaves on Ran(X) 152
1
5.3. Chiral Homology 156
5.4. The Product Formula: First Formulation 162
5.5. Convolution of !-Sheaves 163
5.6. Commutative Factorization Algebras 171
5.7. The Product Formula: Second Formulation 177
6. Calculation of the Trace 181
6.1. The Cotangent Fiber 183
6.2. The Atiyah-Bott Formula 191
6.3. Summable Frob-Modules 194
6.4. The Trace Formula for BG 197
6.5. Calculation of the Trace 201
7. Decomposition of the SheafB 205
7.1. Independence ofG 207
7.2. Construction of the SheavesBS 213
7.3. The Limit of the SheavesBS 218
7.4. Germs ofG-Bundles 224
7.5. Digression: Germs of Equivariant Maps 229
7.6. From Divisors to Open Neighborhoods 235
7.7. The Chiral Homology of the SheavesBS 237
8. The Reduced Cohomology of BunG(X) 243
8.1. Reduced Nonabelian Poincare Duality 244
8.2. The Reduced Product Formula 248
8.3. Application: The Cohomology BunG(X) in Low Degrees 254
8.4. Normalization 260
8.5. Proof of Theorem 8.1.12 270
8.6. Proof of Theorem 8.2.14 279
9. Proof of the Product Formula 282
9.1. Reduction to a Nonunital Statement 283
9.2. Digression: Acyclicity of the Ran Space in Families 287
9.3. The Proof of Proposition 9.1.4 292
9.4. Digression: Verdier Duality on Ran(X) 299
9.5. A Convergence Argument 304
9.6. Proposition 9.4.17: Proof Outline 311
9.7. Construction of a Pullback Square 317
10. The Grothendieck-Lefschetz Trace Formula for BunG(X) 321
10.1. The Trace Formula for a Quotient Stack 324
10.2. Stratifications 328
10.3. The Harder-Narasimhan Stratification (Split Case) 335
10.4. Quasi-Compactness of Moduli Spaces of Bundles 339
10.5. The Harder-Narasimhan Stratification (Generically Split Case) 344
10.6. Comparing Harder-Narasimhan Strata 347
10.7. Reductive Models 359
10.8. Proof of the Trace Formula 362
Appendix A. 368
A.1. G-Bundles 368
A.2. Curves and Divisors 370
A.3. Dilitations 372
A.4. Automorphisms of Semisimple Algebraic Groups 375
A.5. A Relative K¨unneth Formula 381
A.6. Connectivity of the Fat Diagonal 390
References 393
1. Overview
Let K be a number field, let A denote the ring of adeles of K, and let G be a connected semisimple algebraic group overK. A conjecture of Weil (now a theorem, thanks to the work of Kottwitz, Lai, and Langlands) asserts that ifG is simply connected, then the Tamagawa measure µTam(G(K)\G(A)) is equal to 1. Our goal in this paper is to prove an analogous result in the case whereK is the function field of an algebraic curve defined over a finite field.
In this section, we will recall the statement of Weil’s conjecture, translate the function-field analogue into a problem in algebraic geometry, and outline our approach to that problem.
We begin in §1.1 by reviewing the Smith-Minkowski-Siegel mass formula for integral qua- dratic forms (Theorem 1.1.15). We then reformulate the mass formula as a statement about the volumes of adelic groups (following ideas of Tamagawa and Weil) and state the general form of Weil’s conjecture. In§1.2 we consider the function field analogue of Weil’s conjecture.
Reversing the chain of reasoning given in§1.1, we reformulate this conjecture as a problem of counting principal G-bundles on an algebraic curve X defined over a finite field Fq (here we takeGto be a group scheme over the curve X, whose generic fiber is an algebraic group over the function fieldKX).
Principal G-bundles on X can be identified with points of an algebraic stack BunG(X), called themoduli stack ofG-bundles onX. In§1.3, we will state a version of the Grothendieck- Lefschetz trace formula for BunG(X) which reduces the problem of counting G-bundles on X to the problem of computing the trace of the (arithmetic) Frobenius endomorphism of the cohomology ring H∗(BunG(X)×Fq Fq;Z`). Our goal then is to understand the topology of the moduli stack BunG(X). In§1.4, we discuss the analogous problem in the case where X is defined over the field of complex numbers, and describe several “local-to-global” principles which can be used to compute algebro-topological invariants of BunG(X) in terms of the local structure ofGat the points of X. The bulk of this paper is devoted to developing analogous ideas over an arbitrary algebraically closed ground field (such asFq); we provide a brief outline in§1.5.
Acknowledgements. We would like to thank Alexander Beilinson, Vladimir Drinfeld, Bene- dict Gross, and Xinwen Xhu for helpful conversations related to the subject of this paper.
We also thank Brian Conrad for suggesting the problem to us and for offering many helpful suggestions and corrections. The second author would like to thank Stanford University for its hospitality during which much of this paper was written. This work was supported by the National Science Foundation under Grant No. 0906194.
1.1. The Mass Formula and Weil’s Conjecture. LetR be a commutative ring and letV be anR-module. Aquadratic formonV is a mapq:V →Rsatisfying the following conditions:
(a) The construction (v, w)7→q(v+w)−q(v)−q(w) determines anR-bilinear mapV×V → R.
(b) For every elementλ∈Rand everyv∈V, we haveq(λv) =λ2q(v).
Aquadratic space overR is a pair (V, q), whereV is a finitely generated projectiveR-module andqis a quadratic form onV.
One of the basic problems in the theory of quadratic forms can be formulated as follows:
Question 1.1.1. LetR be a commutative ring. Can one classify quadratic spaces overR (up to isomorphism)?
Example 1.1.2. LetV be a vector space over the fieldRof real numbers. Then any quadratic formq:V →Rcan be diagonalized: that is, we can choose a basise1, . . . , en forV such that
q(λ1e1+· · ·+λnen) =λ21+· · ·+λ2a−λ2a+1− · · · −λ2a+b
for some pair of nonnegative integersa, bwitha+b≤n. Moreover, the integersaandbdepend only on the isomorphism class of the pair (V, q) (a theorem of Sylvester). In particular, if we assume thatq is nondegenerate (in other words, thata+b =n), then the isomorphism class (V, q) is completely determined by the dimensionn of the vector space V and the difference a−b, which is called thesignatureof the quadratic formq.
Example 1.1.3. LetQdenote the field of rational numbers. There is a complete classification of quadratic spaces over Q, due to Minkowski (later generalized by Hasse to the case of an arbitrary number field). Minkowski’s result is highly nontrivial, and represents one of the great triumphs in the arithmetic theory of quadratic forms: we refer the reader to [49] for a detailed and readable account.
Let us now specialize to the caseR =Z. We will refer to quadratic spaces (V, q) over Zas even lattices(since the associated bilinear formb(x, y) =q(x+y)−q(x)−q(y) has the property that b(x, x) = 2q(x) is always even). The classification of even lattices up to isomorphism is generally regarded as an intractable problem (see Remark 1.1.17 below). Let us therefore focus on the following variant of Question 1.1.1:
Question 1.1.4. Let (V, q) and (V0, q0) be even lattices. Is there an effective procedure for determining whether or not (V, q) and (V0, q0) are isomorphic?
Let (V, q) be a quadratic space over a commutative ringR, and suppose we are given a ring homomorphism φ : R → S. We let VS denote the tensor product S⊗RV. An elementary calculation shows that there is a unique quadratic formqS:VS →S for which the diagram
V q //
R
φ
VS
qS //S
is commutative. The construction (V, q)7→(VS, qS) carries quadratic spaces overRto quadratic spaces overS; we refer to (VS, qS) as theextension of scalarsof (V, q). If (V, q) and (V0, q0) are isomorphic quadratic spaces overR, then extension of scalars yields isomorphic quadratic spaces (VS, qS) and (VS0, qS0) over S. Consequently, if we understand the classification of quadratic spaces overS and can tell that (VS, qS) and (VS0, q0S) are notisomorphic, it follows that (V, q) and (V0, q0) are not isomorphic.
Example 1.1.5. Let q : Z→ Z be the quadratic form given by q(n) = n2. Then the even lattices (Z, q) and (Z,−q) cannot be isomorphic, because they are not isomorphic after extension of scalars toR: the quadratic space (R, qR) has signature 1, while (R,−qR) has signature−1.
Example 1.1.6. Letq, q0:Z2→Zbe the quadratic forms given by q(m, n) =m2+n2 q0(m, n) =m2+ 3n2.
Then (Z2, q) and (Z2, q0) become isomorphic after extension of scalars to R(since both qua- dratic forms are positive-definite). However, the quadratic spaces (Z2, q) and (Z2, q0) are not
isomorphic, since they are not isomorphic after extension of scalars to the fieldF3=Z/3Z(the quadratic formqF3 is nondegenerate, but qF03 is degenerate).
Using variants of the arguments provided in Examples 1.1.5 and 1.1.6, one can produce many examples of even lattices (V, q) and (V0, q0) that cannot be isomorphic: for example, by arranging thatq and q0 have different signatures (after extension of scalars to R) or have nonisomorphic reductions modulon for some integer n >0 (which can be tested by a finite calculation). This motivates the following definition:
Definition 1.1.7. Let (V, q) and (V0, q0) be positive-definite even lattices. We say that (V, q) and (V0, q0)of the same genusif (V, q) and (V0, q0) are isomorphic after extension of scalars to Z/nZ, for every positive integer n (in particular, this implies thatV and V0 are free abelian groups of the same rank).
Remark 1.1.8. One can also define study genera of lattices which are neither even nor positive definite, but we will restrict our attention to the situation of Definition 1.1.7 to simply the exposition.
More informally, we say that two even lattices (V, q) and (V0, q0) are of the same genus if we cannot distinguish between them using variations on Example 1.1.5 or 1.1.6. It is clear that isomorphic even lattices are of the same genus, but the converse is generally false. However, the problem of classifying even lattices within a genus has a great deal of structure. One can show that there are only finitely many isomorphism classes of even lattices in the same genus as (V, q). Moreover, the celebratedSmith-Minkowski-Siegel mass formulaallows us to say exactly how many (at least when counted with multiplicity).
Notation 1.1.9. Let (V, q) be a quadratic space over a commutative ring R. We let Oq(R) denote the automorphism group of (V, q): that is, the group ofR-module isomorphismsα:V → V such thatq=q◦α. We will refer to Oq(R) as theorthogonal groupof the quadratic space (V, q). More generally, ifφ: R→ S is a map of commutative rings, we let Oq(S) denote the automorphism group of the quadratic space (VS, qS) over S obtained from (V, q) by extension of scalars toS.
Example 1.1.10. Suppose q is a positive-definite quadratic form on an real vector space V of dimension n. Then Oq(R) can be identified with the usual orthogonal group O(n). In particular, Oq(R) is a compact Lie group of dimension n22−n.
Example 1.1.11. Let (V, q) be a positive-definite even lattice. For every integerd, the group Oq(Z) acts by permutations on the setV≤d ={v∈V :q(v)≤d}. Since qis positive-definite, each of the setsV≤dis finite. Moreover, ford0, the action of Oq(Z) onV≤dis faithful (since an automorphism ofV is determined by its action on a finite generating set forV). It follows that Oq(Z) is a finite group.
Let (V, q) be a positive-definite even lattice. The mass formula gives an explicit formula for the sumP
(V0,q0) 1
|Oq0(Z)|, where the sum is taken over all isomorphism classes of even lattices (V0, q0) in the genus of (V, q). The explicit formula is rather complicated in general, depending on the reduction of (V, q) modulo p for various primes p. For simplicity, we will restrict our attention to the simplest possible case.
Definition 1.1.12. Let (V, q) be an even lattice. We will say that (V, q) isunimodularif the bilinear form b(v, w) = q(v+w)−q(v)−q(w) induces an isomorphism of V with its dual Hom(V,Z).
Remark 1.1.13. Let (V, q) be a positive-definite even lattice. The condition that (V, q) be unimodular depends only on the reduction ofqmodulopfor all primesp. In particular, if (V, q) is unimodular and (V0, q0) is in the genus of (V, q), then (V0, q0) is also unimodular. In fact, the converse also holds: any two unimodular even lattices of the same rank are of the same genus (though this is not obvious from the definitions).
Remark 1.1.14. The condition that an even lattice (V, q) be unimodular is very strong: for example, ifqis positive-definite, it implies that the rank ofV is divisible by 8.
Theorem 1.1.15 (Mass Formula: Unimodular Case). Let nbe an integer which is a positive multiple of8. Then
X
(V,q)
1
|Oq(Z)| = Γ(12)Γ(22)· · ·Γ(n2)ζ(2)ζ(4)· · ·ζ(n−4)ζ(n−2)ζ(n2) 2n−1πn(n+1)/4
= Bn/4 n
Y
1≤j<n/2
Bj
4j.
Hereζ denotes the Riemann zeta function,Bj denotes the jth Bernoulli number, and the sum is taken over all isomorphism classes of positive-definite even unimodular lattices(V, q)of rank n.
Example 1.1.16. Letn= 8. The right hand side of the mass formula evaluates to 6967296001 . The integer 696729600 = 21435527 is the order of the Weyl group of the exceptional Lie group E8, which is also the automorphism group of the root lattice ofE8(which is an even unimodular lattice). Consequently, the fraction 6967296001 also appears as one of the summands on the left hand side of the mass formula. It follows from Theorem 1.1.15 that no other terms appear on the left hand side: that is, the root lattice ofE8is theuniquepositive-definite even unimodular lattice of rank 8, up to isomorphism.
Remark 1.1.17. Theorem 1.1.15 allows us to count the number of positive-definite even uni- modular lattices of a given rank with multiplicity, where a lattice (V, q) is counted with mul- tiplicity |O1
q(Z)|. If the rank of V is positive, then Oq(Z) has order at least 2 (since Oq(Z) contains the grouph±1i), so that the left hand side of Theorem 1.1.15 is at most C2, whereCis the number of isomorphism classes of positive-definite even unimodular lattices. In particular, Theorem 1.1.15 gives an inequality
C≥ Γ(12)Γ(22)· · ·Γ(n2)ζ(2)ζ(4)· · ·ζ(n−4)ζ(n−2)ζ(n2)
2n−2πn(n+1)/4 .
The right hand side of this inequality grows very quickly withn. For example, whenn= 32, we can deduce the existence of more than eighty million pairwise nonisomorphic (positive-definite) even unimodular lattices of rankn.
We now describe a reformulation of Theorem 1.1.15, following ideas of Tamagawa and Weil.
Suppose we are given a positive-definite even lattice (V, q), and that we wish to classify other even lattices of the same genus. If (V0, q0) is a lattice in the genus of (V, q), then for every positive integer n we can choose an isomorphism αn : V /nV 'V0/nV0 which is compatible with the quadratic formsqandq0. Using a compactness argument (or some variant of Hensel’s lemma) we can assume without loss of generality that the isomorphisms{αn}n>0are compatible
with one another: that is, that the diagrams V /nV αn //
V0/nV0
V /mV αm //V0/mV0
commute whenevermdivides n. In this case, the data of the family{αn}is equivalent to the data of a single isomorphismα : Zb⊗ZV → Zb ⊗ZV0, where Zb ' lim←−n>0Z/nZ denotes the profinite completion of the ringZ.
By virtue of the Chinese remainder theorem, the ring Zb can be identified with the product Q
pZp, where pranges over all prime numbers and Zp denotes the ring lim
←−Z/pkZ of p-adic integers. It follows that (V, q) and (V0, q0) become isomorphic after extension of scalars to Zp, and therefore also after extension of scalars to the field Qp = Zp[p−1] of p-adic rational numbers. Since the lattices (V, q) and (V0, q0) are positive-definite and have the same rank, they also become isomorphic after extending scalars to the field of real numbers. It follows from Minkowski’s classification that the quadratic spaces (VQ, qQ) and (VQ0, qQ0 ) are isomorphic (this is known as theHasse principle: to show that quadratic spaces overQare isomorphic, it suffices to show that they are isomorphic over every completion ofQ; see§3.3 of [49]). We may therefore choose an isomorphismβ:VQ→VQ0 which is compatible with the quadratic formsq andq0.
LetAf denote the ring offinite adeles: that is, the tensor productZ⊗b ZQ. The isomorphism Zb'Q
pZp induces an injective map
Af 'Zb⊗ZQ,→Y
p
(Zp⊗ZQ)'Y
p
Qp,
whose image is therestricted productQres
p Qp⊆Q
pQp: that is, the subset consisting of those elements{xp}of the productQ
pQp such thatxp∈Zpfor all but finitely many prime numbers p. The quadratic spaces (V, q) and (V0, q0) become isomorphic after extension of scalars toAf in two different ways: via the isomorphismαwhich is defined overZ, and via the isomorphismb βwhich is defined overQ. Consequently, the compositionβ−1◦αcan be regarded as an element of Oq(Af). This element depends not only the quadratic space (V0, q0), but also on our chosen isomorphismsαandβ. However, any other isomorphism between (V
Zb, q
bZ) and (V0
bZ, q0
Zb) can be written in the form α◦γ, where γ ∈Oq(bZ). Similarly, the isomorphism β is well-defined up to right multiplication by elements of Oq(Q). Consequently, the compositionβ−1◦αis really well-defined as an element of the set of double cosets
Oq(Q)\Oq(Af)/Oq(bZ).
Let us denote this double coset by [V0, q0].
It is not difficult to show that the construction (V0, q0) 7→[V0, q0] induces a bijection from the set of isomorphism classes of even lattices (V0, q0) in the genus of (V, q) with the set Oq(Q)\Oq(Af)/Oq(bZ) (the inverse of this construction is given by the procedure of reglu- ing; see Construction 1.2.15). Moreover, if γ∈Oq(Af) is a representative of the double coset [V0, q0], then the group Oq0(Z) is isomorphic to the intersection
Oq(bZ)∩γ−1Oq(Q)γ.
Consequently, the left hand side of the mass formula can be written as a sum X
γ
1
|Oq(bZ)∩γ−1Oq(Q)γ|, (1)
whereγranges over a set of double coset representatives.
At this point, it will be technically convenient to introduce two modifications of the calcu- lation we are carrying out. For every commutative ringR, let SOq(R) denote the subgroup of Oq(R) consisting of those automorhisms of (VR, qR) which have determinant 1 (if R is an integral domain, this is a subgroup of index at most 2). Let us instead attempt to compute the sum
X
γ
1
|SOq(bZ)∩γ−1SOq(Q)γ|, (2)
whereγruns over a set of representatives for the collection of double cosets X= SOq(Q)\SOq(Af)/SOq(bZ).
Ifq is unimodular, expression (2) differs from the expression (1) by an overall factor of 2 (in general, the expressions differ by a power of 2).
Remark 1.1.18. Fix an orientation of theZ-moduleV (that is, a generator of the top exterior power ofV). Quantity (2) can be written as a sum P 1
|SOq0(Z)|, where the sum is indexed by all isomorphism classes oforientedeven unimodular positive-definite lattices (V0, q0) which are isomorphic to (V, q) asorientedquadratic spaces after extension of scalars toZ/nZ, for every integern >0.
Let A denote the ring of adeles: that is, the ring Af×R. Then we can identify X with the collection of double cosets SOq(Q)\SOq(A)/SOq(bZ×R). The virtue of this maneuver is thatAhas the structure of a locally compact commutative ring containing Qas a discrete subring. Consequently, SOq(A) is a locally compact topological group which contains SOq(Q) as a discrete subgroup and SOq(bZ×R) as a compact open subgroup.
Let µ be a Haar measure on the group SOq(A). One can show that the group SOq(A) is unimodular: that is, the measureµ is invariant under both right and left translations. In particular,µdetermines a measure on the quotient SOq(Q)\SOq(A), which is invariant under the right action of SOq(bZ×R). We will abuse notation by denoting this measure also by µ.
Write SOq(Q)\SOq(A) as a union of orbitsS
x∈XOxfor the action of the group SOq(bZ×R).
Ifx ∈X is a double coset represented by an element γ ∈ SOq(A), then we can identify the orbitOx with the quotient of SOq(bZ×R) by the finite subgroup SOq(bZ×R)∩γ−1SOq(Q)γ.
We therefore have X
γ
1
|SOq(bZ×R)∩γ−1SOq(Q)γ| = X
x∈X
µ(Ox) µ(SOq(bZ×R)) (3)
= µ(SOq(Q)\SOq(A)) µ(SOq(bZ×R)) . (4)
Of course, the Haar measure µon SOq(A) is only well-defined up to scalar multiplication.
Rescaling the measureµhas no effect on the right hand side of the preceding equation, since µappears in both the numerator and the denominator of the right hand side. However, it is possible to say more: it turns out that there is a canonical normalization of the Haar measure, known asTamagawa measure.
Construction 1.1.19. LetG be a linear algebraic group of dimension d over the fieldQ of rational numbers. Let Ω denote the collection of all left invariantd-forms onG, so that Ω is a 1-dimensional vector space overQ. Choose a nonzero elementω∈Ω.
The vector ω determines a left-invariant differential form of top degree on the smooth man- ifoldG(R), which in turn determines a Haar measureµR,ω onG(R). For every prime number p, an analogous construction yields a measure µQ
p,ω on the p-adic analytic manifold G(Qp).
Assuming that G is connected and and has no nontrivial characters, one can show that the product of these measures determines a measureµTam on the restricted product
G(R)×
res
Y
p
G(Qp)'G(A).
Letλbe a nonzero rational number. Then an elementary calculation gives µR,λω =|λ|µR,ω µQ
p,λω =|λ|pµQ
p,ω;
here |λ|p denotes the p-adic absolute value of λ. Combining this with the product formula Q
p|λ|p= |λ|1 , we deduce thatµTam is independent of the choice of nonzero elementω∈Ω. We will refer toµTam as theTamagawa measureof the algebraic groupG.
If (Λ, q) is a positive-definite even lattice, then the restriction of the functor R 7→SOq(R) to Q-algebras can be regarded as a semisimple algebraic group over Q. We may therefore apply Construction 1.1.19 to obtain a canonical measureµTam on the group SOq(A). We may therefore specialize equation (4) to obtain an equality
X
γ
1
|SOq0(Z)| = µTam(SOq(Q)\SOq(A)) µTam(SOq(bZ×R)) , (5)
where it makes sense to evaluate the numerator and the denominator of the right hand side independently.
Remark 1.1.20. The constructionR7→Oq(R) also determines a semisimple algebraic group over Q. However, this group is not connected, and the infinite product Q
pµQp,ω does not converge to a measure on the restricted productQres
p Oq(Qp) = Oq(Af). This is one reason for preferring to work with the group SOq in place of Oq.
Remark 1.1.21. The numerator on the right hand side of (5) is called theTamagawa number of the algebraic group SOq. More generally, if G is a connected semisimple algebraic group over Q, we define the Tamagawa number of G to be the Tamagawa measure of the quotient G(Q)\G(A).
The denominator on the right hand side of (5) is computable: if we choose a differential form ωas in Construction 1.1.19, it is given by the product
µR,ω(SOq(R))Y
p
µQ
p,ω(SOq(Zp)).
The first term is the volume of a compact Lie group, and the second term is a product of local factors which are related to counting problems over finite rings. Carrying out these calculations leads to a very pretty reformulation of Theorem 1.1.15:
Theorem 1.1.22(Mass Formula, Adelic Formulation). Let(V, q)be a nondegenerate quadratic space overQ. Then µTam(SOq(Q)\SOq(A)) = 2.
The appearance of the number 2 in the statement of Theorem 1.1.22 results from the fact that the algebraic group SOq is not simply connected. Let Spinq denote the (2-fold) universal cover of SOq, so that Spinq is a simply connected semisimple algebraic group overQ. We then have the following more basic statement:
Theorem 1.1.23. Let(V, q)be a positive-definite quadratic space overQ. Then µTam(Spinq(Q)\Spinq(A)) = 1.
Remark 1.1.24. For a deduction of Theorem 1.1.22 from Theorem 1.1.23, see [43].
Theorem 1.1.23 motivates the following:
Conjecture 1.1.25(Weil’s Conjecture on Tamagawa Numbers). LetGbe a simply connected semisimple algebraic group overQ. ThenµTam(G(Q)\G(A)) = 1.
Conjecture 1.1.25 was proved by Weil in a number of special cases. The general case was proven by Langlands in the case of a split group ([31]), by Lai in the case of a quasi-split group ([29]), and by Kottwitz for arbitrary simply connected algebraic groups satisfying the Hasse principle ([28]; this is now known to be all simply connected semisimple algebraic groups over Q, by work of Chernousov).
The goal of this paper is to address the function field analogue of Conjecture 1.1.25, which we will discuss in the next section.
1.2. Weil’s Conjecture for Function Fields. In this section, we will review the definition of Tamagawa measure for algebraic groupsG which are defined over function fields. We will then state the function field analogue of Weil’s conjecture, and explain how to reformulate it as a counting problem (using the logic of§1.1 in reverse).
Notation 1.2.1. LetFq denote a finite field withqelements, and letX be an algebraic curve over Fq (which we assume to be smooth, proper, and geometrically connected). We let KX
denote the function field of the curveX (that is, the residue field of the generic point ofX).
We will writex∈X to mean thatxis a closedpoint of the curveX. For each pointx∈X, we let κ(x) denote the residue field of X at the point x. Then κ(x) is a finite extension of the finite fieldFq. We will denote the degree of this extension by deg(x) and refer to it as the degreeof x. We letOx denote the completion of the local ring of X at the point x: this is a complete discrete valuation ring with residue fieldκ(x), noncanonically isomorphic to a power series ringκ(x)[[t]]. We letKxdenote the fraction field ofOx.
For every finite set Sof closed point ofX, letAS denote the productQ
x∈SKx×Q
x /∈SOx. We letAdenote the direct limit
lim−→
S⊆X
AS.
We will refer to A as the ring of adeles of KX. It is a locally compact commutative ring, equipped with a ring homomorphismKX→Awhich is an isomorphism ofKX onto a discrete subset ofA. We letA0=Q
x∈XOx denote the ring ofintegral adeles, so thatA0is a compact open subring ofA.
LetG0be a linear algebraic group of dimensionddefined over the fieldKX. It will often be convenient to assume that we are given anintegral modelofG0: that is, thatG0is given as the generic fiber of a smooth affine group schemeGover the curveX.
Remark 1.2.2. If G0 is a simply connected semisimple algebraic group over KX, then it is always possible to find a smooth affine group scheme with generic fiberG0. See, for example, [12] or§7.1 of [11].
Remark 1.2.3. In what follows, it will sometimes be convenient to assume that the group scheme G → X has connected fibers. This can always be arranged by passing to an open subgroupG◦⊆G(which does not injure our assumption thatGis an affine group scheme over X, since the open immersion G◦ ,→ Gis complementary to a Cartier divisor and is therefore an affine morphism).
For every commutative ring R equipped with a map SpecR→X, we letG(R) denote the group ofR-points ofG. Then:
• For each closed point x∈X,G(Kx) is a locally compact group, which containsG(Ox) as a compact open subgroup.
• We can identify G(A) with the restricted product Qres
x∈XG(Kx): that is, with the subgroup of the product Q
x∈XG(Kx) consisting of those elements{gx}x∈X such that gx∈G(Ox) for all but finitely many values ofX.
• The group G(A) is locally compact, containing G(KX) as a discrete subgroup and G(A0)'Q
x∈XG(Ox) as a compact open subgroup.
1.2.1. Let us now review the construction of Tamagawa measure on the locally compact group G(A). Let ΩG/X denote the relative cotangent bundle of the smooth morphism π: G→X. Then ΩG/X is a vector bundle onGof rankd= dim(G0). We let ΩdG/X denote the top exterior power of ΩG/X, so that ΩdG/X is a line bundle on G. Let L denote the pullback of ΩdG/X along the identity sectione: X → G. Equivalently, we can identify L with the subbundle of π∗ΩdG/X consisting of left-invariant sections. LetL0 denote the generic fiber ofL, so thatL0is a 1-dimensional vector space over the function fieldKX. Fix a nonzero elementω∈L0, which we can identify with a left-invariant differential form of top degree on the algebraic groupG0.
For every pointx∈X,ωdetermines a Haar measureµx,ωon the locally compact topological groupG(Kx). Concretely, this measure can be defined as follows. Lettdenote a uniformizing parameter forOx (so thatOx'κ(x)[[t]]), and letGOx denote the fiber product SpecOx×XG.
Choose a local coordinatesy1, . . . , yd for the group GOx near the identity: that is, coordinates which induce a mapu:GOx →AdO
x which is ´etale at the origin ofGOx. Letvx(ω) denote the order of vanishing ofωat the pointx. Then, in a neighborhood of the origin inGOx, we can write ω=tvx(ω)λdy1∧ · · · ∧dyd, whereλis an invertible regular function. Letmxdenote the maximal ideal ofOx, and let G(mx) denote the kernel of the reduction map G(Ox)→ G(κ(x)). Since y1, . . . , yd are local coordinates near the origin, the map uinduces a bijection G(mx) → mdx. The measure defined by the differential formdy1∧· · ·∧dydonG(mx) is obtained by pulling back the “standard” measure onKxdalong the mapu, where this standard measure is normalized so thatOdx has measure 1. It follows that the measure ofG(mx) (with respect to the differential formdy1∧ · · · ∧dyd) is given by |κ(x)|1 d. We then define
µx,ω(G(mx)) =q−deg(x)vx(ω) 1
|κ(x)|d.
The smoothness ofGimplies that the mapG(Ox)→G(κ(x)) is surjective, so that we have µx,ω(G(Ox)) =q−deg(x)vx(ω)|G(κ(x))|
|κ(x)|d .
Remark 1.2.4. Since G(Ox) is a compact open subgroup of G(Kx), there is a unique left- invariant measureµonG(Ox) satisfying
µ(G(Ox)) =q−deg(x)vx(ω)|G(κ(x))|
|κ(x)|d .
The reader can therefore take this expression as thedefinitionof the measure µx,ω. However, the analytic perspective is useful for showing that this measure is independent of the choice of integral model chosen. We refer the reader to [57] for more details.
The key fact is the following:
Proposition 1.2.5. Suppose that G0 is connected and semisimple, and let ω be a nonzero element ofL0. Then the measuresµx,ω on the groups G(Kx) determine a well-defined product measure onG(A) =Qres
x∈XG(Kx). Moreover, this product measure is independent of ω.
Proof. To check that the product measure is well-defined, it suffices to show that it is well- defined when evaluated on a compact open subgroup ofG(A), such asG(A0). This is equivalent to the absolute convergence of the infinite product
Y
x∈X
µx,ω(G(Ox)) = Y
x∈X
q−deg(x)vx(ω)|G(κ(x))|
|κ(x)|d , which we will discuss in§6.5.
The fact that the product measure is independent of the choice of ω follows from the fact that the infinite sum
X
x∈X
deg(x)vx(ω) = deg(L)
is independent ofω.
Definition 1.2.6. LetG0 be a connected semisimple algebraic group overKX. Letddenote the dimension ofG0, and let g denote the genus of the curve X. The Tamagawa measureon G(A) is the Haar measure given informally by the product
µTam=qd(1−g) Y
x∈X
µx,ω
Remark 1.2.7. Equivalently, we can define Tamagawa measure µTam to be the unique Haar measure onG(A) which is normalized by the requirement
µTam(G(A0)) =qd(1−g)−deg(L) Y
x∈X
|G(κ(x))|
|κ(x)|d .
Remark 1.2.8. To ensure that the Tamagawa measureµTam is well-defined, it is important that the quotients |G(κ(x))||κ(x)|d converge swiftly to 1, so that the infinite productQ
x∈X
|G(κ(x))|
|κ(x)|d is absolutely convergent. This can fail dramatically ifG0is not connected. However, it is satisfied for some algebraic groups which are not semisimple: for example, the additive groupGa. Remark 1.2.9. If the groupG0is semisimple, then any left-invariant differential formωof top degree onG0is also right-invariant. It follows that the groupG(A) is unimodular. In particular, the measure µTam on G(A) descends to a measure on the quotient G(KX)\G(A), which is invariant under the action ofG(A) by right translation. We will denote this measure also by µTam, and refer to it asTamagawa measure. It is characterized by the following requirement:
for every positive measurable functionf onG(A), we have Z
x∈G(A)
f(x)dµTam = Z
y∈G(KX)/G(A)
( X
π(x)=y
f(x))dµTam, (6)
whereπ:G(A)→G(KX)\G(A) denotes the projection map.
An important special case occurs when f is the characteristic function of a coset γH for some compact open subgroupH ⊆G(A). In this case, each element ofπ(γH) has exactlyo(γ)
preimages inU, whereo(γ) denotes the order of the finite groupG(KX)∩γHγ−1 (this group is finite, since it is the intersection of a discrete subgroup ofG(A) with a compact subgroup of G(A)). Applying formula (6), we deduce that µTam(π(γH)) = µTamo(γ)(H).
Example 1.2.10. LetG = Ga be the additive group. Then the dimension dof Gis equal to 1, and the line bundleLof left-invariant top forms is isomorphic to the structure sheafOX
of X. Moreover, we have an equality |G(κ(x))| = |κ(x)| for eachx ∈ X. Consequently, the Tamagawa measureµTam is characterized by the formulaµTam(G(A0)) =q1−g. Note that we have an exact sequence of locally compact groups
0→H0(X;OX)→G(A0)→G(KX)\G(A)→H1(X;OX)→0, so that the Tamagawa measure of the quotientG(KX)\G(A) is given by
|H1(X;OX)|
|H0(X;OX)|µTam(G(A0)) = qg
qq1−g= 1.
Remark 1.2.11.One might ask the motivation for the auxiliary factorqd(1−g)appearing in the definition of the Tamagawa measure. Remark 1.2.10 provides one answer: the auxiliary factor is exactly what we need in order to guarantee that Weil’s conjecture holds for the additive groupGa.
Another answer is that the auxiliary factor is necessary to obtain invariance under Weil restriction. Suppose that f :X →Y is a separable map of algebraic curves overFq. LetKY
be the fraction field ofY (so that KX is a finite separable extension of KY), let AY denote the ring of adeles ofKY, and letH0 denote the algebraic group overKY obtained fromG0by Weil restriction along the field extensionKY ,→KX. Then we have a canonical isomorphism of locally compact groupsG0(A)'H0(AY). This isomorphism is compatible with the Tamagawa measures on each side, but only if we include the auxiliary factorqd(1−g)indicated in Definition 1.2.6. See [42] for more details.
1.2.2. Our goal in this paper is to address the following version of Weil’s conjecture:
Conjecture 1.2.12(Weil). Suppose thatG0 is semisimple and simply connected. Then µTam(G(KX)\G(A)) = 1.
Let us now reformulate Conjecture 1.2.12 in more elementary terms. Note that the quotient G(KX)\G(A) carries a right action of the compact group G(A0). We may therefore write G(KX)\G(A) as a union of orbits, indexed by the collection of double cosets
G(KX)\G(A)/G(A0).
Applying Remark 1.2.9, we calculate µTam(G(KX)\G(A)) = X
γ
µTam(G(A0))
|G(A0)∩γ−1G(KX)γ|
= qd(1−g)−deg(L)(Y
x∈X
|G(κ(x))|
|κ(x)|d )X
γ
1
|G(A0)∩γ−1G(KX)γ|. We may therefore reformulate Weil’s conjecture as follows:
Conjecture 1.2.13 (Weil). Suppose that G0 is semisimple and simply connected. Then we have an equality
Y
x∈X
|κ(x)|d
|G(κ(x))| =qd(1−g)−deg(L)X
γ
1
|G(A0)∩γ−1G(KX)γ|,
where the sum on the right hand side is taken over a set of representatives for the double quotientG(KX)\G(A)/G(A0).
Remark 1.2.14. In the statement of Conjecture 1.2.13, the product on the left hand side and the sum on the right hand side are generally both infinite. The convergence of the left hand side is equivalent to the well-definedness of Tamagawa measureµTam, and the convergence of the right hand side is equivalent to the statement thatµTam(G(KX)\G(A)) is finite.
1.2.3. We now give an algebro-geometric interpretation of the sum appearing on the right hand side of Conjecture 1.2.13. In what follows, we will assume that the reader is familiar with the theory of principalG-bundles; we will give a brief review in§A.1.
Construction 1.2.15(Regluing). Letγbe an element of the groupG(A). We can think ofγ as given by a collection of elementsγx∈G(Kx), having the property that there exists a finite setS such thatγx∈G(Ox) whenever x /∈S.
We define a G-bundlePγ onX as follows:
(a) The bundlePγ is equipped with a trivializationφon the open setU =X−S.
(b) The bundle Pγ is equipped with a trivialization ψx over the scheme SpecOx of each point x∈S.
(c) For each x ∈ S, the trivializations of Pγ|SpecKx determined by φ and ψx differ by multiplication by the the element γx∈G(Kx).
Note that theG-bundlePγis canonically independent of the choice ofS, so long asScontains all pointsxsuch thatγx∈/G(Ox).
Remark 1.2.16. Letγ, γ0 ∈G(A). TheG-bundlesPγ andPγ0 come equipped with trivializa- tions at the generic point ofX. Consequently, giving an isomorphism between the restrictions Pγ|SpecKX and Pγ0|SpecKX is equivalent to giving an element β ∈ G(KX). Unwinding the definitions, we see that this isomorphism admits an (automatically unique) extension to an iso- morphism ofPγ withPγ0 if and only ifγ0−1βγ belongs toG(A0). This has two consequences:
(a) The G-bundles Pγ and Pγ0 are isomorphic if and only ifγ andγ0 determine the same element of G(KX)\G(A)/G(A0).
(b) The automorphism group of theG-torsorPγ is the intersectionG(A0)∩γ−1G(KX)γ.
Remark 1.2.17. Let P be a G-bundle on X. Then P can be obtained from Construction 1.2.15 if and only if the following two conditions are satisfied:
(i) There exists an open setU ⊆X such that P|U is trivial.
(ii) For each pointx∈X−U, the restriction of Pto SpecOx is trivial.
By a direct limit argument, condition (i) is equivalent to the requirement that P|SpecKX be trivial: that is, thatPis classified by a trivial element of H1(SpecKX;G0). IfG0 is semisimple and simply connected, then H1(SpecKX;G0) vanishes (see [24]).
If the mapG→X is smooth and has connected fibers, then condition (ii) is automatic (the restrictionP|Specκ(x) can be trivialized by Lang’s theorem (see [30]), and any trivialization of P|Specκ(x)can be extended to a trivialization ofP|SpecOx by virtue of the assumption thatG is smooth.
Suppose now that Ghas connected fibers. Combining Remarks 1.2.16 and 1.2.17, we obtain the formula
µTam(G(KX)\G(A))'qd(1−g)−deg(L)(Y
x∈X
|G(κ(x))|
|κ(x)|d )X
P
1
|Aut(P)|.
Here the sum is taken over all isomorphism classes of generically trivialG-bundles on X. We may therefore reformulate Conjecture 1.2.12 as follows:
Conjecture 1.2.18(Weil). LetG→X be a smooth affine group scheme with connected fibers whose generic fiber is semisimple and simply connected. Then
Y
x∈X
|κ(x)|d
|G(κ(x))| =qd(1−g)−deg(L)X
P
1
|Aut(P)|.
The assertion of Conjecture 1.2.18 can be regarded as a function field version of Theorem 1.1.15. More precisely, we have the following table of analogies:
Classical Mass Formula Conjecture 1.2.18 Number fieldQ Function fieldKX
Quadratic space (VQ, qQ) overQ Algebraic GroupG0
Even lattice (V, q) Integral modelG Even lattice (V0, q0) of the same genus PrincipalG-bundleP
P
q0 1
|Oq0(Z)|
P
P 1
|Aut(P)|.
1.2.4. There are a number of tools that are available for attacking Conjecture 1.2.18 that have no analogue in the case of a number field. More specifically, we would like to take advantage of the fact that the collection of allG-bundles onX admits an algebro-geometric parametrization.
Notation 1.2.19. For everyFq-algebraR, let BunG(X)(R) denote the category of principal G-bundles on the relative curve XR = SpecR ×SpecFq X (where morphisms are given by isomorphisms of G-bundles). The construction R 7→ BunG(X)(R) determines an algebraic stack, which we will denote by BunG(X) and refer to as themoduli stack ofG-bundles onX.
By definition, we can identifyFq-valued points of BunG(X) with principalG-bundles onX. We will denote the sumP
P 1
|Aut(P)|by|BunG(X)(Fq)|: we can think of it as a (weighted) count of the objects of BunG(X)(Fq), which properly takes into account the fact that BunG(X)(Fq) is a groupoid rather than a set.
Remark 1.2.20. One can show that BunG(X) is a smooth algebraic stack overFq. Moreover, for everyG-bundlePonX, the dimension of BunG(X) at the point determined byPis given by the Euler characteristic
−χ(gP) = H1(X;gP)−H0(X;gP),
wheregPdenotes the vector bundle onX obtained by twisting the Lie algebragofGusing the torsorP. Since the generic fiberG0is semisimple, the groupGacts trivially on the top exterior powerVd
g, so that
d
^gP '
d
^g'L∨.
It follows that the vector bundle gP has degree −deg(L), so that so that the Riemann-Roch theorem givesχ(gP) =d(1−g)−deg(L) is independent ofP. Applying the same analysis to anyR-valued point of BunG(X), we conclude that BunG(X) is equidimensional of dimension d(g−1) + deg(L). We may therefore rewrite the right hand side of Conjecture 1.2.18 as a fraction
|BunG(X)(Fq)|
qdim(BunG(X)) .
Heuristically, this is a normalized count of the numberG-bundles onX, where the normaliza- tion factorqdim(BunG(X)) can be regarded as a naive estimate determined by the dimension of BunG(X).
1.2.5. For every closed pointx∈X, letGx denote the fiber product Specκ(x)×XG, so that Gx is a connected algebraic group overκ(x). Let BGx denote the classifying stack ofGx: this is a smooth algebraic stack of dimension−dover Specκ(x). Then BGx(Fq) is the category of Gx-bundles on Specκ(x). IfGxis connected, then Lang’s theorem implies that everyGx-bundle on Specκ(x) is trivial. Moreover, the automorphism group of the trivialGx-bundle is given by Gx(κ(x)) =G(κ(x)). Consequently, we have an identity
|κ(x)|d
|G(κ(x))| = |BGx(κ(x))|
|κ(x)|dim(BGx). We may therefore rewrite Weil’s conjecture in the suggestive form
|BunG(X)(Fq)|
qdim(BunG) = Y
x∈X
|BGx(κ(x))|
|κ(x)|dim(BGx). (7)
Roughly speaking, formula (7) reflects the idea that BunG(X) can be viewed as a “continuous product” of the classifying stacks BGx, where x ranges over the closed points of X. Most of this paper will be devoted to making this heuristic idea more precise.
1.3. Cohomological Formulation. Throughout this section, we let X denote an algebraic curve defined over a finite fieldFq andGa smooth affine group scheme over X. The analysis given in §1.2 shows that Weil’s conjecture can be reduced to the problem of computing the sumP
P 1
|Aut(P)|, where P ranges over all isomorphism classes of G-bundles on X. Roughly speaking, we can think of this quantity as counting the number ofFq-points of the moduli stack BunG(X).
1.3.1. Let us begin by discussing the analogous counting problem where we replace BunG(X) by an algebraic variety Y defined over Fq. Let Fq be an algebraic closure of Fq, and let Y = SpecFq×SpecFqY denote the associated algebraic variety overFq. We let Frob :Y →Y denote the product of the identity map from SpecFq to itself with the absolute Frobenius map from Y to itself. We refer to Frob as the geometric Frobenius map on Y. If Y is a quasi- projective variety equipped with an embedding j : Y ,→ Pn, then the map Frob is given in homogeneous coordinates by the construction
[x0:· · ·:xn]7→[xq0:· · ·xqn]
(this map carriesY to itself, since Y can be described using homogeneous polynomials with coefficients inFq).
Let Y(Fq) denote the finite set of Fq-points of Y. Then Y(Fq) can be identified with the fixed point locus of the map Frob :Y →Y. Weil had the beautiful insight that one should be able to compute the integers|Y(Fq)|using the Lefschetz fixed-point formula, provided that one had a sufficiently robust cohomology theory for algebraic varieties. Motivated by this heuristic, he made a series of famous conjectures about the behavior of the integers|Y(Fq)|.
Weil’s conjectures were eventually proven by the Grothendieck school using the theory of
`-adic cohomology. We will give a brief summary here, and a more detailed discussion in§2. Fix a prime number`which is invertible inFq. To every algebraic varietyV overFq, the theory of
`-adic cohomology assigns`-adic cohomology groups{Hn(V;Q`)}n≥0and compactly supported
`-adic cohomology groups{Hnc(V;Q`)}n≥0, which are finite dimensional vector spaces overQ`.
IfY is an algebraic variety overFq, then the geometric Frobenius map Frob :Y →Y is proper and therefore determines a pullback map from H∗c(Y;Q`) to itself. We will abuse notation by denoting this map also by Frob. We then have the following:
Theorem 1.3.1(Grothendieck-Lefschetz Trace Formula). Let Y be an algebraic variety over Fq. Then the number ofFq-points of Y is given by the formula
|Y(Fq)|=X
i≥0
(−1)iTr(Frob|Hic(Y;Q`)).
1.3.2. For our purposes, it will be convenient to write the Grothendieck-Lefschetz trace formula in a slightly different form. Suppose now thatY is a smooth variety of dimensionn overFq. Then, from the perspective of`-adic cohomology,Y behaves as if it were a smooth manifold of dimension 2n. In particular, it satisfies Poincare duality: that is, there is a perfect pairing
µ: Hic(Y;Q`)⊗Q` H2n−i(Y;Q`)→Q`.
This pairing is not quite Frob-equivariant: instead, it fits into a commutative diagram Hic(Y;Q`)⊗Q`H2n−i(Y;Q`) µ //
Frob⊗Frob
Q`
qn
Hic(Y;Q`)⊗Q`H2n−i(Y;Q`) µ //Q`,
reflecting the idea that the geometric Frobenius map Frob :Y →Y has degreeqn. In particular, pullback along the geometric Frobenius map Frob induces an isomorphism from H∗(Y;Q`) to itself, and we have the identity
q−nTr(Frob|Hic(Y;Q`))'Tr(Frob−1|H2n−i(Y;Q`)).
We may therefore rewrite Theorem 1.3.1 as follows:
Theorem 1.3.2(Grothendieck-Lefschetz Trace Formula, Dual Version). LetY be an algebraic variety overFq which is smooth of dimension n. Then the number ofFq-points ofY is given by the formula
|Y(Fq)|
qn =X
i≥0
(−1)iTr(Frob−1|Hi(Y;Q`)).
1.3.3. We would like to apply an analogue of Theorem 1.3.2 to the problem of counting G- bundles on an algebraic curveX.
Notation 1.3.3. LetCdenote the field of complex numbers, and fix an embeddingι:Z`,→C.
LetM be aZ`-module for whichC⊗Z` M is a finite-dimensional vector space overC. Ifψis any endomorphism ofM as aZ`-module, we let Tr(ψ|M)∈Cdenote the trace ofC-linear map C⊗Z` M →C⊗Z`M determined byψ. More generally, ifψ is an endomorphism of a graded Z`-module M∗, we let Tr(ψ|M∗) denote the alternating sum P
i≥0(−1)iTr(ψ|Mi) (provided that this sum is convergent).
Let BunG(X) denote the moduli stack of G-bundles on X. We let BunG(X) denote the fiber product SpecFq×SpecFqBunG(X), which we regard as a smooth algebraic stack overFq. For everyFq-algebra R, we can identify the category ofR-valued points of BunG(X) with the category of principalG-bundles on the relative curveXR= SpecR×SpecFqX.
Note that if R is an Fq-algebra, then the construction a 7→ aq determines an Fq-algebra homomorphism from R to itself, and therefore induces a map FrobR : XR → XR (which is the identity on X). If P is a principal G-bundle on XR, then Frob∗RP is another principal
