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Yuval Z. Flicker
Drinfeld Moduli Schemes
and Automorphic Forms
The Theory of Elliptic Modules
with Applications
123
YuvalZ.Flicker
DepartmentofMathematics
TheOhioStateUniversity
Columbus,OH,USA
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-1-4614-5887-6 ISBN978-1-4614-5888-3(ebook)
DOI10.1007/978-1-4614-5888-3
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012952719
MathematicsSubjectClassification(2010):11F70,22E35,22E50,11G09,11G20,11G45,11S37,14G10,
11F72,22E55
(cid:2)c YuvalZ.Flicker2013
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Contents
1. Introduction 1
Part 1. Elliptic Moduli 9
2. Elliptic Modules: Analytic Definition 11
3. Elliptic Modules: Algebraic Definition 17
4. Elliptic Modules: Geometric Definition 27
5. Covering Schemes 37
Part 2. Hecke Correspondences 43
6. Deligne’s Conjecture and Congruence Relations 45
6.1. (cid:2)-Adic Cohomology 45
6.2. Congruence Relations 55
Part 3. Trace Formulae 65
7. Isogeny Classes 67
8. Counting Points 73
9. Spherical Functions 79
Part 4. Higher Reciprocity Laws 91
10. Purity Theorem 93
11. Existence Theorem 105
12. Representations of a Weil Group 113
12.1. Weil Groups 113
12.2. (cid:2)-Adic Representations 115
12.3. ε-Factors 117
12.4. Product L-Functions of Generic Representations of GL(n) 119
12.5. Correspondence 123
12.6. Smooth Sheaves 126
12.7. Local and Global Correspondence 131
13. Simple Converse Theorem 139
13.1. Introduction 139
13.2. Generic Representations 140
13.3. The Global Functions U and V 142
13.4. The Integrals I and Ψ 144
13.5. Proof of the Simple Converse Theorem 146
References 149
Index 153
v
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1. Introduction
Let F be a geometric global field of characteristic p > 0, A its ring of
ad`eles, G = GL(r) and π an irreducible admissible representation of G(A),
namely a G(A)-module, over C. Then π is the restricted direct product
⊗vπv over all places v of F of irreducible admissible Gv = G(Fv)-modules
πv. For almost all v the component πv is unramified. In this case there are
nonzero complex numbers z1,v,...,zr,v, uniquely determined up to order by
πv and called the Hecke eigenvalues of πv, with the following property: πv
is the unique irreducible unramified subquotient π((zi,v)) of the Gv-module
I(zv) = Ind(δ1/2zv;Bv,Gv) w(cid:2)hich is normalizedly induced from the unram-
ified character zv : (bij) (cid:3)→ izidvegv(bii) of the upper triangular subgroup
Bv of Gv.
Thefirstmainthemeinthisworkconcernscongruencerelations(seebelow).
The second such theme concerns the following purity theorem for cuspidal
G(A)-modules. Let π be a complex cuspidalG(A)-module; it is anirreducible
admissible representationπ of G(A) which occurs as a direct summand in the
representationofG(A)byrighttranslationonthespaceofcomplex-valuedcus-
pidal functions on G(F)\G(A). If π has a cuspidal component and a unitary
centralcharacter,thentheabsolutevalueofeachHeckeeigenvalueziv ofalmost
all unramified components πv of π is equal to one. This is Theorem 10.8. Its
proof uses neither Deligne’s conjecture nor the congruence relations. The pu-
rity theoremis a representationtheoreticanalogueofRamanujan’sconjecture
concerning the Hec(cid:2)ke eigenvalues (or rather Fourier coefficients) of the cusp
form Δ(z) = e2πiz ∞(1−e2πizn) on the upper half plane Im(z) > 0 for the
1
group SL(2,Z).
The third major theme in this work concerns the higher reciprocity law.
Let σ be a continuous r-dimensional (cid:5)-adic representation σ : W(F/F) →
GL(r,Q ) of the Weil group of F, which is constructible, namely unramified
(cid:3)
for almost all v. Equivalently σ is a smooth (cid:5)-adic sheaf on SpecF which
extends to a smooth (cid:5)-adic sheaf on an open subscheme of the smooth pro-
jective curve whose function field is F. For such v the restriction σv of σ to
the decomposition group W(Fv/Fv) at v factorizes through W(Fv/Fv) (cid:5) Z,
where Fv is the residue fieldofFv. The isomorphismclassofσv is determined
by the eigenvalues {ui,v = ui(σv);1 ≤ i ≤ r} of the (geometric) Frobenius
σv(Frv). Then we say that such σ and the G(A)-module π = ⊗πv correspond
if for almost all v the r-tuple (ui(σv)) is equal, up to order, to the r-tuple
(zi(πv)).
The case of r =1 is class field theory: W(F/F) (cid:5)A×/F×, which in the
ab
localcaseassertsthatW(Fv/Fv)ab (cid:5)Fv×,normalizedbymappingageometric
Frobenius Frv to a local uniformizer πππv in Fv×. Here Frv ∈ W(Fv/Fv) is
any element which maps to the inverse ϕ−1 of the “arithmetic” Frobenius
substitution ϕ:x(cid:3)→xqv, which generates W(Fv/Fv) and is an automorphism
of Fv over Fv. Let ∞ be a fixed place of F.
Y.Z. Flicker, Drinfeld Moduli Schemes and Automorphic Forms: The Theory 1
of Elliptic Modules with Applications, SpringerBriefs in Mathematics,
DOI 10.1007/978-1-4614-5888-3 1, © YuvalZ. Flicker 2013
2 YUVALZ.FLICKER
The special case of the higher reciprocity law which is proven in this work
asserts that the correspondence defines a bijection between the sets of
(1) equivalence classes of cuspidal representations π of GL(r,A) whose com-
ponent π∞ is cuspidal and (2) equivalence classes of irreducible r-dimensional
continuous (cid:5)-adic constructible representations σ of W(F/F), or irreducible
rank r smooth (cid:5)-adic sheaves on SpecF which extend to smooth sheaves on an
opensubschemeofthecurveunderlyingF,whoserestrictionσ∞ toW(F∞/F∞)
is irreducible.
This reciprocity law is reduced in Chaps.11 and 12 to Deligne’s conjecture
(Theorem 6.8), introduced by Deligne in the 1970s for the purpose of this
reduction. The fixed point formula expresses a trace on (cid:5)-adic cohomology
with compact supports and coefficients in a smooth sheaf, in terms of traces
onthe stalksofthe sheafatfixedpoints. Deligne’s conjecture assertsthatthe
fixedpointformularemainsvalidinthecontextofthecompositionofacertain
correspondence on a separated scheme of finite type over a finite field and a
sufficiently high power, depending on the correspondence, of the Frobenius
morphism.
This work existed as an unpublished manuscript since 1983. It is the first
openlycirculatedmanuscriptwhereDeligne’sconjectureappeared. Itsresults,
first discussed in a seminar with Kazhdan in 1982, were announced in the
publication [FK3]. Perhaps this work contributed a little to the interest in
Deligne’s importantconjecture. This wasthe originalpurpose ofthis work,to
motivate Deligne’s conjecture by means of exhibiting some of its applications.
SincethenthisconjecturewasprovenunconditionallybyFujiwara[Fu]in1997
andlaterbyVarshavsky[V]in2007,afterworkbyZink[Z],EdShpiz[Sh],and
Pink [P] in special cases. This completed that part of our work which relied
on Deligne’s conjecture. Our results concern the full local correspondence
(for GL(r) over a local field of positive characteristic) as well as the global
correspondenceforcuspidalrepresentations(ofGL(r,A)overaglobalfunction
field) which have a cuspidal component. They are based on Drinfeld’s theory
of elliptic modules, also named Drinfeld modules by Deligne, introduced by
Drinfeld [D1], [D2] in 1974 and 1977 to prove the reciprocity law when r =2.
Drinfeld later introduced a generalization, which he named Shtukas, to re-
move the restriction that the global cuspidal representations have a square-
integrable component. The work was carried out by Lafforgue [Lf1] and [Lf2]
in 1997 and 2002, who in addition to Deligne’s (proven) conjecture used the
full traceformulaofArthurinthefunctionfieldcase,toobtainthereciprocity
law foranyglobalcuspidalrepresentation. His importantworkis nevertheless
technically very challenging, so on the occasion of teaching a course on the
topicatOSUinwinter2012,weupdatedourworktoincludethereferencesto
theproofsofDeligne’sconjectureandotherworksthatcontinuedandextended
ours. I hope itisstill ofinterestnotonlyasthe firstworkwherethe localcor-
respondence and a major case of the global correspondence were established
in the general rank case but also since our work is considerably simpler than
that of Lafforgue, as we use only a simple case of the trace formula, and the
relatively elementary theory of elliptic (= Drinfeld) modules.
DRINFELDMODULISCHEMESANDAUTOMORPHICFORMS 3
In particular we were led to develop in Chap.13 a “simple” converse the-
orem for GL(r) over a function field, “simple” meaning for cuspidal global
representations with cuspidal local components at a fixed finite set of places
of the global field F. However, note that the converse theorem is not used
in the decomposition of the cohomology (see below), neither in the proof of
the existence theorem (for each π there is a σ) nor in the proof of the local
correspondence. It is used only to show the surjectivity of the map π (cid:3)→σ.
Thus, assuming Deligne’s conjecture (Theorem 6.8), we show in Chap.12
that there exists a unique bijection, denoted πv (cid:3)→ σv or σv (cid:3)→ πv and called
thelocal reciprocity correspondence, betweenthesetsof(1)equivalenceclasses
of irreducible Gv = GL(r,Fv)-modules πv and (2) equivalence classes of con-
tinuous (cid:5)-adic r-dimensional representations σv of W(Fv/Fv), namely rank r
smooth(cid:5)-adicsheavesonSpecFv,withthefollowingproperties. ItpreservesL-
and ε-factors of pairs, relates cuspidal πv with irreducible σv, commutes with
taking contragredient, and relates the central character of πv with the deter-
minantofthe correspondingσv by localclassfieldtheory W(Fv/Fv)ab (cid:5)Fv×,
which is normalized by mapping a geometric Frobenius to a uniformizer in
F×. The local correspondence has the property that π and σ correspond if
v
and only if their components πv and σv correspondfor all v. Again, Deligne’s
conjecture was used in the originalversionof this work as a conjecture, but it
is now proven.
To state our fourth main theme, we introduce some notations. Our work
relies on Drinfeld’s theory of elliptic modules (see [D1, D2]). Their definition
and basic properties are discussed in Chaps.2 and 3. Denote by A the ring
of elements of the function field F which are integral outside the fixed place
∞. Let I (cid:9)={0} be any ideal in A which is contained in at least two maximal
ideals. In Chap.4 we recall the construction of the (Drinfeld) moduli scheme
X = Mr,I of isomorphism classes of elliptic A-modules of rank r with I-level
structure. Itis anaffinescheme offinite rank overA. InChap.5we construct
(cid:3)
a finite ´etale Galois covering X of X, whose Galois group Γ is a quotient of
an anisotropic inner form D∞× of G(F∞).
Put X = X ⊗A F, where F is a separable closure of F. Let ρ be an
irreducible nontrivial representation of Γ and of D∞×. Let π∞ be the corre-
sponding square-integrablerepresentationofG(F∞). InSect.6.1we recallthe
definition of the associated smooth Q -sheaf L = L(ρ) on X and of the Q -
(cid:3) (cid:3)
adic cohomologyspaces Hi(X,L) of X with compactsupport and coefficients
c
in L. Let Af be the ring of F-ad`eles without a component at ∞, U = UI
the congruence subgroup of G(Af) defined by I, and HI the Hecke algebra
of Q(cid:3)-valued UI-biinvariant compactly supported functions on G(Af). An
irreducible HI-module will be regarded here as an irreducible G(Af)-module
which has a nonzero UI-fixed vector. The Galois group Gal(F/F) acts on F,
hence(cid:4)on X and on Hci = Hci(X,L(ρ)); so does the Hecke algebra HI. Put
Hc∗ = i(−1)iHci; it is a virtualH I ×Gal(F/F)-module. Namely it is a sum
of finitely many irreducibles π(cid:3)f ⊗σ(cid:3), with integral multiplicities.
Ourfourth maintheme is the followingexplicit reciprocity law. It underlies
the proofs of the purity theorem and the reciprocity law. Suppose that π∞ is
