The Langlands-Kottwitz method for the modular curve

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MSC

MATHEMATICS

M

ASTER

T

HESIS

The Langlands–Kottwitz Method for the

Modular Curve

Author: Supervisor:

Philip van Reeuwijk

dr Arno Kret

Examination date:

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Abstract

This thesis aims to explain in a self-contained way the "algebraic" side of Kottwitz’s proof (fol-lowing the method proposed by Ihara and Langlands) of the Hasse-Weil conjecture for Shimura varieties of PEL-type, in the special case of GL(2). We define the modular curves X1(N), and

exhibit an explicit formula for their number of points over finite fields. On the "analytic" side (not treated in this thesis), this formula can be rewritten using the Arthur-Selberg trace formula to obtain meromorphicity of theL−function of X1(N).

Title: The Langlands–Kottwitz Method for the Modular Curve Author: Philip van Reeuwijk , ph.v.reeuwijk@gmail.com, 5686202 Supervisor: dr Arno Kret

Second Examiner: Prof. dr Lenny Taelman Examination date: 14th September 2017 Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction:

The Hasse–Weil Conjecture

With any (smooth, projective) algebraic varietyX/Q comes its associated zeta function Z(s) (defined in terms of the so-called local zeta functions ofX at p for primes p of good reduction — these are relatively straightforward functions of #X (Fp) for a suitable model X /Z of X —

and some other factors for the bad and infinite primes).

This zeta function, initially shown to be holomorphic on the half-planeℜs > dim(X )+1, is thought to in fact be meromorphic on the entire complex plane. This is known as theHasse– Weil Conjecture for X : while considered out of reach in general, this conjecture has been proved in various ‘modular’ cases, exploiting techniques and results from the vast area of research stemming from the Langlands program.

In the case whereX is a curve, the local zeta function at a good primes p are given by Z_p(X , s) = exp _∞ X n=1 #X (Fpn) n (p −s₎n .

We will describe a part of the case X = Y₁(N) (or in fact its compactification X₁(N)), a modular curve, or moduli space of certain elliptic curves, for N ≥ 4. In this case, the bad primes are precisely the primes p| N . This leads to the zeta function

ZN(X₁(N), s) =Y

p-N

Z_p(X₁(N), s),

which is the full zeta function ofX₁(N) up to the finitely many factors p | N.

Implicitly, we will proceed by consideringX₁(N) as a Shimura variety associated to the al-gebraic group GL(2); this strategy stems from Langlands, who proved the case of GL(2) him-self in[7, 8] and formulated in [12] the precise conjectures that have allowed Kottwitz [6] to finally reduce the Hasse–Weil Conjecture for Shimura varieties of PEL-type to various cent-ral conjectures in the Langlands program, following decades of work by many. In our case ofG= GL(2) these conjectures are known or vacuous, but in the general case one needs the Fundamental Lemma (since proved by Ngo), Arthur’s conjectures on the discrete spectrum of reductive groups, and the stabilization of the trace formula. We will follow, and try to expand upon, the exposition by Clozel in his Séminaire Bourbaki[1].

Two things need to be mentioned. First, this particular case X = X1(N) had been proved

earlier by Ihara, using a different technique: the virtue of the Langlands–Kottwitz approach we will follow is that it generalizes to other Shimura varieties. Second, we only treat the case of good reduction:X1(N)/k where char(k) - N. The case where the ground field characteristic

dividesN has been investigated by Scholze[13], laying the foundation for his proof of the local Langlands conjectures for general GL(n) in [14].

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The Langlands-Kottwitz method

Slogan: if X can be written as a Shimura variety associated to a group G, the number of points ofX over finite fields, and hence the zeta function of X , are controlled by the automorphic representations of and forms onG. You can use this to prove Hasse–Weil for X .

The general strategy that Langlands proposed and Kottwitz carried out for Shimura varieties of PEL-type can very coarsely be seen a a three-step process:

1. Count the points onX over finite fields using the Honda-Tate theorem and counting lattices in Tate modules.

2. Write the result of 1. in terms of orbital integrals overG.

3. Compare the result of 2. to the geometric side of the trace formula, and conclude. In this thesis, we will confine ourselves to step 1. forX₁(N). The rest of this introduction will be a short high-level overview of the argument, including the (more analytically-flavoured) final steps that lead to the proof of Hasse–Weil forX₁(N).

Elliptic curves and their moduli An elliptic curve is a non-singular curve E of genus 1 to-gether with a distinguished pointO ∈ E (over some ring over which the curve is defined). This uniquely defines a group structure on the points ofE which is abelian with identity elementO : by the general theory of algebraic curves, a distinguished point (prime divisor) defines a map of the points of the curve to the Picard group is the curve byP 7→ [P − O ]: this is a bijection precisely when the genus is 1, and can then by declared to be an isomorphism of groups.

Thus it makes sense to talk about an elliptic curve with a subgroup of orderN , and a gener-ator for such a group. Since base changes preserve the order of elements, we have ourselves a moduli problem: for a schemeS we must functorially parametrize the set of elliptic curves E/S together with a pointP∈ E of order N . It turns out that for N ≥ 4 and for schemes S/Z(_N1), this moduli problem is representable, and we call the representing scheme themodular curve Y₁(N). This means in particular that for any finite field of characteristic p - N, there is a natural bijection between the elliptic curves overk with a point of order N and the set Y₁(N)(k).

We want to count the elements ofY₁(N)(k) for k ' F(pn) a finite field: that is, we want to count the number of elliptic curves overk equipped with a point of order N , up to isomorph-ism. (Heuristically, since the curves are given by cubic equations over k, and contain points with coordinates ink, we have a finite number of curves with a finite number of points, so a finite answer.) To do this, we first count the equivalence classes of curves up to isogeny, and then the number of isomorphism classes within a given isogeny class.

Classifying curves up to isogeny: Honda–Tate A field k of finite cardinality q= pncomes with its Frobenius endomorphism Frob :x 7→ xq _{which in turn induces the Frobenius map}

π ∈ End(E) of an elliptic curve over the field. For an ordinary elliptic curve, that is, a curve

for which the qr−torsion has qr elements over an algebraic closure of k for all r , the en-domorphism ring is a two-dimensional commutative Z−algebra: hence the tensor product EndQ(E) := End(E) ⊗ Q is a quadratic number field.

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Within End(E) and hence EndQ(E), πE is a root of its characteristic polynomialfπ= X2−

aX+q, where a = q +1−#E(k) is a fundamental quantity that in particular satisfies the Hasse bound|a| ≤ 2pq. Hence we get∆(f_π) = a2−4q ≤ 0, but in fact the case ∆ = 0 corresponds to the supersingular (i.e. not ordinary) curves, so we can conclude that∆ < 0; this means that π_E, when considered an element of EndQ(E) ⊂ C, is an imaginary algebraic integer with absolute

valuepq under all complex embeddings.

Such a number is called a Weilq−integer, and it turns out the reverse is also true: such a quadraticq−-integer (up to conjugation by the Galois group) corresponds bijectively to an isogeny class of elliptic curves over the field withq elements, provided it satisfies some extra arithmetic constraints explained in the sequel.

This follows from the more general theorem of Honda and Tate, who show that a similar statement holds for Abelian varieties over finite fields.

We can now identify the quadratic field End_Q(E) with a subset of M₂(Q), which identifies an isogeny class of Frobenii with a conjugacy class. This has the advantage of bringing the discussion into the group we are interested in, with the added benefit that the Frobenii of supersingular curves correspond to central elements, allowing a unified treatment.

Counting isogenous curves up to isomorphism In the complex case, an elliptic curve is the quotient of C by some latticeΛ; over finite fields, the Tate module of an elliptic curve, constructed from its subgroups of p−power order, fulfils a similar role. Over C, two elliptic curves given by latticesΛ,Λ0_{are isomorphic if and only if their lattices are, and similarly an}

isomorphism of Tate modules determines an isomorphism of elliptic curves in characteristicp. Moreover, since the Tate modules are constructed from finite subgroups, they are able to keep track of points of fixed order, so in this formulation we can easily single out isomorphisms that respect a given distinguished point of fixed finite order on the two isomorphic curves.

This reduces the problem of counting the points on the moduli spaceY₁(N)(k) to counting certain explicitly describable lattices in a vector space. The explicit formula that we obtain marks the close if this thesis.

The analytic side The lattice-counting problem defined above can be written as a orbital integral over the adèles of GL(2). These integrals are well-understood, and in particular, the Selberg trace formula relates them to certain well-behaved classes of representations of GL(2).

We end up with a formula of the form Z_p(X₁(N), s) =Y

ρ

L_p(ρ, s + 1/2)n(ρ), where

• Z_pis the local factor at p of the zeta function of X₁(N) • ρ runs over certain irreducible representations of GL(2) • L_p is the local factor at p of the L−function of ρ

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The upshot is that the L−functions Lp(ρ) are known to be meromorphic on C, so by

as-semblingZX₁(N)(s) from its local factors, we see that it is meromorphic too, and the Hasse-Weil

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1 Elliptic Curves and their Moduli

1.1 Moduli Problems, Moduli Spaces

A schemeX over a base scheme S is a morphismϕ : X → S; a morphisms of two schemes over the same base is a morphism of the two schemes commuting with the maps to the base. By a scheme over a ring (in particular, a field), we mean a scheme over its spectrum.

Generally speaking, amoduli problem is the problem of endowing a set of geometric objects with a comparable geometric structure of its own, making it into amoduli space. (A familiar and basic example is the set of one-dimensional subspaces of a vector spaceV , the projective spaceP(V ).) More formally, and in the context of algebraic geometry, we can formulate the following:

Suppose we have a functor

F : Sch/R → Set

that assigns to a schemeS/R over a ring R a set F (S) of certain objects over S (a notion which we will make precise in the case of elliptic curves with level data below). This is an abstract moduli problem, and a solution is a schemeM such that F is naturally isomorphic to Hom(·, M), the functor assigning toS the set of morphisms Hom(S, M).

Theorem 1. If a moduli functor F is representable, the representing scheme M is unique up to isomorphism.

Proof. Suppose M and M0both representF , i.e. we have Hom(·, M) ' F ' Hom(·, M0_).

In particular, this gives a family of bijections

f_X : Hom(X , M) → Hom(X , M0)

for all objectsX . Then f_M(Id_M) : M → M0and f_M−10 (IdM0) : M0→ M are inverses, and M and

M0are isomorphic.

In this case, for a schemeS/R we call

M(S) := Hom(S, M) = F (S)

the S−points of M . In particular, for a field k with a map R → k we have the points M (k) corresponding to the geometric objects described by our functorF over k.

In full generality, many moduli problems of interest fail to be representable by a scheme, and the theory of algebraic stacks becomes necessary. In our particular situation though, we

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can get by using the more light-weight formalism of Katz-Mazur[10], which we will follow closely.

An elliptic curve over a scheme S is a proper, smooth morphism of schemes π : E → S such that the fibres ofπ are curves of genus one, together with a section O : S → E, that is, a morphismO withπ ◦ O = IdS. Intuitively, this amounts to having for allP ∈ S an elliptic

curveE_P := π−1(P) such that O_P := O(P) ∈ E_P is the identity element. In other words, we have a family of elliptic curves varying algebraically over the baseS. Locally, this means the fibres are given by Weierstrass equations:

Theorem 2. For every P ∈ S there is an affine open neighbourhood U ⊂ S of P and regular functions a₁,a₂,a₃,a₄,a₆∈ OS(U) such that for all u ∈ U, π−1(u) is the curve given by

y2+ a₁(u)xy + a₃(u)y = x3+ a₂(u)x2+ a₄(u)x + a₆(u)

Now let Ell denote the category whose objects are elliptic curves over an arbitrary base E→ S and whose morphisms are cartesian squares of such curves: a commutative diagram

E0 //

E

S0 //S

such that the mapE0→ E ×SS0is an isomorphism of curves overS0.

A moduli problem (functor)F : Ell→ Set is (absolutely) representable if it is representable in the sense defined above: there exists an elliptic curveE /S such that

F ' Hom(·, E /S );

it is called relatively representable if for every curveE/S the functor F_E_/S: Sch/S → Set given byF_E_/S(T ) = F ((E ×_ST)/T ) is representable. A representable moduli problem is relatively representable: indeed ifE /S represents F , then (E ×_S S)/S represents FE/Sfor allE/S.

IfE /S represents F , the associated functor Sch → Set given by S7→ {isomorphism classes of (E/S, α ∈ F (E/S))}

is easily seen to be represented by S ; this description of the moduli problem fortifies our intuition of moduli spaces as parametrising "isomorphism classes of curves with structure".

We call a moduli problem F on Ell rigid if for any E/S ∈ Ell and α ∈ F (E/S), the pair (E/S,α) has no non-trivial automorphisms.

Lemma 1. A representable moduli problem is rigid.

Proof. Let the moduli problem F be representable byE /S . A choice of natural isomorphism F ' Hom(·, E /S ) induces a universal element u ∈ F (E /S ) corresponding to Id ∈ Hom(E /S , E /S ) under the chosen bijection. Hence the pair(E/S , u) satisfies: for every E/S ∈ Ell and α ∈ F(E/S), there is a unique ϕ : E → E with F (ϕ)(u) = α. If the pair (E/S,α) admitted a non-trivial automorphism, thisϕ would not be unique and we conclude that representable moduli problems are rigid.

Conversely, it is proved in (4.7) of[10] that an affine relatively representable moduli problem is representable if and only if it is rigid.

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1.2 The Moduli Space X

₁

(N)

The particular moduli problem we will be looking at is the "exact levelN structure"[Γ₁(N)], that is, the functor given by

[Γ1(N)](S) = {isomorphism classes of (E/S, P ∈ E(S) with ord(P) = N)}.

Recall (see[17, 19]) that an isogeny of elliptic curves is a morphism (in particular, group homomorphism) f : E→ E0_{that is surjective and has a finite kernel. The rank of that kernel}

is called thedegree deg(f ). An isogeny f induces a dual isogeny f∗:E0→ E of the same degree (saym), and we have f f∗= [m], where [m] : E → E is the multiplication-by-m endomorph-ism ofE, of degree m2_{. Dualising is an involution on End(E) (f}∗∗_{= f ) and satisfies}

(f + g)∗_{= f}∗_{+ g}∗_.

We have[m][m]∗= [m2], hence [m] = [m]∗(as([m] − [m]∗) is identically 0 on the image of [m], which is surjective).

For an endomorphism f : E→ E, we have

[deg(1 + f )] = (1 + f )(1 + f )∗_{= (1 + f )(1 + f}∗_{) = [1] + [deg(f )] + f + f}∗_.

This means there is an integer, called thetrace tr(f ), such that f + f∗= [tr(f )]. (Henceforth, we will tacitly identifym with[m], tr(f ) with f + f∗etc. when clarity permits.)

Any f ∈ End(E) satisfies f2− ( f + f∗)f + f f∗, so is a root of the polynomial P= X2− tr( f )X + deg( f ).

This polynomial is non-negative on R: it suffices to show it’s non-negative on Q, i.e. that n2− nm tr( f ) + m2deg(f )

is non-negative for n, m ∈ Z. But this expression equals deg(n − m f ), and the degree of an endomorphism is non-negative by definition. On the other hand, this must meanP has a single real root at most so its discriminant tr(f )2_{− 4 deg( f ) must be non-positive: tr( f )}2_{≤ 4 deg( f ).}

Theorem 3. The moduli problemΓ₁(N) affine and relatively representable.

Proof. See[10], Theorem 5.1.1. The proof given there is the result of an axiomatic setup ap-plicable to a wider class of elliptic curve moduli problems, including elliptic curves equipped with a subgroup of ordern and elliptic curves equipped with a basis of the N−torsion. For more on the complex picture and the arithmetic viewpoint,[16] is a good start.

Theorem 4. The moduli problemΓ1(N) is rigid, and therefore representable.

Proof. Suppose we have a curve E with an automorphism f : this is just an isogeny of degree 1. Hence we can say tr(f )2_{≤ 4, or tr( f ) = 0, ±1, ±2. Let N ≥ 4 and suppose f is the identity}

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map on a subgroup of rankN : then this subgroup is a subgroup of the kernel of f − 1, so the rank of this kernel (the degree of(f − 1)) divides N. But this degree is equal to

(f − 1)(f − 1)∗_{= 2 − tr(f ),}

so we have tr(f ) ≡ 2 mod N.

ForN≥ 5, together with the constraint on the trace from above, we conclude that tr( f ) = 2, so f satisfies

f2− 2 f + 1 = ( f − 1)2= 0

so f = 1, the identity on E. For N = 4, another solution is tr(f ) = −2, hence f2+ 2f + 1 = (f + 1)2= 0

and f = −1. However, in this case the f − 1 that was supposed to vanish on our subgroup is equal to -2, so this subgroup is in factE[2] (the kernel of [2], a Klein 4-group that does not contain an element of order 4).

In either case, we find that an automorphism ofE fixing an element of order N ≥ 4 is trivial, proving rigidity of the moduli problemΓ₁(N). This means there is a smooth affine scheme Y₁(N)/Z[1/N] that represents the functor [Γ₁(N)] for n ≥ 4. In particular, for a finite field k, the set of isomorphism classes of elliptic curves overk with a choice of distinguished point of exact orderN is in bijection with Y₁(N)(k).

Conversely,Γ₁(N) is not rigid for N = 2 or 3. For N = 3, the proof above affords the option tr(f ) = −1, hence

f2+ f + 1 = 0

and f is a primitive third root of unity. So look at E : y2_{+ y = x in characteristic greater}

than 3, the curve with j(E) = 0. This curve has an automorphism group of order 6, so an automorphism f : E → E of order 3. On the other hand, over a large enough field we have E[3] ' (Z/3Z)2_{, a group of order 9 containing 8 elements of order 3. This} _{f , being an}

auto-morphism, permutes the points of order 3 amongst themselves, and a permutation of order 3 on a set of 8 elements has a fixed point.

ForN = 2, any elliptic curve has the automorphism -1, which fixes elements of order 2 by definition.

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2 Counting Curves up to Isogeny:

Honda–Tate

2.1 The Tate Module

Letk be a finite field of order q = pa. The degreeq Frobenius endomorphismπ : E → E of an elliptic curveE/k satisfies its own (quadratic, integral) characteristic polynomial

X2− t X + q = 0,

wheret := tr(π) equals q + 1 − #E(k) and satisfies t ≤ 2pq, the Hasse-Weil bound. So π is an algebraic integer in the division algebraM := End_k(E) ⊗ Q.

Let k be an algebraic closure of k and G = Gal(k/k) its absolute Galois group. Since a finite extension of degreee of k is the fixed field of F_qe (thee−th power of the q−Frobenius endomorphismx7→ xq_of_{k), every finite quotient of G is generated by F}

q. Since the absolute

Galois group is the inverse limit of its finite quotients, we haveG' ˆZ (the group of profinite integers), topologically generated by Frobenius.

Let` 6= p be prime. Denote by E[N] the kernel of multiplication-by-N on an elliptic curve E/k over the algebraic closure, i.e. the subgroup

{P ∈ E(k)|[N ]P = 0}.

Whenp- N, we have E[N] ' (Z/NZ)2, and moreover, the action ofG on E descends to E[N]. In particular, for everyn∈ N we have the G−module E[`n] equipped with natural transition mapsE[`n+1] → E[`n] given by multiplication by `. This is called a projective system, and we can form its projective limit

lim ←−E[` n_{] :=} ¨ (P1,P2, . . .) ∈ Y n∈N E[`n_] P_i= [`]P_i₊₁∀i ∈ N « . This projective limit of the system (E[`n_])

n∈N inherits theG−module structure from the

E[`n]; moreover, since the E[`n] are free rank 2 Z/`nZ−modules, their projective limit is a free rank 2 module over

lim ←−Z/`

n_Z_{= Z}

`,

i.e. abstractly isomorphic to Z2_`. It is called the Tate moduleT_`(E) of E. For two curves E, E0 overk there is a natural injective map

ϕ : Homk(E, E0) → HomG(T`(E),T`(E0))

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Theorem 5. (Tate isogeny theorem) The induced map

ϕ : Homk(E, E0) ⊗ZZ`→ HomG(T`(E),T`(E0))

is an isomorphism.

Proof. See[11], Theorem 3.3. Sketch:

It can be shown thatϕ has a torsion-free cokernel. Hence it suffices to show that Hom_k(E, E0) ⊗_ZQ_`→ HomG(V`(E),V`(E0))

is an isomorphism (withV_`(E) := T_`(E) ⊗_Z `Q`).

Assume there is an isogeny f from E to E0, i.e. Hom_k(E, E0) 6= 0. Then every other isogeny is obtained from f by precomposing with an endomorphism in Endk(E), so in other words

Hom_k(E, E0) ⊗_ZQ_`and End_k(E) ⊗_ZQ_`have the same dimension over Q_`. Since the map Hom_k(E, E0) → Hom_G(T_`(E),T_`(E0))

is injective, Hom_G(T_`(E),T_`(E0)) is nonzero as well and by the same reasoning as before, dim_Q

`HomG(V`(E),V`(E

0_{)) = dim}

Q_`EndG(V`(E)).

So we must show that dim_Q

`Endk(E) ⊗ZQ`≥ dimQ`EndG(V`(E)) :

The field Q(π) is a subfield of the division algebra M defined earlier, and lies in its centre since an endomorphism of E is defined of k if and only if it commutes withπ. Hence the image ofπ in End(V_`(E)) ' M₂(Q_`) (also called π) is semisimple.

Ifπ 6∈ Z, we have [Q(π) : Q] = 2, so since End_k(E) ⊗_ZQ_` contains Q_`(π) it is at least 2-dimensional over Q_`. At the same time, since the Galois groupG is topologically generated byπ, the G−equivariant endomorphisms End_G(V_`(E)) are precisely the centraliser of π in the 4-dimensional matrix algebra End(V_`(E)). The dimension of this centraliser must divide 4, and since it contains Q_`(π) it is at least 2. But we cannot have

dim_Q_`End_G(V_`(E)) = 4,

since thenπ would be central in M₂(Q_`), while the centre of a matrix algebra consists of the scalars and we had assumed thatπ is not a scalar. So

dim_Q

`EndG(V`(E)) = 2 and our mapϕ is an isomorphism.

In the caseπ ∈ Z, since π is central in M every endomorphism of E commutes with it, so is defined overk, and E is a supersingular elliptic curve with a 4-dimensional endomorphism algebraM (a quaternion algebra, explained below). Hence

dim_Q

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is automatically at least the dimension of End_G(V_`(E)), and we are done.

All that remains is to show that for HomG(V`(E),V`(E0) 6= 0, there is an isogeny from E to

E0. This is done by constructing appropriate liftings ofE and E0to a number field over which an isogeny exists, and showing that this isogeny descends back tok.

Sinceπ generates G as a profinite group we find that E and E0are isogenous overk if and only if their Frobenii have the same characteristic polynomial as endomorphisms of their Tate modules. This in turn means their traces agree, or in other words #E(k) = #E0_(k).

2.2 The Honda–Tate Theorem for Elliptic Curves

Hence the trace of the Frobenius endomorphism ofE, viewed as an algebraic integerπ, de-terminesE up to isogeny; thisπ is a Weil q−number, an algebraic integer such that |ι(π)| = q for all embeddings ι : Q(π) → C. We call two of them equivalent if there is an isomorph-ism of fields f : Q(π) ' Q(π0) with f (π) = π0(this is the same notion as having the same characteristic polynomial). As Honda and Tate proved, the converse is true too ([5, 18], see [4]):

Theorem 6. (Honda–Tate for elliptic curves) Letπ be the Frobenius endomorphism of an elliptic curve E/k. Then Q(π) is a subfield of M := End_k(E) ⊗ Q such that:

(i) π is a rational or quadratic algebraic integer with |π| = q (ii) there is a prime p above p in Q(π) such that |π|_p6= 1 (iii) |π|q= 1 for all other primes q of Q(π)

Conversely, any such algebraic integer is up to equivalence of Weil q−numbers the Frobenius endomorphism of some elliptic curve over k, determined up to isogeny.

The cases that can occur are the following:

WhenE is ordinary, we have M = Q(π) a quadratic imaginary number field such that (p) = pp0splits in D. In this case|π|p = q, |π|p0 = 1 (up to swapping p and p0) and the trace t is

coprime to p.

A quaternion algebra is a 4-dimensional algebraA over Q such that there are a, b ∈ A satis-fying

• 1,a, b , ab span A over Q

• a2andb2are in Q, and are negative • ab= −ba.

WhenE is supersingular, we haveπn∈ Z for some n, and p|t . Then either Q(π) = Q ⊂ M whereM is a quaternion algebra, or Q(π) = M is imaginary quadratic like before but p ramifies ((p) = p2_; _|π|

p= q) or remains inert ((p) = p; |π|p= pq; q has to be an even power of p for

this to happen).

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Remark: The number-theoretic conditions above are necessary. In full generality, the Honda-Tate theorem provides a bijection betweenall Weil q−integers and the isogeny classes of Abelian varieties overk, and we need the arithmetic data to single out elliptic curves from amongst all Abelian varieties. For example, while x2_{− 2x + 8 defines a quadratic 8−integer π, we have}

(2) = pp0_with_{(π) = p}2_p0_{in Q(π), and indeed π turns out to be the Frobenius of an Abelian}

3-fold.

Remark: While two elliptic curves over k with equivalent Weil numbers are isogenous, there is not necessarily an isogeny between them defined over k. But we can say the following: if

π and π0_{are equivalent, assume without loss of generality that}_{(π) and (π}0_{) are powers of the}

same prime p in Q(π). Then u = π/π0_{is a unit in Q(π), which is Q or an imaginary quadratic}

field. That means its group of units is torsion, sou is a root of unity of order at most 6. This gives us an extensionK/k with [K : k] ≤ 6 such that E and E0become isogenous overK.

To sum up, we have bijectively associated an equivalence class of algebraic integers to an isogeny class of elliptic curves over a finite field. In turn, we embed these integers inM2(Q),

where the notion of equivalence of Weil numbers corresponds to conjugacy. Thus, for an isogeny class of elliptic curves overk we finally end up with a conjugacy class in GL(2,Q).

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3 Counting Isogenous Curves:

Lattices in Cohomology

3.1 The Dieudonné Module

In the previous chapter we have used the Tate moduleT_`(E) of an elliptic curve E over a field of characteristic p for primes` 6= p. For ` = p this construction fails, since the pn−torsion ofE isn’t necessarily well-behaved subgroup of order p2n(it could for instance be a subgroup scheme containing a single point of multiplicity p2n).

For a commutative ringR and a prime p, let(X₀,X₁, . . .) be an infinite sequence of elements ofR and put W_n= ∞ X i=0 piX_ipn−i

(soW₀= X₀,W₁= X₀p+ pX₁, etc.). The setW(R) of all such infinite sequences in R can be turned into a (unique) commutative ring by demanding that:

• addition and multiplication inW(R) are given by universal (i.e. independent of R) in-tegral polynomials

• all polynomialsW_nare homomorphismsW(R) → R.

The setW(R) equipped with these operations is called the ring of Witt vectors of R. We are interested in the caseR = F_pn a finite field: in this case we have W(F_pn) = the unramified

extension of Zp of degreen, and in particular W(Fp) = Zp. For a full treatment of Witt

vectors, see[15].

The correct objects for capturing pn_{−torsion in characteristic p are the Dieudonné-modules}

D[pn]. The technical details can be found in [9, 2, 3], but the facts we need are as follows. For a finite fieldk of characteristic p, let W(k) be its ring of Witt vectors. The p−power Frobenius x7→ xp ofk induces a Frobenius endomorphismσ : W (k) → W (k). Define the Dieudonné ringDkas theW(k)−algebra generated by F,V subject to the conditions

F V = V F = p

F w= σ(w)F ∀w ∈ W (k)

wV = V σ(w) ∀w ∈ W (k)

(remembering thatW(k) is isomorphic to (an extension of) Z_pand hence contains an element p).

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Theorem 7. There is a functor D from the category of commutative group schemes of p−power order over k and the category of Dk−modules of finite W (k)−length such that:

(i) D is an anti-equivalence of categories

(ii) if the order of a group scheme G/k is pe_{, then the W}_{(k)−length of D(G) is e.}

(iii) for a map of fields k→ K, let GK denote the base change of G/k to K. Then

D(G_K) ' D(G) ⊗_W_(k)W(K)

For an elliptic curveE/k, let D(pn) be the D_k−module corresponding to E[ pn] under the anti-equivalence above. TheseD(pn_{) form a projective system, and we denote its projective}

limit D(E). The claim that this generalises the Tate module to ` = p is warranted by the following:

Theorem 8. (D(E) is free of rank 2 and Tate’s theorem holds) (i) D(E) is free of rank 2 as a W (k)−module

(ii) Hom_k(E, E0_{) ⊗}

ZZp' HomD_k(D(E), D(E0))

From now on, we will writeV_`(E) = T_`(E)⊗_Z

`Q`for primes` 6= p and Vp(E) = D(E)⊗Zp

Q_p. We defineT(E) = D(E) × Q_`6=pT_`(E) and V (E) = Q_`V_`(E).

While, as we have seen, T_p(E) and V_p(E) are D_k−modules, for ` 6= p all the T_`(E) and V_`(E) are G = Gal(k/k)−modules. Writing Z[G] for the group ring of G, this gives T (E) and V(E) the structure of a D_k×Z[G]−module; we shall denote this ring by DG (the Dieudonné-Galois ring?). ByDG−invariance of a subset S of a group X that carries the structure of both aG−module and a Dk−module, we mean that S is stable under both G and Dk. While this

does not of course imply invariance under the entire ringDG, we shall still abuse notation and write, for example,XDG _{for the set of}_{DG−invariants in the sense just described.}

3.2 Counting curves

After describing conjugacy classes in chapter 2, it is now time to take the level structure (i.e. distinguished point of order N ) into consideration and count the number of isomorphism classes of isogenous elliptic curves. Fix an elliptic curveE₀/k and denote

I(E₀) = {(E, P) ∈ Y₁(N)(k) | Hom_k(E, E₀) ⊗ Q 6= 0}.

In other words,I(E₀) is the set of elliptic curves over k with a distinguished point P ∈ E(k) isogenous toE₀; the condition Hom_k(E, E₀)⊗Q 6= 0 simply means that Hom_k(E, E₀) 6= 0, i.e. an isogeny exists, but in the sequel it will be useful to formally invert all isogenies by tensoring with Q. Note that we do not impose any condition on how the isogeny treats the distinguished pointP : indeed, the "base curve" E₀does not even carry a distinguished point.

Define an equivalence relation ' on I (E0) by: (E, P) ' (E0,P0) if and only if there is a

k−isomorphism of elliptic curves f : E → E0such thatf(P) = P0. We writeI(E0) = I (E0)/ '.

We have

V(E) =Y

`

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where all theV_`are 2-dimensional vector spaces over Q_`. AnR−lattice in a vector space over a fieldK is the R−span of a K−basis. In particular, by a lattice in V_`(E) we shall mean the Z_`−span (or W (k)−span in the case of ` = p) of a Q_`−basis. Given lattices Λ_`⊂ V_`(E) for all`, we call

Y

`

Λ_`⊂ V (E) a lattice inV(E). Next, we define

L(E0) = (Λ, P) | Λ ⊂ V (E0) DG-invariant lattice, P ∈ ₁ NΛ/Λ DG ,

soΛ is a Galois-invariant lattice in V (E₀) and P is a Galois-invariant lattice point of order N. The division algebraM= Endk(E0) ⊗ Q acts on L(E0), and we write L(E0) = L(E0)/M∗.

We now define a mapψ : I (E₀) → L(E₀) by the following steps:

For(E, P) ∈ I (E₀), pick a non-zero f ∈ Hom_k(E, E₀) ⊗ Q. This f induces (with abuse of notation) a map f : V(E) → V (E₀) which gives us a DG−invariant lattice

Λ = f (T (E)) ⊂ V (E0).

LetTp(E) = Q_`6=pT_`(E) be "the Tate module outside of p". Unpacking the definition of the Tate module, we can also write this as

Tp(E) = lim ←−

p-N

E[N],

the inverse limit over theN−torsion of E for p - N . This allows us to write E[N] = Tp(E)/NTp(E)

forN coprime to p. In other words, when using the Tate module to detect torsion, as long as we limit ourselves to torsion coprime to p, the Dieudonné module does not enter into the description and we might just as well write

E[N] = T (E)/NT (E)

This in turn means that f induces a DG−invariant point f (P ) ∈_N1Λ/ΛDG. This gives us (f (T (E)), f (P)) ∈ L(E0); let the image of (f (T (E)), f (P)) in L(E0) be ψ(E, P). (Remark: this

reasoning clearly breaks down in the case p| N , and the precise structure of the Dieudonné module needs to be taken into account. This has been done in[13].)

This last step ensures thatψ is well-defined: a different choice f0 ∈ Homk(E, E0) ⊗ Q is

related by f0_{= i ◦ f for some i ∈ M}∗_{, so makes no difference in}_L(E 0).

We claim: ψ is a bijection. Theorem 9. The mapψ is injective

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Proof. Supposeψ(E, P) = ψ(E0_,_P0_{): this means there are maps f : E → E}

0and f0:E0→ E0

and ani∈ D∗such that(Λ, f (P)) = (iΛ0,i f0(P0)), where Λ = f (T (E)) ⊂ V (E0) and similarly

forΛ0.

In Hom_k(E, E₀) ⊗ Q resp. Hom_k(E0_,_E

0) ⊗ Q, f and f0are invertible so we can write

ϕ : E−→ Ef 0 i−1

−→ E0 (f0₎−1

−−→ E0,

where a prioriϕ is some formal map in Hom_k(E, E0) ⊗ Q. However, to prove that ϕ is an actual map it suffices to show thatϕ(T (E)) ⊂ T (E0_{) by Tate’s isogeny theorem, and this is}

clearly true: f maps T(E) to Λ, i−1mapsΛ to Λ0, and f0mapsT(E0) to Λ0.

Moreover, reversing our diagram produces a ϕ−1 which is a morphism by the exact same reasoning and clearly invertsϕ, so ϕ : E → E0is in fact an isomorphism of elliptic curves.

To conclude, we note thatf maps P to f(P), i−1sends this to f(P0) and after applying (f0)−1 we end up at P0. So ϕ is an isomorphism with ϕ(P) = P0, or in other words: if ψ(E, P) =

ψ(E0_,_P0_{) then (E, P) and (E}0_,_P0_{) coincide in I (E}

0), and so ψ is injective.

Theorem 10. The mapψ is surjective

Proof. Take a DG−invariant lattice Λ ⊂ V (E0) and a point P ∈

₁

NΛ/Λ

DG

. Since Λ is a lattice, there is anx∈ N such that xΛ ⊂ T (E0). The group M∗contains Q∗, so in particular

x∈ M∗. This means we have(Λ, P) = (xΛ, xP) ∈ L(E₀).

So assume without loss of generality that Λ ⊂ T (E0). By definition of the Tate module,

T(E₀)/Λ is a finite subgroup G ⊂ E₀, so f : E₀ → E := E0/G is an isogeny, and we have a

canonical identificationT(E) = Λ. Hence P gives us a point P0∈ E(k) of order N , and by construction we haveψ(E, P0) = (Λ, P) in L(E₀) and ψ is surjective.

As we have seen, we can replace E₀in the definition ofI₀(E₀) and L₀(E₀) above by the al-gebraic integerπ, since an isogeny class depends only on the Frobenius endomorphism of its members. Furthermore, this Frobenius element, considered as an algebraic integer, is only determined up to conjugacy in M₂(Q), so we can replace π by its conjugacy class γ.

So finally we arrive at the following formula. For a fieldk of q= pn_{elements, and for}_N_{≥ 4}

coprime to p, we have

#X₁(N)(k) = X

γ⊂M2(Q)

#L(γ), where:

• γ runs over the conjugacy classes of Frobenii of elliptic curves (i.e. quadratic q−integers satisfying the arithmetic conditions from the Honda–Tate theorem)

• L(γ) is the set of DG−stable lattices in Q_pQ2_pon which Frobenius acts as prescribed byγ, with a distinguished DG−invariant point of order N, modulo the action of M∗.

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Popular summary

Voor elke keuze van a ∈ R definieert y = x2_{+ a een parabool: je kan dus zeggen dat de}

verzameling R opgevat kan worden als een verzameling van bepaalde parabolen, dat wil zeg-gen, meetkundige krommen. Maar R is zelf ook een (vrij eenvoudige) meetkundige kromme, dus we zien hier dat het soms mogelijk is eenverzameling meetkundige objecten zelf ook als meetkundig object te zien: zo’n verzameling heet dan een moduliruimte, en is van groot belang in de meetkunde.

Een elliptische kromme is een speciaal soort kromme die in veel takken van de wiskunde en daarbuiten bestudeerd wordt. Moduliruimten van elliptische krommen zijn zelf ook weer krommen, de modulaire krommen. Dit zijn vaak ingewikkelde objecten waar we meer van zouden willen begrijpen, om op die manier weer meer over elliptische krommen te weten te komen. Op basis van een modulaire kromme kan je een complexe functie definiëren, de zeta-functie van de kromme, die veel informatie over de kromme bevat maar veel makkelijker te begrijpen zou moeten zijn. Hiervoor is echter wel noodzakelijk dat deze zeta-functie goede analytische eigenschappen heeft en geen al te raar gedrag vertoont. Dat dit ook daadwerkelijk zo is, is het vermoeden van Hasse en Weil.

Om dit vermoeden te bewijzen lopen we in deze scriptie de methode na die Ihara en Lang-lands hiervoor bedacht hebben, en die door Kottwitz is uitgevoerd. Omdat de elliptische krom-men waaruit de modulaire kromme is opgebouwd zulke bijzondere objecten zijn, kunnen we van hun speciale eigenschappen gebruik maken om precies te tellen hoeveel punten de modu-laire kromme bevat. Op basis hiervan kan met analytische methodes (waar we in deze scriptie niet op ingaan) de zeta-functie op zo’n manier bepaald worden, dat blijkt dat deze inderdaad alle gewenste eigenschappen heeft.

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