Lectures on Nx(p)

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Lectures on N

_X

(p)

Jean-Pierre Serre

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Preface

The title of these lectures requires an explanation: what does NX(p)

mean ?

Answer: NX(p) is the number of solutions mod p of a given family X

of polynomial equations in several unknowns, and with coefficients in Z, so that it makes sense to reduce mod p, and count the solutions. For a fixed X, one wants to understand how NX(p) varies with p : what is its

size and its congruence properties ? Can it be computed by closed formulae, by cohomology, and/or by efficient computer programs ? What are the open problems ?

These questions offer a good opportunity for reviewing several basic tech-niques in algebraic geometry, group representations, number theory, cohomo-logy (both `-adic and standard) and modular forms.

This is why I chose this topic for two week-long courses at the National Center for Theoretical Sciences (NCTS) in Hsinchu, Taiwan, in July 2009 and April 2011. A group of people wrote up a set of notes based on my 2009 lectures, and I rewrote and expanded them. Here is the result of that rewriting–expanding.

There are nine chapters. The first four are preliminary, and short : they contain almost no proofs.

Chapter 1 gives an overview of the main theorems on NX(p) that will be

discussed later, and Chapter 2 contains explicit examples, chosen for their simplicity and/or for aesthetic reasons.

Chapter 3 is about the Chebotarev density theorem, a theorem that is essential in almost everything done in Chapters 6 and 7 ; note in particular the “frobenian functions” and “frobenian sets” of §3.3 and §3.4.

Chapter 4 reviews the part of `-adic cohomology that will be used later. Chapter 5 contains results on group representations that are difficult to find explicitly in the literature, for instance the technique consisting of com-puting Haar measures in a compact `-adic group by doing a similar compu-tation in a real compact Lie group.

These results are applied in Chapter 6 in order to discuss the possible relations between two different families of equations X and Y . Here is an example : suppose that |NX(p) − NY(p)| 6 1 for every large enough p ; then

there are only three possibilities :

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ii) there exists a non-zero integer d such that NX(p) − NY(p) = (d_p) for

every large enough p ;

iii) same as ii) with (d_p) replaced by −(d_p).

This looks mysterious at first, but if one transforms it into a statement on group characters, it becomes very simple.

Chapter 7 is about the archimedean properties of the NX(p) – a topic on

which we know much less than in the `-adic case.

Chapter 8 is an introduction to the Sato-Tate conjecture, and its concrete aspects.

Chapter 9 gives an account of the prime number theorem, and of the Chebotarev density theorem, in higher dimension.

The text contains a few complementary results, usually written as exercises, with hints.

It is a pleasure to thank NCTS and its director Winnie Li for their hospitality and for their help during and after these lectures. I also thank K.S. Kedlaya and K. Ribet for their numerous corrections.

Paris, August 2011 Jean-Pierre Serre

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A measure on a compact topological space X is a Radon measure in the sense of [INT] (see also [Go 01]), i.e. a continuous linear form ϕ 7→ µ(ϕ) on the Banach space of continuous functions on X. Most of the measures we consider are positive of mass 1 ; this means that µ(1) = 1 and ϕ _{> 0 ⇒ µ(ϕ) > 0.} The comparison symbols {O, o, ∼, <<} of analytic number theory have their usual meaning ; it is recalled the first time the symbol occurs.

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Chapter 1. Introduction

1.1. Definition of NX(p) : the affine case

Let fα(X1, ..., Xn) ∈ Z[X1, ..., Xn] be a family of polynomials with integer

coefficients. If p is a prime number, let Nf(p) be the number of solutions of

the system of equations fα(x1, ..., xn) = 0 over the field Fp = Z/pZ, i.e.

Nf(p) = |{(x1, ..., xn) : xi ∈ Fp, fα(x1, ..., xn) = 0 in Z/pZ for each α}| .

This can be translated into the language of commutative algebra as fol-lows. Let (fα) be the ideal of Z[X1, ..., Xn] generated by the polynomials fα.

The quotient A = Z[X1, ..., Xn]/(fα) is a ring of finite type over Z. The points

x ∈ (Fp)n with fα(x) = 0 for all α correspond bijectively to the

homomor-phisms A → Fp, or equivalently to the maximal ideals1 of A with residue

field Fp; we are just counting such homomorphisms.

More generally, we shall be interested in the number of solutions of the equations fα(x) = 0 in a finite field of order pe, for e = 1, 2, ... ; this number

will be denoted2 _{by N} f(pe).

1.2. Definition of NX(p) : the scheme setting

Consider a scheme X of finite type over Z ; this assumption means that X has a finite covering by open subschemes of the form Spec A with A of finite type over Z. [In the affine case above, one has X = Spec A, where A = Z[X]/(fα).]

We denote by X/Q the generic fiber of X → Spec Z, i.e. the Q-algebraic

variety deduced from X by the base change Z → Q.

The fiber Xp of X → Spec Z at a prime p ∈ Spec Z is a scheme over Fp.

We denote by NX(p) the number of its Fp-points ; equivalently :

NX(p) = number of closed points x ∈ X whose residue field κ(x) is such

that |κ(x)| = p.

If q = pe _{is a power of p, with e} _{> 1, we define similarly N}

X(q) as the

cardinality of the set X(Fq) of the Fq-points of X (or of Xp - that amounts

1. Recall that, if m is a maximal ideal of A, the quotient A/m is a finite field, cf. e.g. , [AC V-68, cor.1]. Equivalently, if X is a scheme of finite type over Z, a point x of X is closed if and only if its residue field κ(x) is finite.

2. Warning. One should not confuse Nf(pe) with the number Nf(mod pe) of solutions of the equations fα(x) = 0 in the ring Z/peZ ; for examples of computation of Nf(mod pe), see the exercises of §1.3, §2.1.2 and §2.2.3.

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to the same). Recall that an element of X(Fq) is a morphism of Spec Fq into

X ; one may view it as a pair (x, ϕ) where x is a closed point of X and ϕ is an embedding of κ(x) into Fq.

As explained in the Preface, the goal of the present lectures is to describe how NX(p) varies with p, and in particular to relate this variation to the

topology of the analytic space X(C) made up of the complex points of X. [Note that, in the setting of §1.1, X(C) is the set of all x ∈ Cn such that fα(x) = 0 for every α.]

Remarks about NX(pe).

1) NX(pe) is additive in X : if X is a disjoint union of subschemes Xi, we

have NX(pe) = P NXi(p

e

) for every prime p and every e > 1.

2) NX(pe) depends only on the reduced scheme Xred associated with X :

nilpotent elements play no role.

3) If two schemes X and Y become isomorphic over Q, i.e. if X/Q ' Y/Q,

then there exists a prime p0 such that NX(pe) = NY(pe) for every p > p0 and

every e_{> 1. This shows that the knowledge of the Q-variety X}/Q is enough

to determine the NX(pe), for all p but finitely many.

4) There is no need to assume e_{> 1 : there is a reasonable definition of} NX(pe) for every e ∈ Z, see the end of §1.5 ; note however that, when e 6 0,

NX(pe) does not usually belong to N, nor even to Z, but it belongs to Z[1/p].

Remarks 1), 2) and 3) make possible several dévissage arguments ; accor-ding to our needs, we may for instance assume that X is affine, or separated, or projective, or smooth (either over Z or over Q), etc.

Exercise. Let X be a scheme of finite type over Z. Show that there exists a finite set of polynomials f = (fα), as in §1.1, such that NX(pe) = Nf(pe) for every prime p and every integer e.

[Hint. Use the fact that X is noetherian to show that there exists a decreasing sequence of closed subschemes X = X0 ⊃ X1 ⊃ ... ⊃ Xn = ∅ such that each Xi Xi+1 is affine of finite type over Z. If Y is the disjoint union of the Xi Xi+1, then Y is affine, and NX(pe) = NY(pe) for every p, e.]

1.3. How large is NX(p) when p → ∞ ?

Here are some of the results that we shall discuss later (mainly in Chapter 7). The first one is very simple ; it tells us that the empty set can be detected by counting its points mod p :

Theorem 1.1. X(C) = ∅ ⇐⇒ NX(p) = 0 for every large enough p.

Remark. Note first that X(C) = ∅ is equivalent to X/Q = ∅. Suppose that

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is known (see [Ax 67], as well as §7.2.4) that PX has a density3 , which is

a strictly positive rational number. The same is true for the set of p with NX(p2) 6= 0 ; same for NX(p3) 6= 0, etc.

The next theorem (proved in §7.2.1) relates the asymptotic behavior of NX(p) with the complex dimension of X(C) (which is the same as dim X/Q) :

Theorem 1.2. Let d_{> 0.}

(a) dim X(C)_{6 d ⇐⇒ N}X(p) = O(pd) when p → ∞.

(b) Assume dim X(C) _{6 d. Let r be the number of C-irreducible components} of dimension d of X(C). Then

lim sup

p→∞

NX(p)

pd = r.

(c) Assume dim X(C) _{6 d. Let r}0 be the number of Q-irreducible components

of dimension d of X/Q. Then P p6xNX(p) = r0 xd+1 log(xd+1₎+ O(x d+1_{/(log x)}2_{) when x → ∞.}

Recall that the O-notation in (a) means that there exist a prime p0 and a

number C > 0 such that

NX(p) 6 Cpd for all p> p0.

Remark. Theorem 1.2 shows that the asymptotic properties of the function p 7→ NX(p) detect the dimension d of X(C), the number of its irreducible

components of dimension d, and the number of Q-irreducible components of X/Q of dimension d.

Example. The equation x2 + y2 = 0 represents the union of two lines in the affine plane, with slopes i and −i. We have d = 1, r = 2, r0 = 1 and

NX(pe) =      pe if p = 2 2pe− 1 if pe ≡ 1 (mod 4) 1 if pe ≡ 3 (mod 4).

This example shows that “ lim sup ” cannot be replaced by “ lim ” in Theorem 1.2.(b).

3. a “natural density”, in the sense defined later (§3.1.3) ; we shall not be interested in the weaker notion of “Dirichlet density”, which is mostly useful in the equal-characteristic case.

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Remark. In §7.2.1 we shall prove Theorem 1.2 in a refined form, and with a better error term (the same as the one occurring in the Chebotarev density theorem, cf. §3.2.3).

Exercise. Let N (mod pe) be the number of solutions of x2_{+ y}2 _{= 0 in the ring Z/p}e_Z. Show that :

i) N (mod 2e_{) = 2}e_.

ii) If p ≡ 1 (mod 4), then N (mod pe_{) = (e + 1)p}e_{− ep}e−1_.

iii) If p ≡ 3 (mod 4), then N (mod pe_{) = p}e_{if e is even and N (mod p}e_{) = p}e−1_{if e is odd.}

1.4. More properties of p 7→ NX(p)

The next theorem is a kind of rigidity property of the function p 7→ NX(p).

It will be proved in §6.1.3. Here again, the proof uses the Chebotarev density theorem. But it also depends in an essential way on the properties of `-adic cohomology, due to Grothendieck ; we shall recall them in Chapter 4.

Theorem 1.3. Let X, Y be two schemes of finite type over Z. Assume that NX(p) = NY(p) for a set of primes p of density 1. Then there exists a prime

number p0 such that NX(pe) = NY(pe) for all p > p0 and all e> 1.

[If X and Y are separated, the “density 1” hypothesis can be replaced by “ density > 1 − 1/B2 _{”, where B depends only on the cohomology of the spaces}

X(C) and Y (C), cf. Theorem 6.17.]

The same method also gives the following curious-looking result (cf. §6.1.2) : Theorem 1.4. Let X be a scheme of finite type over Z. Let a and m be integers with m _{> 1. The set of primes p such that N}X(p) ≡ a (mod m) has

a density, which is a rational number. If X is separated4 _{and a is equal to}

the Euler-Poincaré characteristic χ(X(C)) of X(C), that density is > 0. Corollary 1.5. For every m _{> 1, the set of p such that}

NX(p) ≡ χ(X(C)) (mod m)

is infinite.

Let us recall what the Euler-Poincaré characteristic is. If T is a locally compact space, let Hi_{(T, Q) be the i-th cohomology group of T with}

coef-ficients in Q ; for i = 0, the Q-vector space Hi_{(T, Q) is the space of locally}

constant functions T → Q. We also have H_ci(T, Q), the cohomology with 4. This assumption insures that X(C) is locally compact, so that its cohomology with compact support is well defined.

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compact support ; here H0

c(T, Q) is the Q-vector space of locally constant

functions T → Q that vanish outside a compact subspace of T .

If the Hi and the H_ci are finite-dimensional vector spaces and vanish for i large, we may define the Euler-Poincaré characteristic of T by the usual formula :

χ(T ) =P

i>0(−1)

i_{dim H}i_{(T, Q),}

together with its variant with compact support : χc(T ) =

P

i>0(−1)idim Hci(T, Q).

These definitions apply5 _{to the locally compact space T = X(C). Moreover,}

by a theorem of Laumon (cf. [La 81]), we have χ(T ) = χc(T ). Hence, in

Theorem 1.4, we may use at will6 _{either χ or χ} c.

Remark. The formula χ(T ) = χc(T ) would become false if T = X(C) were

re-placed by T = X(R). For instance, if X is the affine line and T = X(R) = R, we have χ(T ) = 1 and χc(T ) = −1. More generally, Poincaré duality shows

that, for a real orientable manifold V of (real) dimension d, we have χ(V ) = (−1)d_χ

c(V ).

1.5. The zeta point of view

It is often convenient to pack the information given by NX(pe) into one

single object : the zeta function ζX(s) of the scheme X, i.e. the Dirichlet

series P ann−s defined by the infinite product

ζX(s) =

Y

x∈X

1 1 − |x|−s ,

where x runs through the set X of closed points of X and |x| is the number of elements of the residue field κ(x). The product converges absolutely for Re(s) > dim X, see e.g. [Se 65] ; here dim X is the dimension of the scheme X, not that of X(C) ; for instance dim Spec Z = 1.

5. Indeed, it is enough to prove this when X is quasi-projective (affine would be en-ough), in which case the triangulation theorem of analytic spaces shows that T is homeo-morphic to K L, where K is a finite simplicial complex and L is a closed subcomplex of K. (A proof, in the more general setting of “semi-algebraic sets”, can be found in [BCR 98, §9.2].) This implies that the abelian groups Hi_{(T, Z) and H}i

c(T, Z) are finitely gene-rated and are 0 for i > dimtopT = 2 dim X(C) ; the same is true for the Q-vector spaces Hi_{(T, Q) and H}i

c(T, Q).

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A standard computation shows that the Dirichlet series ζX(s) can be

written as an Euler product

ζX(s) = Y p ζX,p(s), where ζX,p(s) = exp( ∞ X e=1 NX(pe)p−es/e).

Hence ζX(s) is determined by the NX(pe). Conversely, NX(pe) can be

reco-vered from the coefficients an of ζX(s), by expanding the identity

NX(p)t + NX(p2)t2/2 + · · · = log(1 + apt + ap2t2+ · · ·),

where t is an indeterminate. For instance :

NX(p) = ap, NX(p2) = 2ap2 − a2_p, N_X(p3) = 3a_p3 − 3a_pa_p2 + a3_p.

In what follows, we will mostly work with NX(pe), but once in a while we

shall also mention the zeta aspect. Definition of NX(pe) for e 6 0.

By Dwork’s rationality theorem ([Dw 58]), proved by p-adic methods (and reproved `-adically by Grothendieck, see §4.4), ζX,p(s) is a rational function

of p−s. More precisely, there is a product decomposition ζX,p(s) =

Q

z∈C×(1 − zp−s)n(z),

where the integers n(z) are 0 for every z ∈ C× except a finite number. This is equivalent to saying that

(*) NX(pe) =

P

zn(z)z

e _{for every e}

> 1.

By a theorem of Deligne, the z that occur with a non-zero coefficient n(z) are “ p-Weil integers ”, see §4.5 ; moreover, one has n(z) = n(z0) if the algebraic numbers z and z0 are conjugates over Q.

If e ∈ Z is _{6 0, one may take (*) as the definition of N}X(pe) ; hence

it makes sense to write NX(p−1), or NX(p0), and the reader can check that

these numbers enjoy most of the properties of the standard NX(pe) for e > 1 ;

the main differences are :

– NX(p0) belongs to Z, but not always to N (it is equal to the Euler

characteristic of Xp, cf. §4.3) ;

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Chapter 2. Examples

We collect here a few examples where dim X(C) is equal to 0, 1 or 2. Most of them will be reconsidered later from the viewpoint of étale cohomology, cf. §4.6.

2.1. Examples where dim X(C) = 0

We take f ∈ Z[x] with f 6= 0 and consider X = Spec Z[x]/(f ), so that NX(p) is the number of solutions in Z/pZ of f (x) ≡ 0 (mod p).

2.1.1. A quadratic equation Take f = x2+ 1. We have NX(p) =      1 if p = 2 ; 2 if p ≡ 1 (mod 4) ; 0 if p ≡ −1 (mod 4) .

By Dirichlet’s theorem on arithmetic progressions, each of the two cases p ≡ 1 (mod 4) and p ≡ −1 (mod 4) occurs with density 1₂. However, Sar-nak and Rubinstein have shown (assuming the truth of some rather strong conjectures7_{) that there are quite often more primes ≡ −1 (mod 4) than}

primes ≡ 1 (mod 4). (For the precise definition of “quite often”, see their paper [RS 94].)

2.1.2. A typical cubic equation

Take f = x3 − x − 1. This polynomial has discriminant 4 − 27 = −23. When p = 23, the equation f = 0 has a double root mod p and NX(23) is

equal to 2. For p 6= 23, we have (see [Bl 52] and [Se 03]) : NX(p) =      1 if ₂₃p = −1;

3 if ₂₃p = 1 & p is represented by the binary form a2_{+ ab + 6b}2_;

0 if ₂₃p = 1 & p is represented by the binary form 2a2_{+ ab + 3b}2_.

These three sets of primes have density 1/2, 1/6 and 1/3 respectively. Remark. The result can also be expressed in a more compact form by intro-ducing the power series F23=

P∞

n=1anqn defined by the formula :

F23= 1 2( X a,b∈Z qa2+ab+6b2 − X a,b∈Z q2a2+ab+3b2) = q ∞ Y n=1 (1 − qn)(1 − q23n)

7. They assume that the non-trivial zeros of the Dirichlet functions are on the line Re(s) = 1₂, are simple, and that their imaginary parts (normalized to be > 0) are Q-linearly independent.

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= q − q2− q3_{+ q}6 _{+ q}8_{− q}13_{− q}16_{+ q}23_{+ · · ·,}

which is a cusp form of level 23 and weight 1, cf. e.g. [Fr 28, p.472]. It can be proved that

NX(p) = ap+ 1

for every p. Moreover, the zeta function ζX(s) of X is equal to L23(s)ζ(s),

where ζ(s) is the standard zeta function (i.e. that of Spec Z), and L23(s) is

the Dirichlet series L23(s) = X n>1 ann−s = Y p 1 1 − app−s+ (₂₃p) p−2s

whose coefficients anare the same as those of the power series F23. See [Se 02,

§5.3] for an interpretation in terms of Artin L-functions. Exercises.

1) Let NX(mod pe) be the number of solutions of x3− x − 1 = 0 in Z/peZ. Show that : i) NX(mod pe) = NX(p) if p 6= 23.

ii) NX(mod pe) = 1 if p = 23 and e > 1.

2) Let NX(pe) be the number of solutions of x3− x − 1 = 0 in a field with pe elements. Show that NX(pe) = 3 if NX(p) = 1 and e is even, or NX(p) = 0 and e is divisible by 3. Show that NX(pe) = NX(p) otherwise.

2.1.3. Another cubic equation

Take f = x3 + x + 1. This polynomial has discriminant −4 − 27 = −31. The results are almost the same8 _{as those for x}3_{− x − 1, the binary quadratic}

forms a2_{+ ab + 6b}2 _{and 2a}2_{+ ab + 3b}2 _{with discriminant −23 being replaced}

by a2+ ab + 8b2 and 2a2+ ab + 4b2, which have discriminant −31. Here also we have NX(p) = ap+ 1, where ap is the p-th coefficient of the cusp form

F31= 1 2( X a,b∈Z qa2+ab+8b2 − X a,b∈Z q2a2+ab+4b2).

The main difference is that it is not possible to write F31 as a product

qQ

n>1(1 − q

n₎cn _{with bounded exponents c}

n’s (for a more precise statement,

giving the value of lim sup1_nlog |cn|, see exercise below).

Exercise.

8. The main point is that h(−23) = h(−31) = 3, i.e. the quadratic fields of discriminant −23 and −31 have class number 3, cf. §3.3.3.3.

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a) Show that ₋₃₁−6 1

5

is an element of order 3 of the modular group Γ0(31), that fixes the point z0=11+i

√ 3

2·31 of the upper-half plane Im(z) > 0 .

b) Show that the modular form z 7→ F31(e2πiz) vanishes on the Γ0(31)-orbit of z0, and at no other point of the upper-half plane ; show that it is non-zero when Im(z) > Im(z0). c) Use b) to prove that the radius of convergence of the q-series log(1_qF31(q)) is equal to e−π√3/31_.

d) Show that, if one writes F31as qQ(1 − qn)cnwith cn∈ Z (which is possible in a unique way), then lim sup1_nlog |cn| = π

√ 3/31. 2.1.4. Computational problems

If f is given, the problem of computing the corresponding NX(p) for large

p is a P -problem : there is a deterministic algorithm that solves it in time O((log p)A_{) for a suitable exponent A. The method is simple}9 _{: it consists in}

computing the gcd g(x) of f (x) and xp _{− x in F}

p[x] ; the degree of g(x) is

NX(p). This gives an exponent A equal to 3 ; using “fast multiplication” (cf.

[Kn 81, §4.3.3]) one can bring A down to 2 + ε for every ε > 0.

There is a special case where one can do better. When the Galois group of f is abelian, the roots of f belong to a cyclotomic field Q(zm), where zm

is a primitive m-th root of unity. In that case, the value of NX(p) depends

only10 _{on the value of p mod m ; since this value can be computed in time}

O(log p), we can take A = 1.

Problem. Is this the only case where the exponent A can be taken < 2 ? For instance, in the two cubic cases given above, can one prove that there does not exist any deterministic algorithm computing NX(p) in time

O((log p)A_{) with A < 2 ?}

Another natural question is :

Problem. Compute the roots of f mod p in polynomial time.

This is easily done if one accepts probabilistic algorithms (see e.g. [Kn 81, §4.6.2]), but no deterministic polynomial time algorithm seems to be known in the general case, except when deg f = 2, thanks to a theorem of Schoof [Sc 85, §4], and in a few other cases (roots of unity of prime order, cf. [Pi 90]). I do not know what the situation is for the two cubic equations written above.

2.1. Examples where dim X(C) = 1

9. As J-F. Mestre pointed out to me, this method was already known to Libri and to Galois around 1830, cf. [Ga 30].

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2.2.1. Genus 0

Take for X the conic in the projective plane11 _P

2 defined by the

homo-geneous equation

x2+ y2+ z2 = 0.

We have NX(p) = p + 1 for every prime p : this is obvious for p = 2 since

the equation is equivalent to x + y + z = 0, hence has the three solutions (1,0,0), (0,1,0) and (0,0,1). For p > 2, the conic Xp is smooth, and has an

Fp-rational point (by Chevalley-Warning, or by a direct argument), hence is

isomorphic to P1, hence has p + 1 rational points.

[Alternative proof : use Weil’s bound |NX(p) − (p + 1)| 6 2gp

1

2, with g = 0.]

The zeta function of X is ζX(s) = ζ(s)ζ(s − 1).

2.2.2. Genus 1 with complex multiplication

Consider the elliptic curve X in P2 given by the affine equation

y2 = x3− x.

This curve has good reduction outside 2 ; its conductor is 25_{, cf. [Cr 97, p.111,}

case 32A2(A)]. It has complex multiplication : its Q-endomorphism ring is the ring Z[i] of Gaussian integers, with i acting as (x, y) 7→ (−x, iy). One finds that NX(p), for p > 2, is given by

NX(p) = p + 1 − ap,

where ap is as follows :

if p ≡ −1 (mod 4), we have ap = 0, so that NX(p) is p + 1, as if the curve

had genus 0 (but NX(p2) is not p2+ 1 : it is p2+ 2p + 1) ;

if p ≡ 1 (mod 4), we can write p as ππ with π ∈ Z[i] ; hence p = u2+ v2 if π = u + vi with u, v ∈ Z. We can choose π in a unique way12 _{(up to}

conjugation) such that π ≡ 1 (mod (1+i)3_{). Then a}

p = 2u = π+π. Moreover,

NX(pe) is equal to pe+ 1 − (πe+ πe) for every e > 1.

11. Here - and elsewhere too - we use Bourbaki’s notation : the n-dimensional projective space is denoted by Pn, and not by Pn which would suggest that it is the n-th-power of P1_.

12. Every fractional ideal a of Z[i] prime to λ = 1+i has a unique generator πasuch that πa≡ 1 (mod λ3). The map a 7→ πais the Hecke character associated with the elliptic curve X, cf. [De 53]. [Hint. The kernel of the action of λ3_{on X is made up of the following eight} points : the 2-division points (x = 0, 1, −1, ∞) and the four points with x = i, −i. Hence, if p splits in Q(i), the corresponding Frobenius endomorphism, viewed as an element π of Z[i], fixes that kernel, and we have π ≡ 1 (mod λ3_).]

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2.2.3. Genus 1 without complex multiplication

Consider the elliptic curve X in P2 given by the affine equation

y2− y = x3_{− x}2_.

This curve has good reduction outside p = 11 ; its conductor is 11, cf. [Cr 97, p.110, case 11A3(A)].

To compute NX(p) we use the modular form

F11(q) = q ∞ Y n=1 (1 − qn)2(1 − q11n)2 = ∞ X n=1 anqn,

which is a cusp form of level 11 and weight 2, see e.g. [Fr 28, p.432]. Here again, one can prove (Eichler, Shimura, cf. [Sh 66]) the formula

NX(p) = p + 1 − ap.

Example. The q-expansion of F11 is q − 2q2 − q3 + 2q4 + q5 + · · · ; hence

a2 = −2, a3 = −1, a5 = 1 and NX(p) = 5 for p = 2, 3, 5.13

Zeta function. We have

ζX(s) = ζ(s)ζ(s − 1) L(s) , with L(s) =Xan· n−s= Y p 1 1 − app−s+ ε(p) p1−2s ,

where the an’s are the same as above and ε(p) is equal to 0 if p = 11 and to

1 if p 6= 11.

13. Note that NX(p) is divisible by 5 for every p 6= 11, since the group X(Fp) contains a subgroup of order 5, namely the one with (x, y) = (0, 0), (0, 1), (1, 0), (1, 1), (∞, ∞). When p = 11, it is NX(p) − 1 that is divisible by 5 : one has to remove the double point of the cubic in order to get an algebraic group (viz. the multiplicative group Gm).

There is also an explicit formula for the value mod 5 of NX(p)/5 when p 6= 11, namely NX(p)/5 ≡ (p − 1)α(p) (mod 5), where α is the unique homomorphism of (Z/11Z)× into Z/5Z such that α(2) = 1. [Hint. Use the fact that there exists a curve isogenous to X that contains µ5× Z/5Z, namely the curve X0(11). For a different proof, and a generalization, see [Maz 78, p.139].]

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Remark. This kind of relation between zeta functions and modular forms is a special case of the “modularity conjecture”, started by Taniyama in 1955, made more precise by Weil in 196614 and eventually proved by Wiles and others, cf. [Wi 95] and [BCDT 01].

Exercise. Let NX(mod pe) be the number of points of the projective curve X in the ring Z/pe_{Z. Show that :}

i) NX(mod pe) = pe−1NX(p) if p 6= 11. ii) NX(mod pe) = pe− pe−1 if p = 11 and e > 1.

2.2.4. Computation of NX(p) for large p when dim X(C) = 1

It is known that, for a fixed curve X over Q, the computation of NX(p)

can be done in polynomial time with respect to log p. This has been proved by Schoof [Sc 85] when the genus of X is 1, and by Pila [Pi 90] in the general case ; see also [KS08] for more practical aspects of the computation of NX(p)

in the case of hyperelliptic curves.The case of varieties of higher dimension is open, cf. [CL 07] and [CE 11, Epilogue].

2.2. Examples where dim X(C) = 2 2.3.1. Affine quadratic cone

In affine 3-space, consider the quadratic cone X defined by the equation x2 _{= yz. One has N}

X(pe) = p2e for every prime p and every e > 1, as if X

were isomorphic to affine 2-space (which it is not). 2.3.2. Quadrics in 3-space

Take for X the quadric in P3 defined by the homogeneous equation

ax2+ by2+ cz2+ dt2 = 0

where a, b, c, d are non-zero integers. Over C, such a surface is isomorphic to P1× P1. Over a finite field Fq of characteristic 6= 2, this is true if and only

if abcd is a square (assuming that abcd 6= 0 in Fq). In that case the number

of the Fq-points is q2+ 2q + 1.

14. The years 1965-1967 were an especially favorable period for Number Theory : besides Weil’s paper [We 67], and the Sato-Tate conjecture ([Ta 65]), there was the launching of Langlands program [La 67] and the introduction of motives by Grothendieck ([CS 03, pp.173-175]). It was already suspected at that time that these daring theories are but the different facets of the same mathematical object. Half a century later, a lot of progress has been made by Deligne, Faltings, Wiles, Taylor and others, but we still do not know exactly how the pieces fit together.

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If abcd is not a square in Fq, the quadric is isomorphic to the “Weil’s

restriction of scalars” of the projective line P1for the quadratic field extension

Fq2/F_q; the number of its F_q-points is q2+ 1.

2.3.3. Rational surfaces

More generally, consider a smooth projective geometrically15 _irreducible

surface V over a finite field k with |k| = q ; assume that V is geometrically rational, i.e. that it becomes birationally isomorphic to P2 after the ground

field extension k/k. Let NS be the Néron-Severi group of V_/k. The Galois group of k/k acts on NS, and we have (cf. [Man 86, §27]) :

NV(q) = q2+ q Tr(σq)+1,

where Tr(σq) is the trace of the Frobenius element σq ∈ Gal(k/k) acting on

the free Z-module NS.

In the case where V is a smooth quadric, we have NS = Z2, σq acts either

trivially or by permuting the two factors, so that its trace is either 2 or 0, and we recover example 2.3.2.

The case where V is a smooth cubic surface in P3 (due to Weil [We

54, p.588] - see also [Man 86, §27]) is particularly interesting. We then have rank(NS) = 7. The 27 lines of the cubic surface give 27 elements of NS. The group of automorphisms of the incidence graph of these lines is isomorphic to Weyl(E6) = Weyl group of the root system E6. The action of Gal(k/k) on

these lines gives a homomorphism Gal(k/k) → Weyl(E6) that is well defined

up to conjugation. In particular, we have a Frobenius conjugacy class σq in

Weyl(E6), and the formula for NV(q) can be written as

NV(q) = q2+ (1 + a)q + 1,

where a is the trace of σq acting by the reflection representation of Weyl(E6),

which is of dimension 6.

Note that a can only take the values -3, -2, -1, 0, 1, 2, 3, 4, 6, as one sees by looking at the character table of Weyl(E6), cf. [ATLAS, p.27]. Hence :

q2_{− 2q + 1 6 N}

V(q) 6 q2+ 7q + 1.

Remark. For a given q one may ask what the possible values of a in {−3, −2, −1, 0, 1, 2, 3, 4, 6} are. This does not seem to be known. However Swinnerton-Dyer ([Sw 10]) has shown that the minimal value a = −3 is 15. It is customary to say that a scheme X has geometrically a property P if the k-scheme X_/k has property P (as if geometry could only be done over algebraically closed fields).

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always possible, and that the maximal value a = 6 is possible provided q 6= 2, 3, 5.

Exercise. Let V be a smooth cubic surface in P3 over Fq; define a ∈ Z by the formula NV(q) = q2+ (1 + a)q + 1, as above.

a) If q = 2, show that a = 6 is impossible because V would have strictly more Fq-points than P3.

b) If q = 4, show that the smooth cubic surface x2y + xy2 + z2t + zt2 = 0 has q2_{+ 7q + 1 = 45 rational points (i.e. a = 6), and that the automorphism group of the} surface is the unique subgroup of index 2 of Weyl(E6), i.e. is isomorphic to the simple group SU4(F4), cf. [ATLAS, p.26].

[Hint. If x ∈ F4, then x2is the F2-conjugate x of x, so that the equation can be rewritten as xy + xy + zt + zt = 0, and it is invariant by the unitary group SU4(F4).]

c) If a = −3, show that NV(qe) = (qe− 1)2 if e ≡ 1, 2 (mod 3) and NV(qe) = q2e_{+ 7q}e_{+ 1 if e ≡ 0 (mod 3).}

[Hint. Use the fact that the elements of Weyl(E6) of trace −3 have order 3, cf. [ATLAS, p.27] and [Do 07, §10.3.3].]

d) (T. Ekedahl and T. Shioda) If q ≡ 1 (mod 3), show that the equation x3+ y3+ z3+ λt3= 0,

where λ ∈ Fq is not a cube, defines a smooth cubic surface with a = −3. [Hint. Use Galois descent from the Fermat cubic x3+ y3+ z3+ t3= 0.]

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Chapter 3. The Chebotarev density theorem for a number field

We limit ourselves to the standard case where the ground field is a number field. For the higher dimensional case, see §9.3.

3.1. The prime number theorem for a number field 3.1.1. The prime counting function πK(x)

Let K be a number field, i.e. a finite extension of Q, and let OK be the

ring of integers of K. Let VK the set of non-archimedean places of K. The

elements of VK correspond to the maximal ideals of OK; one may identify VK

with Max OK. We denote by pv the maximal ideal corresponding to v ∈ VK,

and by |v| the norm of pv (also called the norm of v), i.e. the number of

elements of the residue field κ(v) = OK/pv. If x is a real number, we put :

πK(x) = number of v ∈ VK with |v|6 x.

When K = Q, VK is the set of prime numbers, and we have

πK(x) = π(x) = number of primes p with p 6 x.

3.1.2. The prime number theorem

The theorem says that πK(x) is asymptotically equal to π(x), i.e. that

πK(x) ∼ x/log x for x → ∞16. More precisely :

Theorem 3.1. There exists c > 0 such that |πK(x)−Li(x)| << x exp(−c

√ log x ) for x_{> 2.}

Recall what the << notation means : let A(x) and B(x) be two complex functions defined on the same set Σ, with B(x) real _{> 0 for all x ∈ Σ. One} writes :

A(x) << B(x) for x ∈ Σ

if there exists a real number C > 0 such that |A(x)|6 C.B(x) for all x ∈ Σ. In most cases, Σ is the set of real numbers x larger than some x0; if x0 is not

specified, then A(x) << B(x) is equivalent to A(x) = O(B(x)).

Recall also that Li(x) is the logarithmic integral of x, i.e.R₂xdt/log t. For every m > 1, one has :

Li(x) = _{log x}x (1 + _{log x}1! + _{(log x)}2! 2 + · · · +

m!

(log x)m + O(

1 (log x)m+1)).

In particular, Li(x) ∼ x/log x for x → ∞. Theorem 3.1 implies : πK(x) = x/log x + x/(log x)2+ O(x/(log x)3) for x → ∞.

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Note that the expression “ logarithmic integral of x ” is sometimes used for the slightly different function li(x) = R₀xdt/log t, where the improper integral R₀x is defined as lim_ε→0(R1−ε

0 +

Rx

1+ε). We have Li(x) = li(x) + li(2) = li(x) + 1.04516... ;

hence most asymptotic statements involving Li(x) remain true for li(x). [Note also that some authors use the notation li(x) for our Li(x), and vice-versa ; there is no universal convention.]

3.1.3. Density

Let P be a subset of ΣK. For every real number x, let πP(x) = number

of v ∈ P with |v|_{6 x. The upper density of P is defined by :} upper-dens(P ) = lim sup πP(x)/πK(x) for x → ∞.

The lower density of P is defined similarly :

lower-dens(P ) = lim inf πP(x)/πK(x) for x → ∞.

One has lower-dens(P )6 upper-dens(P ). When these numbers coincide, they are called the density of P . Hence, P has density λ if and only if :

πP(x) = λx/log x + o(x/log x) for x → ∞.17

When K = Q, one recovers the usual notion of “natural density” of a set of prime numbers.

Exercises.

1) Let P and Q be two subsets of VK. Assume that : a) P has a density that is > 0 ;

b) upper-dens(Q) = 1.

Show that upper-dens(P ∩ Q) = dens(P ) ; in particular, P ∩ Q is infinite.

2) Let P1 = {11, 13, 17, 19, 101, 103, ...} be the set of prime numbers whose first digit in decimal notation is 1.

a) Show that every subset of P1that has a density has density 0. b) Show that upper-dens(P1) = 5/9 and lower-dens(P1) = 1/9. [Hint. Use the prime number theorem, together with the estimate :

Pm=n m=1 10m m = 10n+1 9n + O( 10n n2) for n → ∞.]

3) Let P and Q be two subsets of VK which both have a density. a) Show that

d−(P ∪ Q) + d−_{(P ∩ Q) 6 d(P ) + d(Q) 6 d}+_{(P ∪ Q) + d}+_{(P ∩ Q),} where d, d+ and d− are abbreviations for dens, upper-dens and lower-dens.

If P ∩ Q and P ∪ Q have a density, this implies that d(P )+ d(Q) = d(P ∪ Q)+ d(P ∩ Q). b) Give an example where neither P ∩ Q nor P ∪ Q has a density.

[Hint. Let P1 be the set defined in the previous exercise. Choose for P the set of prime numbers that are ≡ 1 (mod 3). Choose for Q the set of primes that are ≡ 1 (mod 3) if they belong to P1, and that are ≡ −1 (mod 3) if not.]

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3.2. Chebotarev theorem

3.2.1. Decomposition group, inertia group, Frobenius

Let K be as in §3.1, and let E be a finite Galois extension of K. Let G be the Galois group of E/K. The group G acts on the set VE of the

non-archimedean places of E, the quotient being VK.

Let v ∈ VK and choose w ∈ VE lying above v. Let Dw be the

decom-position group of w in G = Gal(E/K), i.e. the subgroup of G fixing w. Let Iw be the inertia subgroup of Dw, i.e. the subgroup made up of the

elements g ∈ Dw that act trivially on the residue field κ(w). We have

Dw/Iw ' Gal(κ(w)/κ(v)). Let σw/v be the canonical generator of Dw/Iw,

i.e. the automorphism x 7→ x|v| of κ(w).

Note that Iw is almost always 1 : it is non-trivial if and only if w is ramified

over v, i.e. if pv divides the discriminant of E/K. When v is unramified, σw/v

can be viewed as an element of Dw; it is the Frobenius element associated

with the pair (w, v)18_{. Its conjugacy class in G only depends on v ; we shall}

denote this class (or any element of it) by σv. When K = Q, we may identify

v with the prime number p = |v|, and we then write σp instead of σv.

3.2.2. Statement of the theorem – qualitative form

Let C be a subset of G stable under inner automorphisms (i.e. a union of conjugacy classes). Let VK,C = {v ∈ VK : v is unramified and σv ∈ C}.

Theorem 3.2. The set VK,C has a density ; that density is equal to |C|/|G|.

Remark. Let Cl G be the set of conjugacy classes of G, and let us put on it the measure µ such that a class C has measure |C|/|G|. We may view Theorem 3.2 as an equidistribution theorem in Cl G with respect to µ. More precisely, let us order the elements v of VK in such a way that v 7→ |v| is increasing19.

Then the Frobenius classes σv are equidistributed in Cl G with respect to µ.

One of the most useful consequences of Theorem 3.2 is : Corollary 3.3. If C 6= ∅, then VK,C is infinite.

Exercise. Refine Corollary 3.3 by proving that VK,C intersects every subset of VK whose upper density is 1.

[Hint. Use Exerc.2 of §3.1.3.]

3.2.3. Statement of the theorem - quantitative form

18. It will later be called the “arithmetic Frobenius”, cf. §4.4 ; its inverse will then be called the “geometric Frobenius”.

19. We follow Bourbaki’s conventions : a function f is said to be increasing (instead of “non-decreasing”) if x_{6 y implies f (x) 6 f (y).}

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If x is a real number _{> 2, let π}C(x) be the number of v ∈ VK,C with

|v| 6 x. Theorem 3.2 can be refined as follows :

Theorem 3.4. (Artin-Chebotarev, cf. [Ar 23], [Ch 25]) There exists a number c > 0 such that

|πC(x) − |C|

|G|Li(x)| = O(x exp(−c

√

log x )) for x → ∞.

If one assumes that the non-trivial zeros of the zeta function of E are on the line Re(s) = 1₂, the right hand side can be replaced by O(x12 log x).

For the history of the theorem, see [LS 96].

For a detailed proof, with an effective (or at least theoretically effective) error term, see [LO 77]. For instance, the number “ c ” of Theorem 3.4 can be taken as c0n

−1₂

E , where c0 is an effectively computable absolute constant,

and nE = [E/Q] ; under GRH, one has

|πC(x) − |C| |G|Li(x)| 6 c1 |C| |G|x 1

2(log |d_E| + n_Elog x) for every x> 2,

where c1 is another effectively computable absolute constant, and dE is the

discriminant of the field E. Such explicit error terms are needed when K is fixed but E varies, see e.g. [Se 81].

3.2.4. Higher moments

We shall later need a variant of Theorem 3.4, where one counts the ele-ments v of VK,C with a weight equal to |v|m for a given exponent m. In order

to state the result, let us call εo(x) the function occurring in the error term,

namely

εo(x) = x exp(−c

p log x ).

Theorem 3.5. For x and m real, define Am(C, x) = P

|v|6x, v∈VK,C|v| m_{. If} m > 0, we have : Am(C, x) = |C| |G|Li(x m+1_{) + O(x}m_ε o(x)) for x → ∞. In particular, if C 6= ∅, then Am(C, x) ∼ |C| |G|. 1 m+1x m+1_{/log x} _{for x → ∞.}

First Proof (analytic number theory style). This follows from Theorem 3.4 by integration by parts.

Second Proof. Write α = |C|_|G| for short. For m = 0, Theorem 3.4 tells us that A0(C, x) = αLi(x) + ε(x) with ε(x) << εo(x). If x > 2 we have

(∗) Am(C, x) = αLi(xm+1) − αLi(2m+1) + xmε(x) −

Rx 2 mt

m−1_ε(t)dt.

Indeed, both sides of this equation have the following properties, which are strong enough to imply that they are equal :

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– they are differentiable at every point x, except possibly when x ∈ N, in which case they are right-continuous and they jump by the same amount, namely xma(x) where a(x) is the number of unramified v with σv ∈ C and

|v| = x ;

– their values at x = 2 are the same, namely 2m_{a(2) ;}

– their derivatives at every x /∈ N are equal to 0 ; this is clear for the left side ; the derivative of the right side is

αxm_{/log x + mx}m−1_{ε(x) + x}m_ε0_{(x) − mx}m−1_{ε(x) = x}m_{(α/log x + ε}0_(x)),

which is 0 because ε0(x) = −α dLi(x)/dx = −α /log x. Formula (*) implies Theorem 3.5, since xm_{ε(x) << x}m_ε

o(x) and : Rx 2 mt m−1_{ε(t)dt <<}Rx 2 t m−1_ε o(t)dt << xmεo(x),

the last inequality being due to the fact that εo(x) is increasing when x is

large enough.

Remark. The proof applies to every εo >> ε which is increasing in a

neighbo-rhood of ∞. In particular, it applies to εo(x) = x

1

2 log x if GRH is true for

the field E.

Variant. It is often convenient to express Theorem 3.5 in terms of class func-tions on G. If f is such a function (with values in C, say), define

Am(f, x) =P_|v|6x|v|mf (σv),

where the sum extends over the v that are unramified in E/K (so that σv is

well defined) and have norm _{6 x. With this notation, Theorem 3.5 can be} reformulated as :

Theorem 3.6. Assume m_{> 0. Then :}

Am(f, x) = <f, 1>G Li(xm+1) + O(xmεo(x)) for x → ∞,

where <f, 1>G = _|G|1

P

g∈Gf (g) is the mean value of f on G.

Indeed, this is the same statement as Theorem 3.5 when f is the cha-racteristic function of a subset C of G that is stable under conjugation ; the general case follows by linearity.

Exercise. With the same notation as in Theorem 3.5, show that A−1(C, x) −|C|_|G|log log x has a limit for x → ∞.

[Hint. Use partial summation.]

3.3. Frobenian functions and frobenian sets 3.3.1. S-frobenian functions and S-frobenian sets

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Let S be a finite subset of VK, and let Ω be a set (with the discrete

topology). Consider a map f : VK S → Ω. We say that f is S-frobenian if

there exists a finite Galois extension E/K, unramified outside S, and a map ϕ : G → Ω, where G = Gal(E/K), such that :

a) ϕ is invariant under conjugation (i.e. ϕ factors through G → Cl G). b) f (v) = ϕ(σv) for all v ∈ VK S.

[Note that ϕ(σv) makes sense because of condition a).]

A subset Σ of VK S is said to be S-frobenian if its characteristic function

is S-frobenian. This means that there exists a Galois extension E/K as above, and a subset C of its Galois group G, stable under conjugation, such that v ∈ Σ ⇐⇒ σv ∈ C ; in that case, Theorem 3.4 shows that Σ has a density,

which is equal to |C|/|G|.

Here is an alternative form of the definition. Let K be an algebraic closure of K and let KS be the maximal subextension ofK that is unramified outside

S. Let ΓS be the Galois group of KS over K ; let Cl ΓS be the set of its

conjugacy classes, with its natural profinite topology (quotient of that of ΓS). We have :

Proposition 3.7.

a) A function f : VK S → Ω is S-frobenian if and only if there exists

a continuous map ϕ : Cl ΓS → Ω such that f is equal to the composition

VK S → Cl ΓS ϕ

→ Ω, where the map on the left is v 7→ σv.

When this is the case, ϕ is unique ; its image is a finite subset of Ω, that is equal to the image of f .

b) A subset P of VK S is S-frobenian if and only if there exists a open

and closed subset U of Cl ΓS such that v ∈ P ⇐⇒ σv ∈ U . When is the

case, U is unique ; it is the closure of the set of σv, for v ∈ P . The set P has

a density that is equal to the Haar measure20 _{of U.}

Proof of a). Since Ω is discrete, the continuity of ϕ means that ϕ is locally constant, i.e. factors through Cl ΓS/N , where N is an open normal subgroup

of ΓS. Hence, the first assertion is just a restatement of the definition. The

uniqueness of ϕ follows from Chebotarev theorem, since the σv’s are dense in

20. We always use the normalized Haar measure, i.e. the Haar measure with total mass 1, see §5.2.1. It is a measure on ΓS; its image by ΓS → Cl ΓS is a measure on Cl ΓS, that we also call the “Haar measure”. If A is a measurable subset of ΓS, that is stable under conjugation, and if cl(A) is its image in Cl ΓS, the Haar measures of A and of cl(A) are the same.

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Cl ΓS; the same argument shows that the images of f and ϕ are the same,

and that they are finite.

The proof of b) is similar ; the assertion about the density is merely a reformulation of Theorem 3.2.

Notation. If f is S-frobenian, the corresponding map ϕ will be denoted by ϕf; we shall view it indifferently as a map of ΓS or of Cl ΓS into Ω. There

exists a maximal open normal subgroup N of ΓS such that ϕf is constant

mod N . Equivalently, there exists a minimal finite Galois extension Ef/K,

unramified outside S, such that ϕf factors through Gal(Ef/K) ; we shall say

that Ef is associated with f .

Remark. Part b) of Proposition 3.7 gives natural bijections between : a) S-frobenian subsets of VK S ;

b) open and closed subsets of ΓS stable under conjugation ;

c) open and closed subsets of Cl ΓS.

These bijections are compatible with taking finite unions, finite intersections, and complements. In particular, the intersection of two S-frobenian sets is S-frobenian.

Proposition 3.8.

i) If an S-frobenian set is non-empty, its density is > 0 ; in particular, it is infinite.

ii) Let P and P0 be two S-frobenian subsets of VK S ; assume that there

are two sets Q and Q0, of density 0, such that P ∪ Q = P0∪ Q0_{. Then P = P}0_.

Proof.

i) If P is S-frobenian and non-empty, the corresponding open and closed subset of ΓS is non empty, hence its Haar measure is > 0. This shows that

dens(P ) > 0.

ii) The set E = P ∪ P0 P ∩ P0 is contained in Q ∪ Q0, hence has density 0. Since E is S-frobenian, this implies E = ∅.

3.3.2. Frobenian sets and frobenian functions

There are cases where one does not want to specify a set S as we did in the previous section. This leads to the following definition :

A subset P of VK is called frobenian if there exists a finite set S such that

P S ∩ P is S-frobenian. Such a set defines an open and closed subset UP of

ΓK = Gal(K/K), closed under conjugation, which one may define either as :

a) the closure of the union of the conjugacy classes of the σv, for v ∈ P S

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b) the inverse image by ΓK → ΓS → Cl ΓS of the set denoted by U in

Proposition 3.7.b).

Two frobenian sets P and P0 are called almost equal if they only differ by a finite set, i.e. if there exists a finite set S such that P P ∩ S = P0 P0∩ S. This is equivalent to UP = UP0. One thus gets a natural bijection between :

a) frobenian subsets of VK, up to the equivalence relation defined by

“almost equality” ;

b) open and closed subsets of ΓK that are stable under conjugation ;

c) open and closed subsets of Cl ΓK.

The properties of S-frobenian sets proved in the previous section imply : Proposition 3.9. i) If a frobenian set is infinite, its density is > 0.

ii) Let P and P0 be two frobenian subsets of VK; assume that there are

two sets Q and Q0, of density 0, such that P ∪ Q = P0∪ Q0_{. Then P and P}0

are almost equal.

Similarly, if S0 is a finite subset of VK, a map f : VK S0 → Ω is called

frobenian if there exists a finite set S containing S0 such that the restriction

of f to VK S is S-frobenian. The fibers of such a map are frobenian subsets

of VK.

Exercises.

1) Let Dens (resp. Frob) be the set of all subsets of VK that have a density (resp. are frobenian). Show that |Frob| = ℵ0 and |Dens| = 2ℵ0. [Hence Frob is much smaller than Dens.]

2) Let Q be a frobenian subset of VK and let α be its density. Show that there exists c > 0 such thatQ v∈Q,|v|6x(1 − 1 |v|) ∼ c (log x)α for x → ∞.

[Hint. Use the exercise in §3.2.4.]

3.3.3. Basic properties of S-frobenian functions Let f : VK S → Ω be an S-frobenian function.

3.3.3.1. If ω is an element of Ω, the set f−1(ω) is S-frobenian, hence is either empty or has a density that is > 0.

If f0 : VK S → Ω is another S-frobenian function, the set of v ∈ VK S

such that f (v) = f0(v) is either empty, or has a density that is > 0 ; this follows from the fact that (f, f0) : VK S → Ω × Ω is S-frobenian.

Similarly, the fibers of a frobenian function are either finite, or of den-sity > 0.

3.3.3.2.Value at 1 and at −1. One can define the value of f at 1 as being ϕf(1), where “ 1 ” means the identity element of ΓS (or of Gal(Ef/K), it

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the image of f , since it belongs to the image of ϕf; hence the set of v ∈ VK S

with f (v) = f (1) is S-frobenian of density > 0.

Similarly, if ι : K → R is a real embedding of K, and cι is the

cor-responding complex conjugation (viewed as an element of Cl ΓS), we can

define the value of f at −1ι as ϕf(cι) ; here again, the set of v ∈ VK S with

f (v) = f (−1ι) is S-frobenian of density > 0. When K = Q, we shall write21

f (−1) instead of f (−1ι).

3.3.3.3. Ψe_{-transforms. Let e be an integer. If ϕ is any function defined}

on a group G, let us denote22 _{by Ψ}e_{ϕ the function g 7→ ϕ(g}e_{). This applies}

in particular to G = ΓS and ϕ = ϕf. Since Ψeϕf is a locally constant class

function on G S, it defines an S-frobenian map of VK S into Ω, which

we shall denote by Ψe_{f . By definition, we have Ψ}e_{f (v) = f (σ}e

v) for every

v ∈ VK S, where σvedenotes the e-th power of the Frobenius element σv. We

have Ψe_{f (1) = f (1), Ψ}e_{f (−1}

ι) = f (1) if e is even, and Ψef (−1ι) = f (−1ι) if

e is odd.

3.3.3.4. Base change. Let K0 be a finite extension of K contained in KS,

hence unramified outside S, and let G0_S = Gal(KS/K0) be the corresponding

subgroup of ΓS.

Let f : VK S → Ω be an S-frobenian map, and let S0 be the inverse

image of S in VK0. There is a unique S0-frobenian map f0 : V_K0 S0 → Ω such

that ϕf0 : G0_S → Ω is the restriction of ϕ_f to G0_S. If v0 ∈ V_K0 S0 has image v

in VK S, and if e is the corresponding residue degree, we have :

f0(v0) = ϕf0(σ_v0) = ϕ_f(σ_ve) = Ψef (v).

Hence f0 can be computed, provided one knows Ψe_{f for e 6 [K}0 _{: K].}

Warning. The map f0 should not be confused with the more natural looking composition VK0 S0 → V_K S → Ω, which is not frobenian in general,

even when [K0 : K] = 2, cf. Exercise 1 below.

3.3.3.5. Mean value. Suppose Ω is a Q-vector space. We may then define 21. These strange notations, which amount to speaking of “the prime 1” and “the prime −1”, lead to very reasonable-looking formulae, as we shall see in §3.4.1.1 and §6.1.2. Note that “ −1 ” was already advertised by Conway in [Co 97 ] as a convenient index for the real place at infinity.

22. The Ψ notation comes from the Adams operations for group representations and K-theory, see §5.1.1 or [Se 78, §9.1, exerc.].

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the mean value mean(f ) of f as that of ϕf, namely mean(f ) = <ϕf, 1> = 1 | G | X g∈G ϕf(g),

where G is any finite quotient of ΓS through which ϕf can be factored. If Ω

is C, or more generally a finite-dimensional vector space over R, Theorem 3.4 implies :

X

|v|6x

f (v) = mean(f ) Li(x) + O(εo(x)) for x → ∞,

where εo(x) is as in §3.2.4, and the summation is restricted to v /∈ S. In

particular, we have mean(f ) = limx→∞ P

|v|6xf (v)

πK(x) .

Similarly, Theorem 3.5 implies : P

|v|6xf (v)|v|m = mean(f ) Li(xm+1) + O(xmεo(x)) for x → ∞,

for every m_{> 0.}

Exercises. We keep the notation of §3.3.3.3 and we denote by π the natural projection VK0 S0 → V_K S.

1) Suppose [K0: K] = 2. Let A be the subset of VK S made up of the v ∈ VK that split completely in K0. Show that A is frobenian, but π−1(A) is not.

[Hint. Observe that π−1(A) has density 1, but that its complement is infinite.]

Use A to construct an S-frobenian function f : VK S → Z/2Z such that f ◦ π is not frobenian.

2) Let E be a Galois extension of K containing K0 and unramified outside S ; put G = Gal(E/K) and G0= Gal(E/K0) so that we have G0⊂ G. Let ψ be a class function on G0 with values in R. Define a function ϕ on G by the formula

ϕ(g) = sup_x∈Gψ(xge(x,g)_x−1_),

where e(x, g) is the smallest integer n > 0 such that xgn_x−1_{∈ G}0_. a) Show that ϕ is a class function on G.

b) Let fϕ: VK S → R and fψ: VK0 S0→ R be the frobenian functions associated

with ϕ and ψ. Show that fϕ(v) = supv0_→vfψ(v0).

c) Let B be an S0-frobenian subset of VK0 S0. Show that A = π(B) is an S-frobenian

subset of VK S.

[Hint. Apply b) to a sufficient large extension of K0 contained in KS, and choose ψ such that fψ is the characteristic function of B. Then fϕis the characteristic function of A.]

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In this section, the ground field K is Q, so that VK is the set P of all

prime numbers.

3.4.1. Dirichlet examples

The following two examples are essentially due to Dirichlet ([Di 39]) : 3.4.1.1. Arithmetic progressions. Let m be an integer > 0 and let S be the set of the prime divisors of m. Let f : P S → (Z/mZ)× be the function p 7→ p mod m. Then f is S-frobenian, the relevant field extension being the m-th cyclotomic field. One has

f (1) = 1 and f (−1) = −1,

where f (1) and f (−1) are the elements of (Z/mZ)× associated with f as explained in §3.3.2.2.

3.4.1.2. Binary quadratic forms. Let B(x, y) = ax2+ bxy + cy2_{be a binary}

quadratic form, with integral coefficients, whose discriminant d = b2_{− 4ac is}

not a square. Let S be the set of the prime divisors of d. Let PB be the set of

primes p /∈ S that can be represented by B, i.e. are of the form p = B(x, y) for some x, y ∈ Z. Then PB is S-frobenian. It is not empty if the obvious

necessary conditions are met : (a, b, c) = 1 and a > 0 if d < 0 ; the density of PB is then 1/h+(d), except if B is ambiguous , i.e. invariant by an element

of GL2(Z) of determinant −1, in which case the density is 1/2h+(d). [Here

h+(d) is the narrow class number, cf. [Co 93, 5.2.7] ; when d is > 0 it may differ by a factor 2 from the usual class number h(d).]

The proofs of these statements rely on the standard dictionary between binary quadratic forms and invertible ideals in quadratic rings (see [Co 93, §5.2]). The relevant Galois extensions are the abelian extensions of Q(√d) known as ring class fields, cf. [Cox 89, §9].

Exercise. Show that the primes represented by 2x2_{+ xy + 9y}2_{have density 1/7.}

3.4.2. The map p 7→ NX(p)

3.4.2.1. The case where dim X/Q 6 0.

Let X be a scheme of finite type over Z. Assume that dimX/Q 6 0, i.e.

that X(Q) is a finite set. The Galois group ΓQ = Gal(Q/Q) acts on X(Q)

via some quotient G = Gal(E/Q), where E is a finite Galois extension of Q. Proposition 3.10.

a) The map f : P 7→ Z defined by f (p) = NX(p) is frobenian.

b) The corresponding map ϕf (cf. §3.3.1) is the map ϕ : G → Z defined by :

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c) Ψe_{f (p) = N}

X(pe) for every large enough p and every e > 1. [For the

definition of Ψef , see §3.3.3.3.]

d) One has f (1) = |X(C)| and f (−1) = |X(R)|.

e) The mean value of f (in the sense of §3.3.3.5) is |X/Q|.

Proof. Let us suppose first that X = Spec OK, where K is a number field.

We then have NX(p) = number of places v of K with |v| = p.

We may choose for E the Galois closure of K ; we have G = Gal(E/Q) ; let us put H = Gal(E/K). We may identify X(Q) with G/H. Let S be the set of prime divisors of disc(K) (or of disc(E) - it amounts to the same). If p /∈ S, let σp ∈ G be its Frobenius element (up to conjugation). We have

NX(p) = ϕ(σp). This shows that the map p 7→ NX(p) is S-frobenian, and that

ϕ is the associated function. This proves a) and b). Assertion c) follows from the easily proved formula NX(pe) = ϕ(σep). Since ϕ(1) = |G/H| = [K/Q] =

|X(C)|, this implies the assertion about f (1) ; a similar method works for f (−1). As for the mean value of f , it is by definition the mean value of the function ϕ, that is equal to 1 by Burnside’s lemma (see e.g. [Se 02, §2.1]). This proves Proposition 3.10 in the case X = Spec OK. The general case

follows by doing the following operations on X :

- making it reduced ; this does not change any NX(pe) ;

- making it normal ; this changes NX(pe) for only finitely many p ;

- decomposing it into irreducible components ; each component is then isomorphic to X0 = Spec OK Σ, where K is a number field and Σ is a closed

finite subset of Spec OK; one then has NX(pe) = NX0(pe) for all large enough

primes p.

Corollary 3.11. If X is a scheme of finite type over Z such that X/Q 6= ∅,

there are infinitely many p with NX(p) > 0.

Proof. Choose a closed point x of X/Q; its closure Xx in X is a subscheme of

X to which one applies part c) of Proposition 3.10. Hence there are infinitely many p with NXx(p) > 0, and a fortiori NX(p) > 0.

Remarks. 1. The hypothesis X/Q 6= ∅ is equivalent to X(Q) 6= ∅ and to

X(C) 6= ∅.

2. By a theorem of Ax and van den Dries ([Ax 67], [Dr 91], see also §7.2.4), the set of p with NX(p) 6= 0 is frobenian ; in particular, it has a density, that

is > 0, as was first proved in [Ax 67].

Example : Number of roots mod p of a one-variable polynomial.

Let us take X = Z[t]/(H), where H is a non-zero element of the polyno-mial ring Z[t]. Let a0tn be the leading term of H and let S be the set of the

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prime divisors of a0disc(H). Then :

The map p 7→ NH(p) is S-frobenian, and its value at 1 (resp. at −1) is the

number of complex (resp. real23) roots of H. For every e _{> 1, the Ψ}e_{-transform of p 7→ N}

H(p) is p 7→ NH(pe).

3.4.2.2. NX(p) mod m. Let X be a scheme of finite type over Z. The map

p 7→ NX(p) is not frobenian (unless dim X(C) 6 0), if only because its image

is infinite, but it is residually frobenian in the following sense :

For every integer m _{> 1, the map f : p 7→ N}X(p) (mod m) is a frobenian

map of P into Z/mZ; its value at 1 (resp. at −1) is equal to the image in Z/mZ of the Euler-Poincaré characteristic χ(X(C)) (resp. χc(X(R))).

These statements will be proved in §6.1.2 ; note that they imply Theorem 1.4 of §1.4.

3.4.3. The p-th coefficient of a modular form

Let us choose a level N , a weight k > 0, a Dirichlet character ε mod N , and a modular form ϕ = P anqn on Γ0(N ) of weight k and type ε, cf. e.g.

[DS 74, §1]. Assume that the coefficients anof ϕ belong to the ring of integers

A of some number field. Then the map p 7→ ap is residually frobenian, in a

similar sense as in §3.4.2.2. More precisely :

Theorem 3.12. For every integer m _{> 1, the map p 7→ a}p (mod mA) is

SmN-frobenian, where SmN is the set of primes dividing mN. Its value at

1 (resp. at −1) is 2a1 (resp. 0) (mod mA).

Corollary 3.13. The set of p with ap ≡ 2a1 (mod m) is SmN-frobenian and

its density is > 0. The same is true for the set of p with ap ≡ 0 (mod m).

The proof of Theorem 3.12 is by reduction to the case where m is a power of a prime `, and ϕ is an eigenfunction of the Hecke operators Tp

for p /∈ S`N ; in that case, one uses the fact that there exists a 2-dimensional

`-adic representation ρ of ΓS`N such that ap = a1.Tr(ρ(σp)) for every p /∈ S`N.

Similarly, if H(N, k) is the ring generated by the Hecke operators Tp for

p not dividing N, the map p 7→ Tp ∈ H(N, k) is residually frobenian, and its

value at 1 (resp. at −1) is 2 (resp. 0). In particular, if ϕ is as above, one has Tpϕ ≡ 0 (mod m) for a set of p that is SmN-frobenian of density δ > 0. As

explained in [Se 76, §4.6], this implies that ϕ is lacunary (mod m), i.e., if we denote by sm(x) the number of n 6 x with an ≡ 0 (mod m), we have

limx→∞sm(x)/x = 1 ; more precisely : x − sm(x) = O(x/(log x)δ).

23. Note the following curious corollary : there are infinitely many p such that H has the same number of roots in Fp as in R.

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Exercise. Let m_{> 1 and ϕ =}P anqn be as above, and let e be an integer> 1. Show that the map p 7→ ape (mod m) is frobenian, that its value at 1 is (e + 1)a₁ and that its value

at −1 is 0 if e is odd and is a1 if e is even.

3.4.4. Examples of non-frobenian sets of primes

The reader should not think that all reasonably defined sets of primes are frobenian. Here is a counterexample :

Choose X such that X/Q is a geometrically irreducible smooth projective

curve of genus g > 0, and let PX be the set of primes p such that NX(p) >

p + 1. It is a consequence of the general Sato-Tate conjecture (cf. §8) that PX

is not frobenian, but that it has a density, which is a strictly positive rational number ₆ 1₂.

When X/Q is an elliptic curve (i.e. g = 1), most of this has been

pro-ved (see the references given in §8.1.5), the density of PX being equal to 1

4 in the CM case and 1

2 in the CM case. In the second case, the

non-frobenian behaviour of PX should be true in the following strong form :

if F is any frobenian set of primes, then F ∩ PX has a density equal to

dens(PX). dens(F ) = 1₂dens(F ) ; loosely speaking, the condition p ∈ PX is

Lectures on Nx(p)

Lectures on N

X

(p)

Jean-Pierre Serre

Preface

Table of Contents

_X