ON THE DISTRIBUTION OF SUPERSINGULAR PRIMES

Etienne FOUVRY (Orsay) and M. Ram MURTY (Montreal)

1. INTRODUCTION

Let E be a fixed elliptic curve over ℚ. Let j_{E} be its j-invariant. A supersingular prime for E is a rational prime p such that E has good reduction at p and End_{\overline{F}_{p}}(E) is a maximal order in a quaternion algebra. Let π_{0}(x) be the number of such primes p ≤ x. If E has complex multiplication, Deuring [De] showed that

π_{0}(x) ∼ \frac{1}{2} \frac{x}{log x}

as x → ∞. If E does not have complex multiplication, then the asymptotic behaviour of π_{0}(x) is at present unknown. Lang and Trotter [L–T] conjecture the existence of a constant C_{E} > 0 such that

π_{0}(x) ∼ \frac{C_{E}\sqrt{x}}{log x}

as x → ∞. The constant C_{E} is defined in terms of representations of the Galois group Gal(ℚ(E_{tor})/ℚ), where ℚ(E_{tor}) is the field obtained by adjoining to ℚ all the torsion points of E. This constant seems to be rather complicated to compute in a general situation.

Elkies[El1] made the first breakthrough in this direction . By an ingenious argument, he proved

π_{0}(x) → ∞

as x → ∞. Elkies and Murty (see [El2] p.21) obtained the lower bound

π_{0}(x) > log_{2}x

for all sufficiently large x, assuming the Riemann Hypothesis for the classical Dirichlet Lfunctions L(s, χ). We denote by log_{k} the k-fold iterated logarithm function. ( Brown[Br1] obtained the weaker estimate π_{0}(x) > log_{3}x assuming the same unproved hypothesis). They also noted that π_{0}(x) = O(x^{\frac{3}{4}} ) follows unconditionally using a result of Kaneko [Ka] ( see [El3] and also [Mu] for a slightly different approach to the lower bound).

Our goal is to prove unconditionally

THEOREM 1. — For any elliptic curve E and for any positive δ, there exists x_{0}(E, δ) such that the inequality

π_{0}(x) ≥ \frac{log_{3} x}{(log_{4} x)^{1+δ}}

holds for x > x_{0}(E, δ).

THEOREM 2. — For any elliptic curve E, we have the equality

π_{0}(x) = Ω(log_{2} x)

Recall that we write f(x) = Ω(g(x)) if there is a constant c > 0, such that the inequality |f(x)| > cg(x) holds for infinitely many x → ∞.

Both of these theorems will follow from the stronger theorem

THEOREM 3. — For any elliptic curve E and any ε > 0, at least one of the two following statements is true

(i) π_{0}(x) = Ω((log x)^{2−ε})

(ii) π_{0}(x) > log_{2} x for x > x_{0}(E).

Instead of considering one fixed curve, one can work with a family of curves and study the behaviour of π_{0}(x) for this family. We thus obtain the average Lang-Trotter Conjecture : let E_{a,b} be the elliptic curve

y^{2} = x^{3} + ax + b

with a, b ∈ ℤ. Denote by π_{0}(x, a, b) the number of supersingular primes of E_{a,b} less than x.

We will prove the following result

THEOREM 4. — Let A ≥ 1, B ≥ 1. Then for every C > 0, we have the equality

∑_{|a|≤A} ∑_{|b|≤B} π_{0}(x, a, b) = \frac{2π}{3} .AB ∫_{2}^{x} \frac{dt}{\sqrt{t} log t} + O((A + B)x^{\frac{3}{2}} + x^{\frac{5}{2}} + AB\sqrt{x}(log x)^{−C})

.

It is easy to see that under the conditions

(1.1)  A > x^{1+ε}; B > x^{1+ε}

we have

(1.2)  ∑_{|a|≤A} ∑_{|b|≤B} π_{0}(x, a, b) ∼_{ε} \frac{4π}{3} .AB.\frac{\sqrt{x}}{log x}

Such a result allows to say that the Lang-Trotter Conjecture is true on average. Nevertheless, we must check that in the sum considered in Theorem 4, the dominant term does not come from the curves E_{a,b} with complex multiplication, which, by Deuring’s Theorem, have many more supersingular primes. It is well known that E has complex multiplication if and only if j_{E} belongs to a set of thirteen values. So, there are thirteen families of CM-curves E_{a,b}, two of them are the families E_{0,b} and E_{a,0} with a and b ∈ ℤ^{∗}. The other eleven families are parametrized by E_{α_{i}t^{2},β_{i}t^{3}}, with t ∈ ℤ^{∗} and (α_{i}, β_{i}) is in an explicit set of eleven pairs of integers. With these remarks, we deduce that

∑_{|a|≤A} ∑_{{|b|≤B \atop E_{a,b} is CM}} π_{0}(x, a, b) = O(\frac{x}{log x}. max(A, B))

which is negligible compared with the main term.

Let

M = {(a, b) ∈ ℤ^{2}; p^{2} | a ⇒ p^{3}∤b}.

The set M (set of minimality) has been introduced to ensure that two different elliptic curves with parameters belonging to M, are never isomorphic over ℚ. We will shortly give the proof of the following theorem, which can be improved in several directions :

THEOREM 5. — Let ε > 0, x, A, B be real numbers satisfying

A, B > x^{1+ε}; AB > x^{2+ε} min(A^{\frac{1}{4}} , B^{\frac{1}{6}} ).

Then, for x → ∞,we have

∑_{|a|≤A} ∑_{{|b|≤B \atop (a,b)∈M}} π_{0}(x, a, b) ∼_{ε} \frac{4π}{3ζ(10)}.AB.\frac{\sqrt{x}}{log x}

A natural question is to weaken the condition (1.1) so that the relation (1.2) continues to be true (the shorter the averaging is, the closer we are to the Lang-Trotter Conjecture itself). Using a particular case of the classical Weil’s bound, for exponential sums (see Lemma 8 below), we will notably improve the condition (1.1), by proving

THEOREM 6. — Let ε > 0, x, A, B be real numbers satisfying

A, B > x^{\frac{1}{2} +ε}; AB > x^{\frac{3}{2} +ε}.

Then, for x → ∞, we have

∑_{|a|≤A} ∑_{|b|≤B} π_{0}(x, a, b) ∼_{ε} \frac{4π}{3} .AB.\frac{\sqrt{x}}{log x}.

In paragraph 8, we develop other types of averagings, which rather depend on algebraic number theory, and in the last paragraph, we discuss the size of least supersingular prime of an elliptic curve.

This work was done when the first author was visiting the Centre de Recherches Mathématiques de Montréal. He thanks the C. R. M. for its hospitality and financial support. The second author thanks the University of Orsay for support during which this work was completed.

2. LEMMAS.

We first recall how to detect supersingular primes(see [El1] pages 561-562). For brevity, we restrict ourselves to odd primes.

Let D ≡ 0 or 3(mod 4) and denote by O_{D} the order

O_{D} = ℤ[\frac{1}{2}(D + \sqrt{−D})]

of discriminant −D. Given an elliptic curve E over ℚ, let p be an odd prime of good reduction for E and E_{p} the reduction of E(mod p). The criterion of Deuring [De] states that p is supersingular if and only if E_{p} has complex multiplication by some O_{D} such that p is ramified or inert in ℚ(\sqrt{−D}). It is well-known that, given D, there exist only finitely many isomorphism classes of elliptic curves over \overline{ℚ} with complex multiplication by O_{D}. Moreover, the j-invariants of these isomorphism classes are conjugate algebraic integers. Let P_{D}(X) be the modular polynomial associated to O_{D}. It is a monic and irreducible polynomial of ℤ[X], the roots of which are the above j-invariants and it makes sense to consider P_{D}(X)(mod p).

By Deuring’s lifting lemma [De,p.259], complex multiplication in characteristic p can be lifted to characteristic zero and so the roots of P_{D}(X) in characteristic p are jinvariants of curves with an endomorphism \frac{1}{2}(D + \sqrt{−D}).

That is, E_{p} has complex multiplication by some O_{D′} for some D′|D with D/D′ a perfect square if and only if P_{D}(X)(mod p) has j_{E} as a root. If, moreover, −D is a quadratic non-residue (mod p) or the highest power of p dividing D is odd, then p is a supersingular prime for E. We summarize this in :

LEMMA 1. (Deuring). — Let p an odd prime of good reduction for E. Then p is a supersingular prime for E if and only if there exists some D ≡ 0 or 3(mod 4) such that D divides the numerator of P_{D}(j_{E}) and (\frac{−D}{p}) = −1 or the highest power of p dividing D is odd.

We continue by recalling the following lemma from Elkies([El1], Proposition) :

LEMMA 2. — Let l be a prime ≡ 3(mod 4). There are polynomials R(X), S(X) ∈ ℤ[X], such that, modulo l, P_{l}(X) and P_{4l}(X) factor into (X −12^{3})R(X)^{2} and (X −12^{3})S(X)^{2}.

Remark. Another proof of Lemma 2 appears in Kaneko[Ka].

It is easily seen that for l prime ≡ 3(mod 4), j(\frac{1}{2}(1 + \sqrt{−l})) and j(\sqrt{−l}) are the only real roots of P_{l} and P_{4l} respectively, the other falling into complex conjugate pairs. From the Fourier expansion of j, we see

j(z) = e^{−2πiz} + O(1)

as Im(z) → ∞, and hence as l → ∞, the real root of P_{l} (P_{4l} resp.) goes to −∞ (goes to +∞ respectively). Thus, for l sufficiently large, we have P_{l}(j_{E})P_{4l}(j_{E}) < 0. See [Mu] for explicit estimates.

LEMMA 3. — Let p_{1}, p_{2}, ...p_{k} a given set of primes. The number of primes l ≡ 7(mod 8) such that (\frac{p_{i}}{l}) = 1 for 1 ≤ i ≤ k and l ≤ x is

≫ \frac{1}{log x}.\frac{x}{(p_{1}p_{2}...p_{k})^{3}}

provided x ≥ (p_{1}p_{2}...p_{k})^{B}, for some absolute constant B > 0.

Proof. This essentially follows from the proof of a classical theorem of Linnik(see [Bo] page 56) : let (a, m) = 1 and denote by π(x; m, a) the number of primes p ≤ x with p ≡ a(mod m). For x ≥ m^{10}, we have

π(x; m, a) ≫ \frac{1}{log x}.\frac{x}{m^{3}} .

In our context, the l’s we seek lie essentially in some arithmetic progression mod 8p_{1}...p_{k}, and so the result follows from Linnik’s Theorem. Note also that any improvement of the exponents B or 3 in the statement of Lemma 3 has almost no influence on Theorems 1, 2 and 3.

LEMMA 4. — Let l_{1} and l_{2} be two distinct primes ≡ 3(mod 4). If p divides the numerator of both P_{l_{1}}(j_{E})P_{4l_{1}}(j_{E}) and P_{l_{2}}(j_{E})P_{4l_{2}}(j_{E}), then p ≤ 4l_{1}l_{2}.

A lemma of that type was firstly proved by Gross and Zagier [G–Z], then generalized by Dorman [Do]. Our Lemma is an easy consequence of Theorem 2 of Kaneko [Ka]. This Theorem has the advantage of giving a result for the prime divisors of the resultant of the polynomials P_{D_{1}}(X) and P_{D_{2}}(X), when D_{1} and D_{2} are distinct discriminants, not necessarily fundamental.

LEMMA 5. — Let h denote the class number of ℚ(\sqrt{−l}). Then, there exists an absolute constant c_{0}, such that the inequality

| P_{l}(j_{E})P_{4l}(j_{E}) |≤ 2^{4h} exp(c_{0}C\sqrt{l} log^{2} l)

is true with C = log(| j_{E} | +745).

Proof. From [Mu] Lemma 5, we have the inequality

| P_{l}(j_{E}) |≤ 2^{2h} exp(C\sqrt{l} ∑ \frac{1}{a_{i}})

where the sum is over the classical set of representatives of h classes of quadratic forms of discriminant −l, which means quadratic forms

a_{i}x^{2} + b_{i}xy + c_{i}y^{2}

satisfying b_{i}^{2} − 4a_{i}c_{i} = −l, (a_{i}, b_{i}, c_{i}) = 1, −a_{i} < b_{i} ≤ a_{i} < c_{i} or 0 ≤ b_{i} ≤ a_{i} = c_{i}. Since the congruence x^{2} + l ≡ 0(mod a_{i}), has at most 2^{ω(a_{i})} solutions (ω :number of prime divisors ), we get the following inequality

∑_{1≤i≤h} \frac{1}{a_{i}} ≤ 2 ∑_{1≤a≤\sqrt{l}} \frac{2^{ω(a)}}{a} ≤ ∏_{p≤\sqrt{l}}(1 + \frac{2}{p − 1}) ≪ (log l)^{2}.

To bound |P_{4l}(j_{E})|, we start from [Mu] Lemma 6 (where the factor 2^{h} has to be replaced by 2^{3h}) and follow the same technique as above. Hence Lemma 5 folows. Therefore this Lemma implies that the numerator of P_{l}(j_{E})P_{4l}(j_{E}) satisfies

Num(P_{l}(j_{E})P_{4l}(j_{E})) ≪ exp(C′(E)\sqrt{l} (log l)^{2})

with C′(E) depending only on E (an easy consequence of the inequalities deg(P_{l}(X)P_{4l}(X)) = O(h) and h = O(\sqrt{l} log l)).

The following lemma (see Jutila [Ju] Lemma 8 ) gives an upper bound on average for a sum of characters at prime arguments. We have

LEMMA 6. — Let

S(D, X) = ∑_{|d|≤D} | ∑_{3≤n≤X} Λ(n)(\frac{d}{n}) |

where d is a non square integer. Then with D_{0} = exp(c_{0}(log X)^{\frac{1}{2}} ), we have, for 3 ≤ D ≤ D_{0}, the inequality

S(D, X) ≪ X(exp(−c_{1}D^{−ε} log X) + exp(−c_{2}(log X)^{\frac{1}{2}} ))

where c_{1} = c_{1}(ε) and c_{2} are positive constants. Also, S(D, X) ≪ XD^{ε}

for D_{0} < D < X^{\frac{1}{4}} , and

S(D, X) ≪ (XD)^{\frac{4}{5} +ε}

for X^{\frac{1}{4}} ≤ D ≤ X^{\frac{49}{50}} .

In particular, for every C > 0, uniformly for 3 ≤ D ≤ X^{\frac{49}{50}} , we have

S(D, X) ≪ XD(log X)^{−C}.

The above estimations of S(D, X) remain valid if the variable of summation n satisfies n ≡ 3(mod 4) (resp. n ≡ 1(mod 4)).

To prove the last part of this lemma, we can, for instance, detect the odd primes n ≡ 3(mod 4) by the function \frac{1}{2}(1 − (\frac{−1}{n})), and then apply the first part of the lemma.

LEMMA 7. — Let p be a prime, α and β two integers. Define M(A, B, α, β, p) =

|{(a, b); a ≡ α (mod p), b ≡ β (mod p), (a, b) ∈ M, |a| ≤ A, |b| ≤ B}|.

Then, for every positive ε, we have the equality

M(A, B, α, β, p) = \frac{4AB}{p^{2}.ζ(10)}.(1 + O(p^{−1})).(1 + O((log AB)^{−4}))+

O(\frac{A + B}{p} + (AB)^{ε} + min(A^{\frac{1}{4}} , B^{\frac{1}{6}} )).

Note that this lemma is interesting only when A and B are large enough compared with p, for instance

A, B > p^{1+ε}; \frac{AB}{p^{2+ε}} > min(A^{\frac{1}{4}} , B^{\frac{1}{6}} ).

Proof. We start from the formula M(A, B, α, β, p) =

∑_{d} μ(d)|{(a, b); ab ≠ 0, a ≡ α (mod p), b ≡ β (mod p), d^{4}|a, d^{6}|b, |a| ≤ A, |b| ≤ B}|

+O(\frac{A + B}{p} + 1).

Note that in the above formula, we may suppose that d satisfies

d ≤ min(A^{\frac{1}{4}} , B^{\frac{1}{6}} )

When d has all its prime factor less than \sqrt{log AB}, we use the formula

(2.1)  |{(a, b); a ≡ α (mod p), b ≡ β (mod p), d^{4}|a, d^{6}|b, |a| ≤ A, |b| ≤ B}| =

(\frac{2A}{φ(α, p, d^{4})} + O(1)).(\frac{2B}{φ(β, p, d^{6})} + O(1))

where φ(α, p, d^{4}) is equal to pd^{4} if p∤d, d^{4} if p|d and α = 0(mod p) and φ(β, p, d^{6}) defined similarly. When d has a prime factor greater than \sqrt{log AB}(so d is greater than this bound) we use, for the cardinality studied in (2.1), the trivial bound

≤ (\frac{2A}{[p, d^{4}]} + 1)(\frac{2B}{[p, d^{6}]} + 1),

where [m, n] is the least common multiple of m and n.

In the case p∤αβ, by a classical computation, we have M(A, B, α, β, p) =

\frac{4AB}{p^{2}} ∏_{{q prime ≠p \atop q<\sqrt{log AB}}} (1 − \frac{1}{q^{10}} )

+O((AB)^{ε} + (\frac{A + B}{p}) + \frac{AB}{p^{2}} (log AB)^{− \frac{9}{2}} + min(A^{\frac{1}{4}} , B^{\frac{1}{6}} ))

which gives lemma 7 in that case. In the other cases (p|αβ), the computation leads, in the above formula, to a slightly different product over q which is nevertheless also of the form ζ(10)^{−1}.(1 + O(p^{−1})).(1 + O((log AB)^{−4})). This completes the proof of the lemma.

The following lemma is a particular case of [Sc] Corollary 2F :

LEMMA 8. — Let u and v be integers such that at least one of them is not divisible by the prime p. Then, we have the inequality

| ∑_{n=0}^{p−1} e(\frac{un^{4} + vn^{6}}{p}) |≤ 5\sqrt{p}

3. Proof of Theorem 3.

Let p_{1}, ..., p_{k} be the first k supersingular primes for E. By Lemma 3, we can find

≫_{E} \frac{1}{log x}.\frac{x}{(p_{1}...p_{k})^{3}}

primes l ≡ 7 (mod 8) such that l ≤ x, (\frac{p_{i}}{l}) = 1 for 1 ≤ i ≤ k, and (\frac{q_{i}}{l}) = 1 for 1 ≤ i ≤ t(where the q_{i}’s are the primes where E has bad reduction) provided x ≥ (p_{1}...p_{k})^{B} and B a sufficiently large constant, depending only on E. Note that these congruence conditions are compatible when 2 is one of the p_{i}’s or q_{i}’s. For such l, we have by Lemma 2 and by the fact that 2 divides deg(P_{l}(X)P_{4l}(X)), the equality

(\frac{Num(P_{l}(j_{E})P_{4l}(j_{E}))}{l}) = 1.

On the other hand , by the remark following Lemma 2, Num(P_{l}(j_{E})P_{4l}(j_{E})) is a negative rational integer = −N_{l} (say). Then, we have

(\frac{N_{l}}{l}) = −1.

By the choice of l in the lemma, not all the primes dividing N_{l} can be among 2, p_{1}, ..., p_{k}, q_{1}, ..., q_{t}. Hence, by Lemma 1, there is a prime p_{k+1}(l) which is a new (odd) supersingular prime for E.

Thus, to each l, we can associate a supersingular prime p_{k+1}(l). If all these are distinct, then we have

π_{0}(T) ≫_{E} \frac{1}{log x} \frac{x}{(p_{1}...p_{k})^{3}}

where

T = max_{l≤x} N_{l}.

By the remark following Lemma 5, we have

T ≪ exp(C′(E)\sqrt{x} (log x)^{2})

Choosing x = (p_{1}...p_{k})^{A}, for a sufficiently large A depending on E, we find the lower bound

(3.1)  π_{0}(y) ≫ (log y)^{2−ε}

for y = exp(A(p_{1}...p_{k})^{\frac{A}{2}} (log(p_{1}...p_{k}))^{2}).

If for infinitely many k, the p_{k+1}(l), constructed as above, are all distinct, then (3.1) is established for infinitely many y → ∞, hence the first part of Theorem 3.

Suppose now that it is not the case. Then, for all k sufficiently large, there are l_{1} and l_{2}, such that Num(P_{l_{1}}(j_{E})P_{4l_{1}}(j_{E})) and Num(P_{l_{2}}(j_{E})P_{4l_{2}}(j_{E})) have, at least, a common prime factor called p_{k+1}. By Lemma 4, this new supersingular prime satisfies

p_{k+1} ≤ 4l_{1}l_{2}

and by Lemma 3, we find

p_{k+1} ≤ 4(p_{1}...p_{k})^{2A}.

This recursive inequality yields

π_{0}(x) ≫ log_{2} x

for all sufficiently large x. This completes the proof of Theorem 3. Theorem 2 is an immediate corollary.

4. Proof of Theorem 1.

Actually, the proof of Theorem 3 investigates two opposite situations : in the first case a construction of a lot of new supersingular primes, but apparently very far from the old ones, and in the second case, the construction of only one supersingular prime but rather close to the old ones. In some sense, Theorem 1 covers both these cases.

Let δ > 0, then Theorem 2 implies that there exists an arbitrarily large real number x_{0}, such that

π_{0}(x_{0}) ≥ \frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}}

so, maybe by forgetting some supersingular primes less than x_{0}, we assert that there exist p_{1}, ..., p_{k}, supersingular primes less than x_{0}, with

\frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} ≤ k < \frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} + 1.

We follow now the proof of Theorem 3, and define x_{1} = y in the first case, or x_{1} = 4(p_{1}...p_{k})^{2A} in the second one. In both cases, it is easy to see, using the inequality

log(p_{1}...p_{k}) < (\frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} + 1) log x_{0}

that x_{1} satisfies the inequality

log x_{1} < (log x_{0})(log_{2} x_{0}).

In the first case, we have trivially

(4.1)  π_{0}(x_{1}) ≥ \frac{log_{2} x_{1}}{(log_{3} x_{1})^{1+δ}}

and, in the second one, by construction of p_{k+1}, we know that

π_{0}(x_{1}) ≥ π_{0}(x_{0}) + 1 ≥ \frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} + 1,

but, in that case, since

x_{1} ≤ exp(3A. log x_{0}.(\frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} + 1)),

we have

\frac{log_{2} x_{1}}{(log_{3} x_{1})^{1+δ}} ≤ \frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} + O((log_{3} x_{0})^{−δ}) ≤ \frac{log_{2} x_{0}}{(log_{3} x_{0})^{1+δ}} + 1 ≤ π_{0}(x_{1}).

So, in both cases, (4.1) is valid. By replacing x_{0} by x_{1} and so on, we construct an infinite sequence (x_{n})_{n≥0} tending to infinity, such that :

(4.2)  π_{0}(x_{n}) ≥ \frac{log_{2} x_{n}}{(log_{3} x_{n})^{1+δ}} ,

and

(4.3)  log_{2} x_{n} ≤ (log x_{n−1})(log_{2} x_{n−1}).

Now, if x is any large real number, it satisfies, for some n, the inequality

x_{n} ≤ x < x_{n+1}

from which we deduce, by (4.2),

π_{0}(x) ≥ π_{0}(x_{n}) ≥ \frac{log_{2} x_{n}}{(log_{3} x_{n})^{1+δ}} .

we have also, by (4.3),

\frac{log_{3} x}{(log_{4} x)^{1+2δ}} ≤ \frac{log_{3} x_{n+1}}{(log_{4} x_{n+1})^{1+2δ}} ≤ \frac{log_{2} x_{n}}{(log_{3} x_{n})^{1+2δ}} (1 + o(1)) ≤ \frac{log_{2} x_{n}}{(log_{3} x_{n})^{1+δ}} ≤ π_{0}(x).

Hence the end of the proof of Theorem 1 with δ replaced by 2δ.

5. Lang-Trotter conjecture on average.

It is well known that the total number of equivalence classes of elliptic curves over 𝔽_{p} with p + 1 points is equal to the total number of classes of ideals of O_{4p}. The latter quantity is the Kronecker class number H(−4p) (see, for instance[Bi], pages 58-59). Now an elliptic curve over 𝔽_{p} can be written as (for p ≠ 2, 3), E_{a,b} with a, b ∈ 𝔽_{p}. The curves isomorphic to E_{a,b} are the curves E_{au^{4},bu^{6}}, with u ∈ 𝔽_{p}^{∗}. So,the number of curves isomorphic to the elliptic curve (over 𝔽_{p}) E_{a,b} is

\frac{p−1}{6} for a = 0, b ≢ 0 (mod p) and p ≡ 1 (mod 6)

\frac{p−1}{4} for b = 0, a ≢ 0 (mod p) and p ≡ 1 (mod 4)

\frac{p−1}{2} in the remaining cases.

Since there are O(1) isomorphism classes over 𝔽_{p} containing a curve of the form E_{0,b} or E_{a,0}, we deduce that the number of elliptic curves E_{a,b} with 0 ≤ a, b < p having p + 1 points over 𝔽_{p} is equal to

\frac{p}{2}.H(−4p) + O(p)

It remains to study the curves E_{a,b} with |a| ≤ A and |b| ≤ B, to write, under the assumptions of Theorem 4,

∑_{|a|≤A} ∑_{|b|≤B} π_{0}(x; a, b) = \frac{1}{2} ∑_{p≤x}(\frac{2A}{p} + O(1)).(\frac{2B}{p} + O(1)).p.(H(−4p) + O(1)) + O(AB)

(5.1)  = 2AB ∑_{p≤x} \frac{H(−4p)}{p} + O((A + B)\frac{x^{\frac{3}{2}}}{log x} + AB log log x + \frac{x^{\frac{5}{2}}}{log x})

where the error O(AB) comes from the primes 2 and 3 which are supersingular for some E_{a,b} and from the p which are supersingular for curves with non-minimal equation E_{a′p^{4},b′p^{6}}.

The equality

H(−4p) = h(−4p) + h(−p)

and the Dirichlet class number formula

h(−d) = \frac{w\sqrt{d}}{2π}L(1, χ_{−d})

for d ≡ 0 or 3 (mod 4), w = 6, 4, or 2 when d = 3, 4 or d ≥ 7, and χ_{−d} the Kronecker symbol (\frac{−d}{.}) transform the study of (5.1) into a sum of Dirichlet series at the point s = 1. The right hand-side of (5.1) becomes

(5.2) \frac{2AB}{π} ( ∑_{{p≤x \atop p≡3(mod 4)}} \frac{L(1, χ_{−p})}{\sqrt{p}} + 2 ∑_{p≤x} \frac{L(1, χ_{−4p})}{\sqrt{p}}) + O((A + B)x^{\frac{3}{2}} + AB log x + x^{\frac{5}{2}} ).

By partial summation and Polya-Vinogradov inequality, we have, for any parameter U > 1, the equality

(5.3)  L(1, χ_{−p}) = ∑_{n≤U} \frac{χ_{−p}(n)}{n} + O(\frac{\sqrt{p} log p}{U})

and the same equality for −4p.

We choose

U = x^{\frac{3}{4}} .

To obtain cancellations on the summation over p, we introduce the Legendre symbol, so we recall the formula

If p ≡ 3(mod 4) then χ_{−p}(n) = (\frac{n}{p})

and if p ≥ 3 then

(5.4)  χ_{−4p}(n) = (\frac{2}{n})^{2}(−1)^{\frac{p−1}{2} . \frac{n−1}{2}} (\frac{n}{p})

the right hand side of this formula being understood to be 0 when n is even. We will mainly concentrate on the sum

S^{(1)}(U, X) = ∑_{n≤U} \frac{1}{n} ∑_{{p≤x \atop p≡3(mod 4)}} \frac{(\frac{n}{p})}{\sqrt{p}}.

If n is a perfect square, the inner sum is equal to

\frac{1}{2} ∫_{2}^{x} \frac{dt}{\sqrt{t} log t} + O(\sqrt{x} exp(−\sqrt{log x}))

by the prime number theorem and partial summation. This gives rise to the following main term for S^{(1)}(U, X) :

(5.5)  \frac{π^{2}}{12} . ∫_{2}^{x} \frac{dt}{\sqrt{t} log t} + O(\sqrt{x}(U^{− \frac{1}{2}} + exp(−\sqrt{log x})))

We now estimate the sum when n is not a perfect square. If x_{1} = x exp(−c\sqrt{log x}), for an appropriate constant c, we begin by noting, that we trivially have

(5.6)  ∑_{n≤U}′ \frac{1}{n} ∑_{p≤x_{1}}∗ \frac{(\frac{n}{p})}{\sqrt{p}} ≪ \sqrt{x}(log U) exp(−\frac{c}{2}\sqrt{log x})

where the prime on the summation indicates (henceforth) that n is not a square and the star that p ≡ 3(mod 4).

It therefore remains to estimate

∑_{n≤U}′ \frac{1}{n} ∑_{x_{1}<p≤x} ∗ \frac{(\frac{n}{p})}{\sqrt{p}}.

Using dyadic decomposition, (that is, decomposing the sum into intervals of the form (V, 2V)), we see that the above sum is

(5.7)  ≪ (log x) | S^{(1)}(V) |

for some V satisfying 3 ≤ V ≤ U and

S^{(1)}(V) = ∑_{V ≤n<2V}′ \frac{1}{n} ∑_{x_{1}<p≤x} ∗ \frac{(\frac{n}{p})}{\sqrt{p}}.

By partial integration, we get

V S^{(1)}(V) ≪ \frac{1}{\sqrt{x} log x}. ∑_{V ≤n<2V}′ | ∑_{x_{1}<p≤x} ∗ log p (\frac{n}{p}) | + ∫_{x_{1}}^{x} ψ(t) ∑_{V ≤n<2V}′ | ∑_{x_{1}<p≤t} ∗ log p (\frac{n}{p}) | dt

where ψ(t) is the derivative of the function \frac{1}{\sqrt{t} log t}. A direct application of Lemma 6 gives the bound

(5.8)  S^{(1)}(V) ≪ \frac{\sqrt{x}}{(log x)^{C}}

for every positive C and every V ≤ U.

It remains to apply the same technique for the sum in (5.2) containing L(1, χ_{−4p}), using now the formula (5.4). The main term coming from this sum is

(5.9) 2 ∑_{{n≤U \atop n is an odd square}} \frac{1}{n} ∑_{p≤x} \frac{1}{\sqrt{p}} = \frac{π^{2}}{4} . ∫_{2}^{x} \frac{dt}{\sqrt{t} log t} + O(\sqrt{x}(U^{− \frac{1}{2}} + exp(−\sqrt{log x}))).

Gathering the formulas (5.1), (5.2), (5.3), (5.5), (5,6), (5.7), (5.8)and (5.9), we complete the proof of Theorem 4.

6. Sketch of the proof of Theorem 5.

Let p a prime less than x, and let A, B and x satisfying the assumptions of Theorem 5. Then the quantity M(A, B, α, β, p) studied in lemma 7, satisfies

(6.1)  M(A, B, α, β, p) = \frac{4AB}{p^{2}ζ(10)}(1 + O(p^{−1}) + O((log x)^{−4})),

now, following the beginning of the proof of Theorem 4, we have the equality

∑_{|a|≤A} ∑_{{|b|≤B \atop (a,b)∈M}} π_{0}(x, a, b) = ∑_{p≤x} ∑_{α} ∑_{β} M(A, B, α, β, p)

where the inner sum is made over 0 ≤ α, β < p, E_{α,β} is an elliptic curve (mod p) with exactly p + 1 points.

By (6.1), this last quantity is equal to

\frac{2AB}{ζ(10)} ∑_{p≤x} \frac{H(−4p)}{p}(1 + O(p^{−1}) + O((log x)^{−4})) + O(AB log x).

The proof now follows the evaluation of (5.1).

7. Use of exponential sums. Proof of Theorem 6.

Let p be fixed. In each of the H(−4p) equivalence classes of elliptic curves over 𝔽_{p} , with p + 1 elements, we choose a curve

E_{α,β} : y^{2} = x^{3} + αx + β

with 0 ≤ α, β < p. There are H(−4p) − O(1) of these classes with E_{α,β} such that αβ ≠ 0.

To prove Theorem 6, we may suppose that A and B are integers plus \frac{1}{2}. We dissect the interval [−A, A] into subintervals of length p, the first one being [−A, −A + p], the last one, if not complete is denoted by A. The same procedure is applied to [−B, B], the last interval is called B. With these notations and the notation [x] for the integer part of x, we can write under the assumption αβ ≠ 0, the equality

(7.1)  | {(a, b); |a| ≤ A, |b| ≤ B, E_{a,b}(mod p) is isomorphic to E_{α,β}} |

= [\frac{2A}{p}].[\frac{2B}{p}].\frac{p − 1}{2} + \frac{1}{2}.[\frac{2A}{p}] | \{u ∈ 𝔽_{p}^{∗}, βu^{6} ∈ B(mod p)\} |

+\frac{1}{2}.[\frac{2B}{p}] | \{u ∈ 𝔽_{p}^{∗}, αu^{4} ∈ A(mod p)\} |

+\frac{1}{2} | \{u ∈ 𝔽_{p}^{∗}, αu^{4} ∈ A(mod p), βu^{6} ∈ B(mod p)\} | +O(min(A/p^{4}, B/p^{6}))

where the error term comes from the curves E_{αu^{4}p^{4k},βu^{6}p^{6k}}, (k ≥ 1). We write the characteristic function of A(mod p)as

\frac{1}{p} ∑_{h=0}^{p−1} ∑_{a∈A} e(\frac{h(t − a)}{p})

and the last term in (7.1) becomes

\frac{1}{2p^{2}} ∑_{h=0}^{p−1} ∑_{a∈A} ∑_{l=0}^{p−1} ∑_{b∈B} ∑_{u=1}^{p−1} e(\frac{h(αu^{4} − a) + l(βu^{6} − b)}{p}).

Since A and B are intervals, we deduce that this term is equal to

\frac{|A||B|}{2}.\frac{p − 1}{p^{2}}

+O(\frac{|A|}{p^{2}} ∑_{l=1}^{p−1} ||\frac{l}{p}||^{−1} | ∑_{u=1}^{p−1} e(\frac{lβu^{6}}{p}) |)

+O(\frac{|B|}{p^{2}} ∑_{h=1}^{p−1} ||\frac{h}{p}||^{−1} | ∑_{u=1}^{p−1} e(\frac{hαu^{4}}{p}) |)

+O(\frac{1}{p^{2}} ∑_{l=1}^{p−1} ∑_{h=1}^{p−1} ||\frac{l}{p}||^{−1}||\frac{h}{p}||^{−1} | ∑_{u=1}^{p−1} e(\frac{hαu^{4} + lβu^{6}}{p}) |)

where ||x|| is the distance between x and the nearest integer.

An application of Lemma 7 says that this quantity is equal to

\frac{|A||B|}{2}.\frac{p − 1}{p^{2}} + O(\frac{|A| log p}{\sqrt{p}} + \frac{|B| log p}{\sqrt{p}} + \sqrt{p} log^{2} p),

and applying a similar technique to the second and the third term of (7.1), we get

| {(a, b); |a| ≤ A, |b| ≤ B, E_{a,b}(mod p) is isomorphic to E_{α,β}} |

= \frac{p − 1}{2}([\frac{2A}{p}].[\frac{2B}{p}] + [\frac{2A}{p}].\frac{|B|}{p} + [\frac{2B}{p}].\frac{|A|}{p} + \frac{|A||B|}{p^{2}})

+O((\sqrt{p} log^{2} p)(1 + \frac{A}{p} + \frac{B}{p})).

Using the equality [\frac{2A}{p}] + \frac{A}{p} = \frac{2A}{p}, we get

= \frac{p − 1}{2}.\frac{4AB}{p^{2}} + O((\sqrt{p} log^{2} p)(1 + \frac{A}{p} + \frac{B}{p})).

Now, summing over the H(−4p) − O(1) classes of isomorphism with αβ ≠ 0(mod p), we arrive at the formula

∑_{|a|≤A} ∑_{|b|≤B} π_{0}(x, a, b) = 2AB ∑_{p≤x} \frac{H(−4p)}{p} + O(x^{2} log^{3} x + AB log x + (A + B)x log^{3} x)

where the error term O(AB log x) comes from the curves E_{a,b} with p|ab.

The proof follows the proof of Theorem 4.

8. Other types of averagings.

In Theorems 4, 5, and 6, we considered a very large family of curves in two parameters a and b. The aim of this paragraph is to present examples of studies of the function π_{0}(x, a, b) over a thinner family of elliptic curves which is parametrized in one variable only. Some of these results will depend on GRH (i.e. the hypothesis that the zeta-function of any number field has no zero with a real part greater than \frac{1}{2}) but all of them are based on the fact that the study of the number of zeroes of P_{D}(X)(mod p), denoted by ν(D, p),on average, via Chebotarev Theorem, requires the introduction of H_{D}, the Hilbert class field of the field ℚ(\sqrt{−D}).

We will prove

THEOREM 7. — Let a_{0} and b_{0} be two non-zero integers . If we suppose that GRH is satisfied for all the ζ_{H_{D}}, then there exist three constants c_{1}^{+}, c_{2}^{+} and c_{2}^{−} > 0,such that we have

(8.1)  ∑_{|b|≤B} π_{0}(x, a_{0}, b) ≤ c_{1}^{+}B \frac{\sqrt{x}}{log x}

uniformly for B ≥ x ≥ x_{0}(a_{0})

(8.2)  c_{2}^{−}A\frac{\sqrt{x}}{log x} ≤ ∑_{|a|≤A} π_{0}(x, a, b_{0}) ≤ c_{2}^{+}A\frac{\sqrt{x}}{log x}

uniformly for A ≥ x ≥ x_{0}(b_{0}).

In the case where no particular hypothesis is assumed, there exists a θ > 0, such that the following lower bound holds

(8.3)  ∑_{|a|≤A} π_{0}(x, a, b_{0}) ≥ c_{1}^{−}A(log x)^{θ}

uniformly for A ≥ x ≥ x_{0}(b_{0}).

Note that (8.1) and (8.2) give for π_{0}(x, a, b) average upper and lower bounds which are compatible with the Lang-Trotter Conjecture and that (8.3) produces a much better bound than Theorem 1 but always on average ( compare also with Theorem 3i).

Since P_{D}(X) is an irreducible polynomial, Nagell’s Theorem asserts that on average over p, ν(D, p) behaves like 1. But we will work with an unbounded number of polynomials P_{D}(X), so we require a rather large uniformity over D. By classical techniques from analytic and algebraic number theory, we will prove

LEMMA 9. — Under GRH, uniformly for x ≥ 2 and D ≡ 0, 3(mod 4), we have the equalities

∑_{p≤x} ν(D, p) = li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx),

∑_{p≤x, ( \frac{−D}{p})=−1} ν(D, p) = \frac{1}{2}li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx)

and

∑_{{p≤x, ( \frac{−D}{p} )=−1 \atop p≡2(mod 3)}} ν(D, p) = \frac{1}{4}li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx) (for D ≠ −3m^{2}).

All these equalities remain true if ν(D, p) is replaced by ν^{∗}(D, p), the number of distinct roots of P_{D}(X)(mod p).

Note that the above equalities give an asymptotic formula for D ≤ x^{1−ε} which is quite satisfactory, since we will use Lemma 9 for D ≪ \sqrt{x}.

The proof of this lemma requires two lemmas :

LEMMA A. — Let f(X) ∈ ℤ[X] be an irreducible polynomial of degree n. Let n(f, p) be the number of solutions of f(x) ≡ 0(mod p). Assuming GRH, we have

∑_{p≤x} n(f, p) = li x + O(x^{\frac{1}{2}} n(δ(f) + log x))

where

δ(f) = ∑_{p|D(f)} log p + log n

and D(f) is the discriminant of f.

Proof Let K = ℚ(θ) with f(θ) = 0. If d_{K} is the discriminant of K, then the standard methods of analytic number theory yield that the number of prime ideals of degree 1 in K is

π_{K}(x) = li x + O(x^{\frac{1}{2}} log d_{K}x^{n}).

(see e.g. [Ho] p. 55–56, [Dave] , [He]).

The result stated in Lemma A follows from two observations. First, n(f, p) is the number of prime ideals of degree 1 in K, lying over p, provided p∤D(f). Thus

∑_{p≤x} n(f, p) = π_{K}(x) + O(n ∑_{p|D(f)} 1)

( see [He] p.229 for instance). Second, by a result of Hensel (see [Se] Proposition 6, for instance), we write

log d_{K} ≤ n ∑_{p|D(f)} log p + n log n

which completes the proof.

LEMMA B. — If K is a number field, f(X) ∈ O[X] is irreducible over K, of degree n and n(f, 𝔓) denotes the number of solutions of f(X) ≡ 0(mod 𝔓), then assuming GRH, we have

∑_{N𝔓≤x} n(f, 𝔓) = li x + O(x^{\frac{1}{2}} [K : ℚ]n(δ(f) + log x))

where

δ(f) = ∑_{𝔓|D(f)} log N𝔓 + log n.

In that expression N𝔓 is the absolute norm of the prime ideal 𝔓 and the implied constant is absolute.

Proof. The number of prime ideals of K of relative degree from K to ℚ greater or equal to 2 and of absolute norm less than x is

O([K : ℚ]x^{\frac{1}{2}} ).

The number of prime ideals of K(θ) lying over a given prime is at most n. Thus the result follows from Lemma A.

Proof of Lemma 9. The first part is immediate from Lemma A, since only prime divisors of D divide δ(P_{D}).

If we take K = ℚ(\sqrt{−D}) in lemma B, we obtain

∑_{N𝔓≤x} n(P_{D}, 𝔓) = li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx)).

Thus

(∗)  ∑_{p≤x, ( \frac{−D}{p})=1} ν(D, p) = \frac{1}{2}li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx).

Subtracting from the first assertion gives the result.

Finally, if K = ℚ(\sqrt{−3}), we note that P_{D}(X) is still ireducible over ℚ(\sqrt{−3}, \sqrt{−D}) ( this is because ramification implies that the fields are disjoint when D ≠ −3m^{2}). Hence working over ℚ(\sqrt{−3}, \sqrt{−D}), we find

(∗∗)  ∑_{{p≤x, ( \frac{−D}{p} )=1 \atop p≡1(mod 3)}} ν(D, p) = \frac{1}{4}li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx).

Subtracting (**) from (*) yields

(∗ ∗ ∗).  ∑_{{p≤x, ( \frac{−D}{p} )=1 \atop p≡2(mod 3)}} ν(D, p) = \frac{1}{4}li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx)

Now consider

∑_{{p≤x \atop p≡1(mod 3)}} ν(D, p).

This can be interpreted as follows :

Consider K = ℚ(\sqrt{−3}) and look at the number of solutions over this field of P_{D}(X) ≡ 0(mod𝔓). By lemma B, this is

2 ∑_{{p≤x \atop p≡1(mod 3)}} ν(D, p) = li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx).

Thus, we have, after subtracting from the first part of Lemma 9, the equality

∑_{{p≤x \atop p≡2(mod 3)}} ν(D, p) = \frac{1}{2}li x + O(x^{\frac{1}{2}} D^{\frac{1}{2}} log Dx).

To complete the proof, it remains to subtract from (***) the above expression.

The following result of Kaneko gives a bound for the least D such that Deuring criterion is satisfied (Lemma 1). We have

LEMMA 10([Ka], Theorem 1). — Let E an elliptic curve over ℚ and p an odd supersingular prime for E. Then there exists a D ≤ \frac{4}{\sqrt{3}} \sqrt{p} such p|Num(P_{D}(j_{E})), (\frac{−D}{p}) = −1 or the highest power of p dividing D is odd.

a. Proof of (8.1).

To control the size of the supersingular primes we are counting, it is sufficient to prove for X ≤ x the upper bound

(8.4)  B(X, B) = ∑_{|b|≤B}(π_{0}(X, a_{0}, b) − π_{0}(\frac{X}{2} , a_{0}, b)) = O(B \frac{\sqrt{X}}{log X})

Since for X large enough, any D ≪ \sqrt{X} has no prime divisor greater than \frac{X}{2} , Lemma 10 implies that B(X, B) satisfies the inequality

B(X, B) ≤ ∑_{D≪\sqrt{X}} ∑_{|b|≤B} ∑_{{\frac{X}{2} <p≤X \atop p|P_{D} (j_{E_{a_{0},b}})}} 1

(8.5)  ≪ ∑_{D≪\sqrt{X}} ∑_{\frac{X}{2} <p≤X}(  ∑_{{α(mod p) \atop P_{D} (α)≡0(mod p)}} ∑_{b(mod p)} 1)\frac{B}{p}

where the variable of summation satisfies the equation

1728\frac{4a_{0}^{3}}{4a_{0}^{3} + 27b^{2}} ≡ α(mod p).

This quadratic equation in b has at most O(1) solutions (more precisely, at most 2, if p∤3a_{0}). Hence, by (8.5), we find the relation

B(X, B) ≪ B ∑_{D≪\sqrt{X}} ∑_{\frac{X}{2} <p≤X} \frac{ν(D, p)}{p}.

Lemma 9 is applied under the form

∑_{\frac{X}{2} <p≤X} ν(D, p) = O(\frac{X}{log X})

which gives (8.4) and this proves (8.1). b. Proof of (8.2).

The proof of the upper bound works like in the case of (8.1), except we meet the cubic equation

1728\frac{4a^{3}}{4a^{3} + 27b_{0}^{2}} ≡ β(mod p).

which has at most three roots if p is large enough. For the lower bound we use the inequality

A(x, A) = ∑_{|a|≤A}(π_{0}(x, a, b_{0}) − π_{0}(\frac{x}{2}, a, b_{0}))

(8.6)  ≥ ∑_{\frac{X}{2} <D≤X} ∑_{{\frac{x}{2} <p≤x;( \frac{−D}{p} )=−1 \atop p∤P_{D} (1728)}}(  ∑_{{β(mod p) \atop P_{D} (β)=0(mod p)}} ∑_{a(mod p)} 1)\frac{A}{p}

with X = \frac{\sqrt{x}}{10}, where the variable of summation a satisfies

(8.7)  1728\frac{4a^{3}}{4a^{3} + 27b_{0}^{2}} ≡ β(mod p); 4a^{3} + 27b_{0}^{2} ≢ 0(mod p).

The value of X has been choosen to ensure that Lemma 1 produces different supersingular primes by Lemma 4. The relations (8.7) are equivalent to the unique equation

(8.8)  4(1728 − β)a^{3} − 27βb_{0}^{2} ≡ 0(mod p)

for p large enough. If we impose the conditions p ≡ 2(mod 3) and β ≢ 1728(mod p), (8.8) has at least one root ; we deduce the lower bound

(8.9)  A(x, A) ≥ A ∑_{\frac{X}{2} <D≤X} ∑_{{\frac{x}{2} <p≤x;( \frac{−D}{p} )=−1 \atop p≡2(mod 3)}} \frac{ν^{∗}(D, p)}{p} − E

In that expression E is the error term coming from the contribution of terms with p|P_{D}(1728). For D large enough and for an absolute C, we have the inequalities

1 ≤ |P_{D}(1728)| = O(exp(C\sqrt{D} log^{2} D))

(the proof is the same as for the inequality for P_{l}(j_{E}) mentionned in Lemma 5) and

ν^{∗}(D, p) ≤ h(−D)

from which we deduce

(8.10)  E ≪ Ax^{−1} ∑_{D≤X} h(−D)\sqrt{D} log^{2} D ≪ Ax^{ε}.

It is easy to deduce from (8.6), (8.9), (8.10) and Lemma 9 the lower bound of (8.2).

c. Without GRH.

To obtain (8.3), we proceed by using the unconditional version of the Prime Ideal Theorem. By standard analytic number theory, we obtain the relation

π_{K}(x) = li x + O(li x^{β}) + O(x exp(−\sqrt{\frac{log x}{log d_{K}}}))

with the notation as in the proof of Lemma A. Using Stark’s bound for the exceptional Siegel’s zero β ([St] Thm 1’, [MMS] p.279) :

β < max(1 − \frac{1}{4 log d_{K}} , 1 − \frac{c_{1}}{dK^{\frac{1}{n_{K}}}} )

we obtain a uniform result for the sum

∑_{{p≤x, ( \frac{−D}{p} )=−1 \atop p≡2(mod 3)}} ν(D, p)

only for D < (log x)^{θ}, for some θ > 0.

Remark : With more care, we can remove the influence of the Siegel zeroes, thus improving the value of θ.

9. On the least supersingular prime.

The aim of this paragraph is to deal with the following question :

Let E_{a,b} be a given elliptic curve. What is the size of p_{1}(a, b), which is the least supersingular prime of E_{a,b}? Actually, using techniques of the large sieve, we will only prove that p_{1}(a, b) is very small for almost all elliptic curves. In some sense, this result has to be compared with the result concerning the size of the least non quadratic residue (mod p) (see, for instance [Bo], page 7). We will prove the following

THEOREM 8. — Let 2 ≤ y ≤ \sqrt{x}. Then we have the inequality

(9.1)  |\{(a, b); |a| ≤ x, |b| ≤ x, p_{1}(a, b) > y\}| ≪_{κ} x^{2}y^{− \frac{1}{2}} log^{κ} y

for every κ > 1 − \frac{π}{6} .

This theorem asserts that in this set of ≃ x^{2} elliptic curves, almost all of them have their least supersingular prime rather small, less than y (say), for instance with y = \sqrt{x}, the number of exceptions is O(x^{\frac{7}{4}} \sqrt{log x}). We are concerned by bounds y, which are much smaller than the bound which would emerge by closely following the proof presented in paragraph 3 (with k = 0). Note also that the proof of (9.1) is quite straightforward for κ ≥ 1.

The starting point of our proof is a generalisation in several dimensions of the large sieve ; we have :

LEMMA 11. — For each prime p, let Ω(p) be a subset of cardinality ω(p) of the group ℤ^{n}/pℤ^{n} of n-dimensional vectors modulo p.

Let X > 1 and E(X) be the number of x = (x_{1}, ..., x_{n}) with max |x_{i}| ≤ X, for which x(mod p) ∉ Ω(p), for each prime p.

Then, for X ≥ P^{2},we have the inequality

E(X) ≪ X^{n}/(∑_{q≤P} μ^{2}(q) ∏_{p|q} \frac{ω(p)}{p^{n} − ω(p)}).

Such a result is Lemma A of [Ga] and follows from an n-dimensional analogue of the large sieve inequality (see for instance [Hu] Theorem 1 or [Hl]).

To go from Lemma 11 to Theorem 8, we choose n = 2; P = y and X = x. For each p, we define

Ω(p) = {(α, β)(mod p); E_{α,β}(mod p) is an elliptic curve with p + 1 points}

so we have

ω(p) = \frac{p}{2}H(−4p) + O(p)

and we note the implication,

(a, b)(mod p) ∈ Ω(p) ⟹ p is a supersingular prime for E_{a,b} .

(We could get an equivalence above with some care about the minimality of the equation, that is, by considering (ap^{−4k}, bp^{−6k}) mod p where k is the largest possible integer.) So, we have by Lemma 11 the relation

|{(a, b); |a| ≤ x, |b| ≤ x, p_{1}(a, b) > y}| ≪ x^{2}/H(y)

with

H(y) = ∑_{q≤y} μ^{2}(q) ∏_{p|q} \frac{ω(p)}{p^{2} − ω(p)},

and ω(2) = ω(3) = 0. In paragraph 5, we proved the relation

(9.2)  ∑_{p≤y} \frac{H(−4p)}{2p} = (\frac{π}{3} + o(1))\frac{\sqrt{y}}{log y} (y → ∞) ,

and, if we use the trivial inequality

H(y) ≥ ∑_{p≤y} \frac{ω(p)}{p^{2} − ω(p)}

we obtain Theorem 8 for any κ = 1. The improvement comes from taking into account the contribution in H(y), of integers q which are not prime. We define the multiplicative function g(n) by the formula

g(n) = μ^{2}(n) ∏_{p|n} \frac{ω(p)}{p^{2} − ω(p)}.\sqrt{n},

then, by partial summation of (9.2), we see that g satisfies

(9.3)  ∑_{p≤y} g(p) log p = (a + o(1))y (y → ∞) ,

with a = \frac{π}{6} . The problem is now to find a lower bound for the summatory function G(x) = ∑_{n≤x} g(n). The problem of the upper bound is more popular in the literature, and we were unable to find a published result, which fits to our requirement.

The proof of the following lemma was communicated by G.Tenenbaum :

LEMMA 12. — Let g a multiplicative function satisfying (9.3) (for a strictly positive a) and the relation

(9.4)  g(p) ≥ 0 for every p and ∑ \frac{g^{2}(p)}{p^{2}} < ∞.

Then we have the equality

G(x) = ∑_{n≤x} μ^{2}(n)g(n) = x(log x)^{a−1+o(1)}.

The upperbound for G(x) is treated by the inequality of Halberstam and Richert under the form

G(x) ≪ \frac{x}{log x} ∑_{n≤x} \frac{μ^{2}(n)g(n)}{n} ≪ \frac{x}{log x} exp ∑_{p≤x} \frac{g(p)}{p} ≪ x(log x)^{a−1+o(1)}

the last inequality coming from (9.3) after a partial summation.

For the lower bound, we start from the inequality

G(x) ≥ ∑_{m≤x^{\frac{1}{3}}} ∑_{x^{\frac{1}{3}} <p≤x/m} μ^{2}(m)g(m)g(p) ≫ ∑_{m≤x^{\frac{1}{3}}} μ^{2}(m)\frac{g(m)}{m}.\frac{x}{log x}

by (9.3). Put z = x^{\frac{1}{3}} , t = z^{ε} and denote by P (m) the greatest prime factor of m , then we have

(9.5)  ∑_{m≤z} μ^{2}(m)\frac{g(m)}{m} ≥ ∑_{P (m)≤t} μ^{2}(m)\frac{g(m)}{m} − ∑_{{P (m)≤t \atop m>z}} μ^{2}(m)\frac{g(m)}{m}.

By (9.4), the first sum is greater than

∏_{p≤t}(1 + \frac{g(p)}{p}) ≥ exp(∑_{p≤t} \frac{g(p)}{p} − O(∑ \frac{g^{2}(p)}{p^{2}})) ≫ exp(∑_{p≤t} \frac{g(p)}{p}).

We appeal to Rankin’s method to bound from above the second sum, we choose α = \frac{1}{log t} and write

∑_{{P (m)≤t \atop m>z}} μ^{2}(m)\frac{g(m)}{m} ≤ ∑_{P (m)≤t} μ^{2}(m)\frac{g(m)}{m}(\frac{m}{z})^{α} ≤ z^{−α} ∏_{p≤t}(1 + \frac{g(p)}{p}p^{α})

≤ e^{−1/ε} exp\{∑_{p≤t} \frac{g(p)}{p} + O(α ∑_{p≤t} \frac{g(p)}{p} log p)\} ≪ e^{−1/ε} exp ∑_{p≤t} \frac{g(p)}{p}

by (9.4). We fix a very small value to ε, and by (9.3) and (9.5), we get

∑_{m≤z} μ^{2}(m)\frac{g(m)}{m} ≫ exp ∑_{p≤t} \frac{g(p)}{p} = (log x)^{a+o(1)}

This ends the proof of Lemma 12.

For the proof of Theorem 8, we are concerned by a lower bound for H(y), in that context the function g(p) satisfies g(p) = O(p^{ε}), so (9.4) is satisfied. The trivial inequality H(y) ≥ \frac{G(y)}{\sqrt{y}} ends the proof of Theorem 8.

Concluding remarks. These results can be generalized to the context of supersingular Drinfeld modules. This has been done by C. David [Davi] in her doctoral thesis. Indeed, since we have the analogue of the Riemann hypothesis in the Drinfeld context, stronger results can be established unconditionally which improve upon Brown’s results [Br2].

REFERENCES

[Bi] B. BIRCH. — How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43, (1968), 57–60.

[Bo] E.BOMBIERI. — Le grand crible en théorie analytique des nombres, Astérisque (Soc.Math. de France)-18, 1974/1987.

[Br1] M.L. BROWN. — Note on supersingular primes of elliptic curves over ℚ, Bull. London Math. Soc. 20, (1988), 293–296.

[Br2] M.L. BROWN. — Singular Moduli and Supersingular Moduli of Drinfeld Modules, Inv. Math. 110, (1992), 419–439.

[Dave] H. DAVENPORT. — Multiplicative Number Theory (Second Edition), Graduate Texts in Mathematics(74).Springer Verlag, 1980.

[Davi] C. DAVID. — Supersingular Reduction of Drinfeld Modules, Ph. D. Thesis, McGill

University Montreal, 1993.

[De] M. DEURING. — Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14, (1941), 197–272.

[Do] D. DORMAN. — Special values of the elliptic modular function and factorisation formulae, J. Reine Angew. Math. 383, (1988), 207–220.

[El1] N. ELKIES. — The existence of infinitely many supersingular primes for every elliptic curve over ℚ, Inv. Math. 89, (1987), 561–567.

[El2] N. ELKIES. — Supersingular primes of a given elliptic curve over a number field, Ph. D. Thesis, Harvard University, 1987.

[El3] N. ELKIES. — Distribution of Supersingular Primes, Astérisque–Journées Arithmétiques de Luminy 1989 198–199–200, (1991), 127–132.

[Ga] P.X. GALLAGHER. — The large sieve and probalistic Galois theory, Proceedings of Symposia in Pure Mathematics XXIV, (1973), 91–101.

[G-Z] B. H. GROSS and D.ZAGIER. — On singular moduli, J.Reine Angew. Math. 335, (1985),

191–220.

[He] H. HEILBRONN. — Zeta functions and L–functions, Algebraic Number Theory (Edited by J.W.S. CASSELS and A. FRÖHLICH at Academic Press) , (1967), 204–230.

[Hl] M.E. HLAWKA. — Bemerkungen zum grossen Sieb von Linnik, Österreich Akad. Wiss. Math.-Natur. S.B. II 178, (1970), 13–18.

[Ho] C. HOOLEY. — Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts in Mathematics(70)., 1976.

[Hu] M. HUXLEY. — The large sieve inequality for algebraic number fields, Mathematika 15, (1968), 178–187.

[Ju] M. JUTILA. — On the mean-value of L(\frac{1}{2}, χ), Analysis 1, (1981), 149–161.

[Ka] M. KANEKO. — Super singular j-invariants mod p, Osaka J. Math. 26, (1989), 849–855. [L–T] S. LANG and H. TROTTER. — Frobenius in GL_{2} extensions, Lecture Notes in Mathemat-

ics(504).Springer Verlag, 1976.

[Mu] R. MURTY. — Recent developments in the theory of elliptic curves, Proceedings of the

Ramanujan Centennial International Conference , (1987), 45–54.

[MMS] R. MURTY, K. MURTY and N. SARADHA. — Modular forms and the Chebotarev density

theorem, American Journal of Mathematics 110, (1988), 253–281.

[Sc] W.M. SCHMIDT. — Equations over finite fields. An elementary approach, Lecture Notes in Mathematics(536).Springer Verlag, 1976.

[St] H.M. STARK. — Some effective Cases of the Brauer–Siegel Theorem, Inv. Math. 23, (1974), 135–152.

[Se] J-P. SERRE. — Quelques applications du théorème de densité de Chebotarev, Publ. Math. IHES 54, (1982), 123–201.

Etienne FOUVRY Mathématique- Bâtiment 425 Université de Paris–Sud F-91405 ORSAY Cedex

Ram MURTY Department of Mathematics Mc GILL University MONTREAL, PQ CANADA H3A 2K6