
TL;DR
This paper introduces arithmetic characteristic curves for arithmetic Higgs torsors over number fields, extending concepts from algebraic geometry to arithmetic settings, with applications to Hitchin fibrations and the fundamental lemma.
Contribution
It constructs arithmetic characteristic curves using Chevalley morphisms and bases, bridging algebraic geometry and number theory in a novel way.
Findings
Defines arithmetic torsors and Higgs torsors over number fields.
Constructs arithmetic characteristic curves based on Chevalley morphisms.
Sets the stage for future work on Hitchin fibrations and fundamental lemma applications.
Abstract
For a split reductive group defined over a number field, we first introduce the notations of arithmetic torsors and arithmetic Higgs torsors. Then we construct arithmetic characteristic curves associated to arithmetic Higgs torsors, based on the Chevalley characteristic morphism and the existence of Chevalley basis for the associated Lie algebra. As to be expected, this work is motivated by the works of Beauville-Narasimhan on spectral curves and Donagi-Gaistgory on cameral curves in algebraic geometry. In the forthcoming papers, we will use arithmetic characteristic curves to construct arithmetic Hitchin fibrations and study the intersection homologies and perverse sheaves for the associated structures, following Ngo's approach to the fundamental lemma.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Advanced Algebra and Geometry
Arithmetic Characteristic Curves
Lin WENG
Abstract
For a split reductive group defined over a number field, we first introduce the notations of arithmetic torsors and arithmetic Higgs torsors. Then we construct arithmetic characteristic curves associated to arithmetic Higgs torsors, based on the Chevalley characteristic morphism and the existence of Chevalley basis for the associated Lie algebra. As to be expected, this work is motivated by the works of Beauville-Narasimhan on spectral curves and Donagi-Gaistgory on cameral curves in algebraic geometry. In the forthcoming papers, we will use arithmetic characteristic curves to construct arithmetic Hitchin fibrations and study the intersection homologies and perverse sheaves for the associated structures, following Ngo’s approach to the fundamental lemma.
1 Chevelley’s Characteristic Morphism
1.1 Over Number Fields
Let be a number field with its ring of integers. Denote by the associated uncompleted arithmetic curve.
Let be a split reductive group over . Fix a split maximal subtorus and a maximal split quotient torus of . Denote the Lie algebra of by , and set be the associated commutative subalgebra of .
Recall that, with respect to the adjoint action
[TABLE]
admits a natural decomposition
[TABLE]
where
[TABLE]
for running through a finite subset of the space
[TABLE]
of rational characteristics of .
For a fixed minimal split parabolic subgroup of containing , set . Then is proper, and there exists a finite subset of , the so-called set of positive roots of , such that
- (1)
, 2. (2)
, and 3. (3)
admits a subset of simple roots associated to , such that
- (i)
, where {\mathbb{Z}}_{\geq 0}:=\big{\{}n\in{\mathbb{Z}}:\,n\geq 0\big{\}}, and 2. (ii)
forms a basis of the -linear space
[TABLE]
Let be the Weyl group of , defined as the finite quotient group
[TABLE]
where , resp. , denotes the normalizer subgroup, resp. the centralizer subgroup, of in . It is well known that is canonically isomorphic to the subgroup of the automorphism group of generated by the reflections
[TABLE]
It is a canonical result due to Chevalley that, over the base field , the space of -invariant polynomials on coincides with the space of the -invariant polynomials of . That is to say,
[TABLE]
where the actions of both sides are defined by
[TABLE]
Similarly, in terms of the Lie algebras, we have
[TABLE]
where acts on in terms of the adjoint action , namely, is defied as the differential of the conjugation morphism of .
In terms of geometry, the isomorphism (1) naturally induces the scheme theoretic morphism
[TABLE]
or equivalently, for the associated Lie structures,
[TABLE]
where and similarly .
Example 1**.**
For , we have , the symmetric group on symbols and . Then coincides with the morphism
[TABLE]
where denotes the one dimensional vector space generated by the -th elementary symmetric polynomials, and denotes the unity matrix of size . That is to say, assigns a matrix to the associated eigen polynomial. In particular, by restricting to which consists of diagonal matrices , we conclude that
[TABLE]
For this reason, the morphism for general is called the Chevelley characteristic morphism. From above, after restricting to , the Chevelley characteristic morphism is simply equivalent to the assignments of the unordered eigenvalues.
1.2 Over Integral Bases
The diagram in (2) associated to a split reductive group only works over the pointed base . In this section, we construct a natural extension to the integral base , whose generic fiber (over the generic point ) coincides with that of (2).
To start with, we recall the so-called Chevelley basis for . For simplicity, we assume that for the time being.
Definition 1** (See e.g. Ch. VII, §25 of [6]).**
A Chevelley basis for is a basis for the -linear space , consisting of \big{\{}x_{\alpha}:\alpha\in\Phi\big{\}}\bigsqcup\big{\{}h_{\alpha}:\alpha\in\Delta\big{\}} which satisfy the following properties:
- (a)
for all , . 2. (b)
for all , so that and span a three dimensional simple subalgebra of which is isomorphic to via
[TABLE] 3. (c)
for , if , then
- (i)
. 2. (ii)
where the constants are defined by the -string through .
We have the following well-known
Theorem 2** (Chevelley).**
Over , we have
- (1)
There always exists a Chevelley basis \big{\{}x_{\alpha}:\alpha\in\Phi\big{\}}\bigsqcup\big{\{}h_{\alpha}:\alpha\in\Delta\big{\}} on , 2. (2)
All the structural constants lie in . That is to say,
- (i)
for all , . 2. (ii)
for all , , where . 3. (iii)
for , is a -linear combination of ’s (). 4. (iv)
If are independent roots and is the -string through , then
Obviously, once is fixed, are uniquely determined if . In addition, for a general , if is replaced by , deduced from the conditions in (c) of Definition 1, are bounded by the constrains:
- (i)
for all , , 2. (ii)
for all , if , then .
Conversely, it is clear that if satisfies (these two conditions)i) and (ii) just mentioned, then \big{\{}x_{\alpha}:\alpha\in\Phi\big{\}}\bigsqcup\big{\{}h_{\alpha}:\alpha\in\Delta\big{\}} forms a Chevelley basis of as well.
To treat the Lie algebra associated to the split reductive group , it suffices to use the decomposition
[TABLE]
where denotes the center of . Obviously, the integral bases for are parametrized by . Hence, it is natural to define a Chevelley basis of to be the union of a Chevelley basis of as above and an integral basis of . Denote by , resp. , resp. , the associated lattice of , resp. of , resp. of . Obviously, admits a natural Lie structure and does not depend on the chosen integral Chevalley basis.
Moreover, working with , we obtain the following structural diagram over :
[TABLE]
Obviously, associated to the base change , we recover the diagram in (2).
The same construction works for a general number field instead of . That is to say, we may use the base change to obtain the following diagram over the integral base :
[TABLE]
whose general fiber is the similar diagram over , which can be obtained from (7) by replacing the integer ring with its associated number field .
To end this discussion, via the Minkowski embedding , we obtain the Lie algebra and similarly and . Furthermore, induced from the base change , we obtain the following diagram on :
[TABLE]
In particular, similar to the morphisms and , is a finite -morphism, even supposed to be highly ramified in general. For later use, we denote and by and , respectively.
2 -Torsors on
Let be a number field with its ring of integers and its adelic ring. Denote by the set of inequivalent normalized valuations of , and by , resp. , the subsets of consisting of non-archimedean, resp. archimedean, valuations. For each , denote by the -completion of . When , is isomorphic to either or ; and if , is a discrete valuation fields. Accordingly, the valuation is called real, or complex, or -adic. For -adic valuations, denote by the associated valuation ring of , by its maximal ideal, and by its residue field. It is well-known that , being a discrete valuation ring, admits only two prime ideals, namely, and , and is a finite extension of , for a certain prime number . We call the residue extension degree of . Based on all these, we have where denotes the restricted product of the ’s with respect to the ’s. That is to say, an element belongs to if and only if for all but finitely many . It is well known that, induced from the locally compact topologies on the ’s, is locally compact.
Let be a split reductive group over with a pinning . Here is a maximal split subtorus of , is a minimal split parabolic subgroup of containing , and denotes the set of simple roots of the root system associated to and denotes a non-zero vector of the proper subspace of the Lie algebra corresponding to the eigenvalue , where denotes the unipotent radical of . We assume that can be extended to a Chevelley basis of . Set then
[TABLE]
2.1 Torsors over Local and Global Fields
Let be the algebraic closure on contained in , and set be the absolute Galois group of . For each , fix an algebraic closure of , and denote by the local absolute Galois group of at .
Definition 3**.**
For or , a -scheme is called a -torsor if is equipped with a faithful, transitive and -compatible action of on .
Example 2**.**
For or , set . Then
[TABLE]
gives a -torsor structure on over .
Since is defined over , there is a continuous action of on . As usual, set be the collection of -invariant points of and be the set of equivalence classes of 1-cocycles, where , resp. , denotes the set of 1-cocycles, resp. 1-coboundaries, of on , i.e. a continuous map
[TABLE]
satisfying
[TABLE]
and two 1-cocycles and are said to be equivalence if there exists an element of such that
[TABLE]
In other words, is an 1-coboundary if .
Theorem 4** (See e.g. §2 of [7]).**
For or , there exists a natural bijection between the set of isomorphism classes of -torsors on and the set , which sends the trivial -torsor on to the trivial class in .
Here, naturality means that the bijection is compatible with the changes of the field and the group . For reader’s convenience, we sketch a proof.
Proof.
Let be a -torsor on . Fix a point . Then, for any , there exists a unique such that since acts on which itself is a group. It is not difficult to check that is an 1-cocycle and hence induces an element in .
Conversely, if is an 1-cocycle, then on , we obtain a new action of through . Since is quasi-projective, by Weil’s theorem on descent, there exists a -scheme , or better, a -torsor on , such that is -equivalent (with respect to the twisted action of on ). ∎
There is a canonical morphism induced by the inclusion for each , which themselves then induce a natural morphism
[TABLE]
Denote by the kernel of this morphism. It is a result of Borel-Serre that is finite.
Corollary 5**.**
There are only finite many -torsors on , up to isomorphisms, such that, for all , the induced -torsors on are trivial.
2.2 -Torsors on
We begin with the following
Definition 6**.**
A -torsor on is a scheme , equipped with a flat surjective morphism and a family of flat surjective morphism , together with actions of on for all , such that the induced morphism
[TABLE]
are isomorphisms for all points , closed or generic.
In particular, if is a -torsor on , the fiberwise acts on with respect to , and the action on each generic fiber is faithful and transitive. Moreover, for each , over the local integral base, induced by the natural inclusion , we obtain a composition of morphisms . Thus, for the -torsors on , it is not too difficulty to deduce the following result, whose proof we left to the reader.
Lemma 7**.**
- (1)
For each finite place , induced from the natural morphisms and , we have
[TABLE] 2. (2)
For each infinite place , induced by the natural embedding , we have
[TABLE]
In particular, .
2.3 Inner Form
When working over integral base , our choice of a Chevalley basis \big{\{}x_{\alpha}:\alpha\in\Phi\big{\}}\bigsqcup\big{\{}h_{\alpha}:\alpha\in\Delta\big{\}} determines a pinning of . To deal with the associated compatibility problem, in the automorphism group of , we consider the so-called outer automorphism group defined as the collection of the automorphisms of which preserves the pinning . There is a natural split short exact sequence
[TABLE]
Indeed, is identified with the image of under the adjoint representation of ,111The adjoint group of is the Zariski connected component of , and may also be identified with the connected component of the automorphism group of . In particular, coincides with the group of inner automorphisms of G(F) defined over F, and acts simply transitively on the triples of all pinnings of . and hence also fits into the short exact sequence
[TABLE]
where denotes the automorphic of the root system . In addition, the pinning identifies with and hence introduces a section of the morphism in (8).
By taking Galois cohomology, we obtain the morphisms
[TABLE]
Hence, naturally associated to an element are the elements and . It is not too difficult to see that this element belongs to . Denote the induced -torsor on by , which, for later use, we call the inner form of associated to .
For example, an element induces naturally a - torsor on . Hence, if we take as the trivial -torsor, namely, itself to start with, then the element corresponding to defines a split group on .
3 Compatible Metrics
In this subsection, we assume that our base field is the field of real numbers, unless otherwise stated explicitly.
3.1 Maximal Compact Subgroup
We first recall some basic facts on maximal compact subgroups of a real reductive group, mainly following [2].
Let be a real Lie group with finitely many connected components. Then any compact subgroup of is contained in a maximal compact subgroup. Moreover, if is a maximal subgroup of , then is diffeomorphic to the direct product of with a euclidean space, any maximal compact subgroup is conjugate to in and , where denotes the connected component of containing the unit element.
In addition, if is a closed normal subgroup of admitting only finitely many connected components, then the maximal compact subgroups of are the intersections of with maximal compact subgroups of . Similarly, if is a closed subgroup of with finitely many connected components such that all maximal compact subgroups of are conjugate by elements of , the maximal compact subgroups of are the intersections of with the maximal compact subgroups of . Consequently, in both cases, by taking a maximal compact subgroup of containing a maximal compact subgroup of , we conclude that is a maximal compact subgroup in for at one and hence for all maximal compact subgroups of by conjugacy.
More generally, if is a surjective morphism (of Lie groups) whose kernel admits only finitely many connected components, then the maximal compact subgroups of are the images of the maximal compact subgroups of .
3.2 The Cartan Involution
We here recall some basic facts on the Cartan involution associated to an algebraic group, mainly following [5].
Let be an algebraic group defined over a base field . Denote by the radical of and the unipotent radical of and the so-called split radical of , namely, the greatest connected -split subgroup of . By definition, a Levi subgroup of is a maximal reductive -subgroup of . Let
[TABLE]
with , the group of -morphism of into . Then is a normal subgroup of , and is defined over . Note that for a character in , its restriction to is of order , hence is trivial on . Consequently,
[TABLE]
Since any character in is trivial on , we have for any Levi subgroup of . Hence, if is a maximal -split torus of , then
- (i)
and 2. (ii)
contains all compact subgroups of . More generally, 3. (iii)
if and are two -tori in such that is -split, is a torus and is finite, then here exists a normal -subgroup of containing and such that .
Therefore, if is a parabolic -subgroup of and is a maximal -split torus of the split radical of , then, for a maximal compact subgroup of , we have
- (a)
is a maximal compact subgroup of , and 2. (b)
. Furthermore, 3. (c)
if for some , then and the map sending to characterized by is real analytic.
As a direct consequence, when is a reductive group, there exists one and only one involutive automorphism of associated to satisfying the following properties.
- (1)
is ”algebraic,” i.e. the restriction to of an involutive automorphism of algebraic groups of the Zariski-closure of in . 2. (2)
The fixed point set of is . 3. (3)
If is a normal -subgroup of G, then . 4. (4)
leaves and stable. Here, we use the same to denote the induced involution on , and set denotes the center of . 5. (5)
If is the (-1)-eigenspace of in , then is a split component of . 6. (6)
If is the (-1)-eigenspace of in , then, for , there is a decomposition and
[TABLE] 7. (7)
The map is an isomorphism of analytic manifolds of onto .
Note that in the case when is semi-simple, is the usual Cartan involution. Motivated by this, we call the Cartan involution of with respect to . Moreover, the existence of the Cartan involution implies an existence of a non-degenerate -symmetric bilinear form on satisfying the follows.
- (a)
is invariant under and , and is real on . 2. (b)
The quadratic form of is positive definite on and negative definite on . In particular, if we set
[TABLE]
then is a positive definite -invariant and -invariant scalar product for with its associated norm. 3. (c)
For and and the restriction of and to and , respectively, we have that inherit all the properties of above.
In addition, since is -invariant, the following infinitesimal invariance holds.
- (d)
is characteristic, namely,
[TABLE]
Therefore, and are mutually orthogonal with respect to . In addition, since is -invariant, and is mutually orthogonal. Conversely, we may reconstruct the bilinear form using all the above conditions. To be more precise, starting with the Cartan-Killing form on , we may extend it to obtain as the direct sum of the Cartan-Killing form with a symmetric non-degenerate bilinear form on , which is negative definite on and positive definite on . Finally, we may extend this latest to the total space . For later use, we call the canonical form on associated to . For later use, when is viewed as a linear form from to , we write it as .
Finally, let us point out that the Cartan involution can be applied in many ways. For example, if is a -subgroup of containing such that all maximal compact subgroups of are conjugate under , then, for a Levi subgroup of , it makes sense to take about the Cartan involution of with respect to . Moreover, in this case, the subgroup is the unique -stable Levi subgroup of contained in . Consequently, if is a parabolic -subgroup of , and is a maximal compact subgroup of and is a Levi subgroup of containing , then contains one and only one tLevi subgroup of stable under . For this reasons, we will fix a maximal compact subgroup of in the sequel.
3.3 Fine Involutions for Maximal Compact Subgroups
We are now ready to introduce new structures called fine involutions and their associated compatible metrics for general reductive groups, which may be viewed as natural generalizations of the known structures for semi-simple groups (see e.g. [4]).
Let be a reductive group defined over a subfield . Fix a maximal compact subgroup of . Motivated by the Cartan involution associated to the maximal compact group of , we give the following:
Lemma 8**.**
Let be a positive definite real symmetric bilinear form on . Assume is -compatible with respect to the Lie structure of . Then, for ,
- (1)
. 2. (2)
. That is to say, . 3. (3)
Let , resp. , be the - eigenspace, resp. the -eigenspace, of on . Then
- (a)
. 2. (b)
On , resp. , , resp. , is negative definite, resp. positive definite. 3. (c)
* is an orthogonal decomposition with respect to .* 4. (4)
* is compatible with , i.e. is an isometry with respect to the metric on and the metric on .*
Proof.
(1) By the infinitesimal invariance of , namely, two relations in (12), we have since is compatible with the Lie structure on . On the other hand, . Therefore .
(2) This is a direct consequence of (1). Indeed, since , we have . Therefore, and hence .
(3) (a) This is a standand result in linear algebra.
(b) This is a direct consequence of (2). Indeed, since , we have that on , resp. on , by the fact that, on , resp. on , . Consequently, is negative definite on and positive definite on .
(c) This is a direct consequence of (b) and (c).
(4) This is a reinterpretation of (2) and (3). Indeed, by (3), is an isomorphism. Moreover, by (2), , we obtain the following commutative diagram of isomorphisms
[TABLE]
This implies that is an isometry with respect to the metric on and the metric on . ∎
Definition 9**.**
An element in the automorphisms group of the Lie algebra is called a fine involutions of (with respect to the Lie structure on ) if there exists a positive definite real symmetric bilinear form on such that and satisfies all the properties (1), (2), (3) and (4) in Lemma 8. Here, denotes the linear isomorphism associated to the bilinear form on induced from the Cartan involution associated with . Moreover, if this is the case, is called a -compatible metric on (with respect to its Lie structure), 222Here as usual, we view the bilinear as a linear map from to . Hence is indeed a linear endomorphism of . and we denote by .
From the discussion above, it is not difficult to see that fine involutions and admissible metrics on associated to works exactly in the same way for as well, since is reductive and all maximal compact subgroups of are contained in . Indeed, the corresponding constructions on may be viewed as the restrictions of the structures from to . For later use, set
[TABLE]
3.4 Compatible Metrics for Maximal Compact Subgroups
Denote by , resp. the moduli space of -compatible euclidean metrics on , resp. on . Since they contains (the isometric class of) in (11), both and are not empty. Moreover, , resp. , admits a natural interpretation as a subspace of the space of (isomety classes of) euclidean metrics on , resp. on . Our main result of this section is the following:
Proposition 10**.**
Let be a reductive group defined over a subfield and let be a maximal compact subgroup of . Set . We have
- (1)
There are natural actions of on and . 2. (2)
The action of on in (1) induces a natural diffeomorphism
[TABLE]
Proof.
(1) Recall that, for , we have since is characteristic by (12). Hence, for any , we have
[TABLE]
This implies that belongs to as well. Consequently, the assignment
[TABLE]
defines a natural action of on . Here, as usual, for , denotes its adjoint. This proves (1).
(2) In terms of the fine involutions , the action above is equivalent to
[TABLE]
Recall that, for a general , if we set
[TABLE]
Then is a split component of . 333Denote by the adjoint action of on . Then acts trivially on . A split component of is defined to be a maximal closed linear subspace of . By Proposition 2.1.5 of [5], if is a split component of , then . Moreover, by Proposition 2.1.10 of [5], the assignment gives a bijection from the set of fine involutions associated to to the set of split components of . In particular, when , this map gives a bijection from the set of fine involutions to the set of maximal compact subgroups of . Hence, in our case, since is connected and reductive, all maximal compact subgroups of are conjugate to each other. This implies that and hence acts transitively on . Thus, to complete our proof, it suffices to show that the stabilizer group of in is exactly itself. For this we choose to be a Cartan involution satisfying . By definition,
[TABLE]
On the other hand, for , if and only if is in the center of , the connected component of containing the unit element. But this center is trivial by our assumption, hence belongs to the stabilizer group of if and only if . ∎
From the proof, we conclude that in (13) is identified with for a certain compatible of . As a direct consequence, we obtain the following:
Corollary 11**.**
Denote by be the moduli space of euclidean metric on . Then
- (1)
, where denotes the orthogonal group of degree . 2. (2)
The natural map defined by
[TABLE]
is bijecive.
4 Arithmetic -Torsors on
4.1 Integral Structures on Lie Algebras
Let be an algebraic number field with the ring of integers, and let be a connected split reductive group over with its Lie algebra. Our aim here is to introduce a -invariant projective -module in which is closed under the Lie operation.
For simplicity, assume . Since is defined over , its Lie algebra admits a natural rational structure and the adjoint representation is a morphism defined over . Consequently, there always exist -invariant integral structures in , since, for any integral structure in , the image under the action of is again an integral structure in . Obviously, the summation of two -invariant integral structures in is again a -invariant integral structure. Moreover, if is a -invariant integral structure in , we have . Hence, by clearing up denominators, we can instead assume that from the beginning. In this way, we obtain a unique maximal -invariant integral structure in satisfying the condition that .
Put this in a more concrete form, since our reductive group is defined over , we may use the structural decomposition
[TABLE]
where denotes the rational structure on induced by the semi-simple Lie sub-algebra of . Since , this decomposition is compatible with induced by (13). Moreover, since is abelian, we obtain a natural decomposition
[TABLE]
Now by applying the Chevalley (integral) basis for semi-simple Lie algebras, we obtain a canonical integral structure on . Hence, what is left is to introduce an integral structure on which is compatible with the decomposition (17). But this is trivial since is an abelian sub Lie algebra defined over . We thus obtain an induced integral structure on , which we denoted by . Similar arguments then lead to the integral structures and on and , respectively. Consequently, we have
[TABLE]
The above discussion works well if we replace by a split reductive group with a general number field. To indicate the dependence on , we rewrite the associated Lie algebra by . Since it admits a natural -linear space structure, through the Minkowski embedding , we obtain a Lie algebra
[TABLE]
We introduce an -lattice structure on by setting
[TABLE]
Then
[TABLE]
where and . Using a similar argument as above, we conclude that is a projective -submodule in such that . In the sequel, will be called the canonical infinitesimal -structure of .
4.2 Arithmetic -Torsors
Let be a split reductive group over a number field . For each , we fix a maximal compact subgroup of , set . We denote a real, resp. complex, by , resp. . By §3.4, we obtain the moduli spaces , resp. , of the compatible metrics with respect to on , resp. , and natural isomorphisms
[TABLE]
Here , resp. , denotes the orthogonal group, resp. the unitary group, of degree , and , resp. , denotes the number of real, resp. complex, places of . For later use, set
[TABLE]
where, to simplify our notations, we use as a running symbol for and .
Definition 12**.**
Let be a connected (split) reductive group and let be a family of maximal compact subgroups of . By a -compatible arithmetic -torsor over , or simply over , we mean a tuple consisting of a -torsor on and an element of .
Even apparently not quite related, may be viewed as a family of -compatible metrics on the tangent bundles of the -torsor . To explain this, we first recall that admits a natural -torsor structure. This, via the Minkowski embedding, induces a natural -torsor structure on . Hence, for a fixed base point of , the the tangent space of at this point is canonically identified with . Consequently, we obtain a natural metric on this tangent space. Moreover, since is a -torsor, its tangent bundle is a flat bundle. Thus, with the help of the so-called parallel transforms, we obtain a natural metric on the tangent bundle of . This metric is uniquely determined by .
Furthermore, working over , at infinite places, is a -compatible metric on . Thus for an -lattices in whose projective -module component also admits a natural Lie structure over because, by our construction, .
Definition 13**.**
Let be a pair consisting of a projective -module and a family of -compatible metrics on . If is -invariant under the adjoint action, and , then is called a -compatible principle -lattices over .
Denote by , resp. , the moduli stack of -compatible arithmetic -torsors, resp. -torsors, on . Then we have the following
Theorem 14**.**
Let be a split reductive group on . Then there exist the following natural identifications:
- (1)
. 2. (2)
{\mathcal{M}}_{G,F}^{\mathrm{tot}}\simeq\displaystyle{\prod_{\xi\in\mathrm{Ker}^{1}(F,G)}}\!\!\!G^{\xi}\!(F)\backslash G({\mathbb{A}})/\big{(}\!\!\!\!\prod_{v\in S_{\mathrm{fin}}}\!\!\!\!G(\mathcal{O}_{v})\times K_{\infty}\!\times O_{r_{G,\sigma}^{G^{\mathrm{ss}}}}\!({\mathbb{R}})^{r_{1}}\!\times U_{r_{G,\sigma}^{G^{\mathrm{ss}}}}\!({\mathbb{C}})^{r_{2}}\!\big{)}.**
Proof.
(1) For each , let and set be the complementary open subset of in . Then for each element of the affine Grassmannian , we obtain a -torsor on equipped with a trivialization on with the trivial torsor, so that if an automorphicm of is trivial on , then it is necessarily trivial. Recall that there exists a natural morphism which maps on the tuple consisting of at and the unit elements of for all places . Then, by the compatibility condition in Lemma 7, and the fact that is affine and is Dedekind domain, we conclude that the local maps induces an identifications of with the moduli spaces of -torsors on , by adopting a result of Beauville-Laszlo [BZ] in the case when and that of Heinloth in [He] for general cases. Therefore, to conclude our proof, it suffices to apply Proposition 10 to take care of the factor of -compatible metrics , since is nothing but .
(2) is a direct consequence of (1) and its proof, if we apply Corollary 11. ∎
In the sequel, to simplify our presentations, we will use arithmetic -torsors instead of the full version of -compatible arithmetic -torsors over , if no confusion arises.
4.3 Slopes of Arithmetic -Torsors
Let be the maximal split torus in the center of and let be the maximal split quotient torus of . Then
[TABLE]
Here denotes the maximal abelian quotient of . It is well known that the composition
[TABLE]
is an isogeny, i,.e. a morphism with finite kernel and cokernel. Consequently, if we set
[TABLE]
then is a free abelian group of the same rank. Moreover, since , there is a non-degenerate pairing
[TABLE]
Here or . Set now
[TABLE]
then (25) induces a non-degenerating pairing
[TABLE]
Obviously, there is a natural action of on , and
[TABLE]
For our own convenience, we denote the invariant space by .
Definition 15**.**
Let be a -compatible arithmetic - torsor over . An element is called the slope of , denoted by , if, for all , we have
[TABLE]
where denotes an arithmetic -torsor on induced by the reduction of structure group , and denotes the arithmetic degree.
This definition makes sense, since an arithmetic -torsor on is simply a metrized line bundle on . Hence its arithmetic degree is well-defined.
Remark 1**.**
- (1)
Arithmetic -torsors are first introduced in my book [10]. There is a serious overlap between this section and §16.2 of [10], even the context here is much clearer. 2. (2)
As to be expected, the slope can be use to definite stability of arithmetic -torsors ([10]). We omit the details, since it will not be used in our current work.
5 Arithmetic Characteristic Curves
Let be a split reductive group with pinning such that can be extended to a Chevalley basis of .
Let be an arithmetic -torsor on . Denote by the induced locally free sheaf on , or equiva;ently, the associated projective -module in .
Let be a metrized invertible sheaf on . Denote by its associated rank one projective -module. Since is Dedekind, we may identify with a certain fractional ideal of . Denote this fractional ideal by the same letter , by an abuse of notations.
For an element , we denote its images under the Chevalley characteristic morphisms and by and , respectively. It is not difficulty to see that .
The element can also be used to identify with the horizontal section associated to in . In this way, we obtain a morphism .
Definition 16**.**
Let be a split reductive group.
- (1)
The pair consisting of an arithmetic -torsor and an element with an invertible line sheaf is called an arithmetic Higgs -torsor. 2. (2)
The characteristic arithmetic curve associated to is defined to be the scheme of arithmetic dimension one obtained from the morphism through the base change . That is to say,
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induced from the product diagram
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Obviously, the morphism is a finite -covering, even highly ramified in general. Here, as usual, denotes the Weyl group of .
Remark 2**.**
- (1)
The above construction is motivated by the construction of spectral curve (for ) and cameral curve (for general reductive ) by Beauville-Narasimhan ([1]) and Donagi- Gaistgory ([3]), respectively. 2. (2)
When , with the identification , the Chevalley characteristic morphism may be viewed as the assignment for diagonal matrices to their unorded eigenvalues. For this reason, we sometimes also call the arithmetic eigen curve of associated to .
In the forthcoming papers ([11], [12]), we will use arithmetic characteristic curves to construct arithmetic Hitchin fibrations and study the intersection homologies and perverse sheaves for the associated structures, following (Laumon-)Ngo’s approach to the fundamental lemma ([9]) using Hitchin fibrations ([8]).
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