Average Frobenius Distributions of Elliptic Curves

Chantal David^{∗}, Concordia University Francesco Pappalardi^{†}, Università Roma Tre

Abstract Let E be an elliptic curve defined over ℚ. For p a prime of good reduction, let a_{p}(E) be the trace of the Frobenius morphism of the reduced curve E_{p}. The Lang-Trotter conjecture predicts the number of primes with a fixed value of a_{p}(E). We prove in this paper average estimates which fit the estimate predicted by the conjecture, with an explicit constant confirming the conjectural constant of Lang and Trotter.

1 Introduction.

Let E be an elliptic curve defined over the rationals. For any prime p of good reduction, let E_{p} be the elliptic curve over 𝔽_{p} obtained by reducing E mod p. Let a_{p}(E) be the trace of the Frobenius morphism of E/𝔽_{p}. Then, #E(𝔽_{p}) = p + 1 − a_{p}(E), and |a_{p}(E)| ≤ 2\sqrt{p}. The case a_{p}(E) = 0 corresponds to supersingular reduction mod p.

For a fixed r ∈ ℤ, what can be said about the number of primes p such that a_{p}(E) = r? If E has complex multiplication, Deuring showed that half of the primes are primes of supersingular reduction [3]. More precisely, let

π_{E}^{r}(x) = # {p ≤ x : a_{p} (E) = r} .

Then, if E has complex multiplication, π_{E}^{0}(x) ∼ \frac{1}{2} π(x) as x → ∞. If E has complex multiplication and r ≠ 0, the primes with a fixed trace of the Frobenius morphism are primes in quadratic progressions. For example, consider the elliptic curve E : Y^{2} = X^{3} − X with complex multiplication by ℤ[i]. It is easy to see that a_{p}(E) = ±2 if and only if p = 1+n^{2} for some integer n. If q(n) is a quadratic progression, and

Q(x) = # {p ≤ x : p = q(n) for some n} ,

∗Research partially supported by CICMA and by NSERC.

†Research partially supported by CICMA and by Consiglio Nazionale delle Ricerche.

it was conjectured by Hardy and Littlewood [9] that Q(x) ∼ C \frac{\sqrt{x}}{log x}as x → ∞. This conjecture is part of a more general conjecture of Lang and Trotter [11].

Conjecture 1.1 (Lang-Trotter Conjecture) Let E be an elliptic curve defined over ℚ, and let r be an integer. Except for the case where r = 0 and E has complex multiplication, there is a constant C_{E,r} such that

π_{E}^{r}(x) ∼ C_{E,r} \frac{\sqrt{x}}{log x} as x → ∞.  (1)

Using their probabilistic model, Lang and Trotter gave an explicit description of the conjectural constant C_{E,r} (see Section 2). The constant C_{E,r} can be 0, and the asymptotic relation is then interpreted to mean that there is only a finite number of primes such that a_{p}(E) = r. It was shown by Elkies that this cannot happen when r = 0, i.e. that for any E/ℚ, there are infinitely many primes of supersingular reduction [5]. However, if r ≠ 0, there could be only finitely many primes p such that a_{p}(E) = r. For example, if E/ℚ has a rational torsion point of order t, then t divides #E(𝔽_{p}) = p + 1 − a_{p}(E) for all primes of good reduction, which imposes conditions on the values of a_{p}(E).

We prove in this paper average estimates related to the Lang-Trotter conjecture. The average distribution fits the one predicted by the conjecture, and the conjectural constant C_{E,r} of Lang and Trotter is confirmed by our results, as seen in Section 2. Average estimates for the case r = 0 were already obtained by Fouvry and Murty [6], and we obtain a generalization of their results for any r ∈ ℤ. The techniques of Fouvry and Murty do not seem to extend to the general case r ∈ ℤ. Our proof then differs significantly from theirs.

In the following, we fix r ∈ ℤ and we denote by E(a, b) the elliptic curve Y^{2} = X^{3}+aX +b with a, b ∈ ℤ. Then

π_{E(a,b)}^{r}(x) = # {p ≤ x : a_{p} (E(a, b)) = r} .

Following [11], we define

π_{1/2}(x) = ∫_{2}^{x} \frac{dt}{2\sqrt{t} log t} ∼ \frac{\sqrt{x}}{log x}.

Theorem 1.2 Let r be an integer, A, B ≥ 1. For every c > 0, we have

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} π_{E(a,b)}^{r}(x) = C_{r} π_{1/2}(x) + O ((\frac{1}{A} + \frac{1}{B} ) x^{3/2} + \frac{x^{5/2}}{AB} + \frac{\sqrt{x}}{log^{c} x}) (2)

where

C_{r} = \frac{2}{π} ∏_{l|r} (1 − \frac{1}{l^{2}} )^{−1} ∏_{l |̸ r} \frac{l(l^{2} − l − 1)}{(l − 1)(l^{2} − 1)}.  (3)

The constants in the O-symbol depend only on c and r.

As the infinite products of (3) converge to positive numbers, the constant C_{r} is non-zero, even if some C_{E,r} can be 0 as mentioned above.

From the last theorem, we immediately obtain that the Lang-Trotter conjecture is true “on average”, i.e.

Corollary 1.3 Let ϵ > 0. If A, B > x^{1+ϵ}, we have as x → ∞,

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} π_{E(a,b)}^{r}(x) ∼ C_{r} \frac{\sqrt{x}}{log x}. □

In analogy with the classical terminology, we can say that the average order of π_{E(a,b)}^{r}(x) is C_{r} \frac{\sqrt{x}}{log x}. Using the same techniques, we can also prove that the normal order of π_{E(a,b)}^{r}(x) is C_{r} \frac{\sqrt{x}}{log x}. Then, π_{E(a,b)}^{r}(x) ∼ C_{r} \frac{\sqrt{x}}{log x} for “almost all” E(a, b) rather than on average (see Corollary 1.5). We are grateful to A. Granville for suggesting this application of our techniques.

Theorem 1.4 Let ϵ > 0. If A, B > x^{1+ϵ}, then for every c > 0, we have

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} |π_{E(a,b)}^{r}(x) − C_{r}π_{1/2}(x)|^{2} = O (\frac{x}{log^{c} x} + (\frac{1}{A} + \frac{1}{B} ) x^{3} + \frac{1}{AB} x^{5}) (4)

where the constant in the O-symbol depends only on c and r.

The following Corollary is a standard application of the Turán normal order method.

Corollary 1.5 Let ϵ > 0 and fix c > 0. If A, B > x^{2+ϵ}, then for all d > 2c and for all elliptic curves E(a, b) with |a| ≤ A and |b| ≤ B with at most O(AB/ log^{d} x) exceptions, we have the inequality

|π_{E(a,b)}^{r}(x) − C_{r} π_{1/2}(x)| ≪ \frac{\sqrt{x}}{log^{c} x}. □

In Section 2, we compare the constant C_{r} with the constants C_{E,r} predicted by Lang and Trotter. Sections 3 and 4 contain the proof of Theorem 1.2, and Section 5 contains the proof of Theorem 1.4.

2 The Lang-Trotter constant C_{E,r}.

To formulate their conjecture, Lang and Trotter considered a probabilistic model compatible with the Cebotarev Density Theorem and with the Sato-Tate conjecture. From the model, they obtained an explicit description of the constant C_{E,r} in terms of Galois representations, as described below. We compare in this section the conjectural constants C_{E,r} of Lang and Trotter with the constant C_{r} of Theorem 1.2.

Let ρ_{E,m} be the Galois representation

ρ_{E,m} : Gal (\overline{ℚ}/ℚ) → Aut(E[m])

where E[m] is the subgroup of m-torsion points of E(\overline{ℚ}). Since E[m] ≃ (ℤ/mℤ)^{2}, after choosing a basis for E[m], we can identify Aut(E[m]) with GL_{2}(ℤ/mℤ). Let G(m) be the image of ρ_{E,m} in GL_{2}(ℤ/mℤ), and for any subgroup G of GL_{2}(ℤ/mℤ), let G_{r} be the subset of elements of G of trace r modulo m.

Let E be an elliptic curve without complex multiplication. Serre proved in [13] that the image of the Galois representation on the full torsion subgroup of E(\overline{ℚ}) is an open subgroup of GL_{2}(ℤ̂). It follows that there exists an integer m_{E} such that ρ_{E,l} is surjective for all primes l not dividing m_{E}, and such that the image in GL_{2}(ℤ̂) of the Galois representation on the torsion subgroup of E(\overline{ℚ}) is the full inverse image of G(m_{E}). The Lang-Trotter constant C_{E,r} is then defined as [11, p. 36]

C_{E,r} = \frac{2}{π} \frac{m_{E} |G(m_{E})_{r}|}{|G(m_{E})|} ∏_{l |̸ m_{E}} \frac{l |G(l)_{r}|}{|G(l)|}

= \frac{2}{π} \frac{m_{E} |G(m_{E})_{r}|}{|G(m_{E})|} ∏_{l|r^{l |̸ m_{E}}} (1 − \frac{1}{l^{2}} )^{−1} ∏_{l |̸ r^{l |̸ m_{E}}} \frac{l(l^{2} − l − 1)}{(l − 1)(l^{2} − 1)}.  (5)

The second equality follows from the easy estimates

\frac{l |GL_{2}(𝔽_{l})_{r}|}{|GL_{2}(𝔽_{l})|} = \{ \begin{matrix}\frac{l^{3}(l − 1)}{l(l − 1)^{2}(l + 1)} & when r ≡ 0 (l); \\ \frac{l^{2}(l^{2} − l − 1)}{l(l − 1)^{2}(l + 1)} & when r ≢ 0 (l). \end{matrix}

Comparing (3) and (5), we see that the local factors are exactly the same for the primes l |̸ m_{E}. More precisely, is that true that

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} C_{E(a,b),r} ∼ C_{r} as A, B → ∞?

There are partial results for the case r = 0 due to Fouvry and Ullmo [7]. The recent estimates of Duke [4], who showed that for “most” elliptic curves E/ℚ, ρ_{E,l}(G) = GL_{2}(𝔽_{l}) for all primes l, are also relevant to this problem. But this does not imply that m_{E} = 1 (and then C_{E,r} = C_{r}) for those curves. In fact, we never have m_{E} = 1 as shown in [13, Proposition 22].

3 An average of special value of L-series.

We will show in Section 4 how the average

\frac{1}{4AB} ∑_{|a|≤A |b|≤B} π_{E(a,b)}^{r}(x)

can be rewritten as an average of special values of Dirichlet L-series by counting the number of curves over the finite fields 𝔽_{p} with a_{p} = r. The proof of Theorem 1.2 is then obtained by studying this average of L-series, which we evaluate in this section.

Let B(r) = max (3, r, r^{2}/4). In particular, p > B(r) ensures |r| ≤ 2\sqrt{p}, a necessary condition for a_{p}(E) = r.

For d ≡ 0, 1(4), d not a perfect square, and n ≠ 0, let χ_{d}(n) = (\frac{d}{n}) be the Kronecker symbol (see for example [10, p. 304]). The Kronecker symbol is a real character modulo |d|, and for n > 0

d_{1} ≡ d_{2} (n) ⇒ (\frac{d_{1}}{n} ) = (\frac{d_{2}}{n} ) for n odd

d_{1} ≡ d_{2} (4n) ⇒ (\frac{d_{1}}{n} ) = (\frac{d_{2}}{n} ) for n ∈ ℕ.

(6)

We give the proof of Theorem 1.2 for r odd. The proof is similar when r is even therefore we omit it.

The main result of this section is

Theorem 3.1 Let r be an odd integer, and let

K_{r} = ∑_{(2r,f)=1^{f=1}}^{∞} ∑_{n=1}^{∞} \frac{c_{f}^{r}(n)}{nfφ (nf^{2})} with c_{f}^{r}(n) = ∑_{{a(4n)^{∗} \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n}) ,  (7)

where ∑_{a(4n)^{∗}} is the sum over a complete set of invertible residues mod 4n.

Furthermore let

S_{f}(x) = \{B(r) < p ≤ x | 4p ≡ r^{2} (f^{2}), and d = \frac{r^{2} − 4p}{f^{2}} ≡ 0, 1 (4)\} .

Then for any c > 0

∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p∈S_{f} (x)} L(1, χ_{d}) log p = K_{r}x + O (\frac{x}{log^{c} x}) .  (8)

Proof of Theorem 3.1: As r is odd, if f^{2} | r^{2} − 4p then f is odd, and d = (r^{2} − 4p)/f^{2} ≡ 1 (4). Furthermore, since (r, f) | p, and p > B(r), we have (r, f) = 1.

For a fixed parameter U > 0 to be chosen later, we have

L(1, χ_{d}) = ∑_{n≥1} (\frac{d}{n}) \frac{1}{n} = ∑_{n≤U} (\frac{d}{n}) \frac{1}{n} + O (\frac{\sqrt{|d|} log |d|}{U})

= ∑_{n≤U} (\frac{d}{n}) \frac{1}{n} + O (\frac{\sqrt{p} log p}{fU})  (9)

using the Polya-Vinogradov inequality [2, p. 135].

Using (9), we can rewrite the left hand side of (8) as

∑_{(2r,f)=1^{f≤2\sqrt{x}}} \frac{1}{f} ∑_{n≤U} \frac{1}{n} ∑_{p∈S_{f} (x)} (\frac{d}{n}) log p + O (\frac{x^{3/2} log x}{U}) .  (10)

For a fixed parameter V with 1 ≤ V ≤ 2\sqrt{x} to be chosen later, the first part of (10) is

∑_{(2r,f)=1^{f≤V}} \frac{1}{f} ∑_{n≤U} \frac{1}{n} ∑_{p∈S_{f} (x)} (\frac{d}{n}) log p + ∑_{(2r,f)=1^{V <f≤2\sqrt{x}}} \frac{1}{f} ∑_{n≤U} \frac{1}{n} ∑_{p∈S_{f} (x)} (\frac{d}{n}) log p.

The summation for large values of f is easily evaluated as

| ∑_{(2r,f)=1^{V <f≪2\sqrt{x}}} \frac{1}{f} ∑_{n≤U} \frac{1}{n} ∑_{p∈S_{f} (x)} (\frac{d}{n}) log p | ≤ log x log U ∑_{V <f≤2\sqrt{x}} \frac{1}{f} ∑_{n≡4^{∗}r^{2}(f^{2})^{n≤x}} 1

≪ x log x log U ∑_{V <f≤2\sqrt{x}} \frac{1}{f^{3}} ≪ \frac{x log x log U}{V^{2}}

where 4^{∗} is an integer such that 4^{∗} · 4 ≡ 1 (f^{2}).

Therefore we can rewrite the left hand side of (8) as

∑_{(2r,f)=1^{f≤V}} \frac{1}{f} ∑_{n≤U} \frac{1}{n} ∑_{p∈S_{f} (x)} (\frac{d}{n}) log p + O (\frac{x^{3/2} log x}{U} + \frac{x log x log U}{V^{2}} ) .  (11)

The sum over “small values” of f and n leads to the main term. It is evaluated by splitting the sum according to the residue of d mod 4n. Since d = (r^{2} − 4p)/f^{2} is odd, and (\frac{d}{n}) = 0 when (d, n) > 1, using(6) we get

∑_{(2r,f)=1^{f≤V}} \frac{1}{f} ∑_{n≤U} \frac{1}{n} ∑_{p∈S_{f} (x)} (\frac{d}{n}) log p = ∑_{(2r,f)=1^{n≤U,f≤V}} \frac{1}{fn} ∑_{a(4n)^{∗}} (\frac{a}{n}) ∑_{d≡a(4n)^{p∈S_{f} (x)}} log p.  (12)

In the above sum, the two conditions p ∈ S_{f}(x) and d = (r^{2} − 4p)/f^{2} ≡ a (4n) are equivalent to B(r) < p ≤ x and p ≡ (r^{2} − af^{2})/4 (nf^{2}). Furthermore, as (2r, f) = 1, ((r^{2} − af^{2})/4, nf^{2}) = 1 ⟺ (r^{2} − af^{2}, 4n) = 4.

We use the following standard notation

ψ_{1}(x; n, a) = ∑_{p≡a (n)^{p≤x}} log p

E_{1}(x; n, a) = ψ_{1}(x; n, a) − \frac{x}{φ(n)} for (a, n) = 1.

Lemma 3.2 (Theorem of Barban, Davenport and Halberstam) With the notation above, for any K > 0 and x/ log^{K} x ≤ Q ≤ x, we have

∑_{n≤Q} ∑_{a(n)^{∗}} E_{1}^{2}(x; n, a) ≪ Qx log x. □

This classical result can be found, for example, in Davenport [2, p. 169].

We rewrite (12) as

∑_{(2r,f)=1^{n≤U,f≤V}} \frac{1}{fn} ∑_{a(4n)^{∗}} (\frac{a}{n}) ψ_{1} (x; nf^{2}, \frac{r^{2} − af^{2}}{4} ) + O (U log V) ,

where the term O(U log V) comes from the primes less than B(r), and the O-constant depends on r only. Using the notation defined above, we rewrite the last equation as

x ∑_{(2r,f)=1^{n≤U,f≤V}} \frac{c_{f}^{r}(n)}{fnφ(nf^{2})} + O(U log V)

+ ∑_{(2r,f)=1^{n≤U,f≤V}} \frac{1}{fn} ∑_{(r^{2}−af^{2},4n)=4^{a(4n)^{∗}}} (\frac{a}{n}) E_{1} (x; nf^{2}, \frac{r^{2} − af^{2}}{4} ) .  (13)

The second sum of (13) is dominated by the error term. Indeed, using the Cauchy-Schartz inequality, we bound it by

∑_{(2r,f)=1^{f≤V}} \frac{1}{f} (∑_{n≤U} \frac{φ(4n)}{n^{2}})^{1/2} (∑_{n≤U} ∑_{(r^{2}−af^{2},4n)=4^{a(4n)^{∗}}} E_{1}^{2} (x; nf^{2}, \frac{r^{2} − af^{2}}{4} ))^{1/2}

≤ (log U)^{1/2} ∑_{(2r,f)=1^{f≤V}} \frac{1}{f} (∑_{n≤U} ∑_{(r^{2}−af^{2},4n)=4^{a(4n)^{∗}}} E_{1}^{2} (x; nf^{2}, \frac{r^{2} − af^{2}}{4} ))^{1/2}

≤ (log U)^{1/2} ∑_{(2r,f)=1^{f≤V}} \frac{1}{f} (∑_{n≤U} ∑_{b(nf^{2})^{∗}} E_{1}^{2}(x; nf^{2}, b))^{1/2} ,

as a_{1} ≢ a_{2} (4n) ensures b_{1} = \frac{r^{2} − a_{1}f^{2}}{4} ≢ b_{2} = \frac{r^{2} − a_{2}f^{2}}{4} (nf^{2}).

Fix any c > 0. Then the last sum is bounded by

≤ log V (log U)^{1/2} ( ∑_{n≤UV^{2}} ∑_{a(n)^{∗}} E_{1}^{2}(x; n, a))^{1/2} ,

which is

≤ log V (log U)^{1/2} \frac{x}{log^{c+2} x}  (14)

when

UV^{2} ≤ \frac{x}{log^{B(c)} x}  (15)

from Lemma 3.2, with B(c) = 2c + 6.

Finally, using (11), (13) and (14), we obtain

∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p∈S_{f} (x)} L(1, χ_{d}) log p = x ∑_{(2r,f)=1^{n≤U,f≤V}} \frac{c_{f}^{r}(n)}{fnφ(nf^{2})}

+O (U log V + \frac{x log V log^{1/2} U}{log^{c+2} x} + \frac{x^{3/2} log x}{U} + \frac{x log x log U}{V^{2}} )  (16)

for any U, V satisfying (15).

In order to find the asymptotic behavior of the main term, we have to estimate the growth of

c_{f}^{r}(n) = ∑_{{a(4n)^{∗} \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n}) .

Let κ(n) be the multiplicative arithmetic function generated by the identity

κ(l^{α}) = \{ \begin{matrix}l & α odd \\ 1 & α even \end{matrix}

for any prime l and any positive integer α. Then for a positive integer n, κ(n) is the smallest integer dividing n such that n/κ(n) is a square.

Lemma 3.3 The following hold:

1. if n is odd, then c_{f}^{r}(n) = ∑_{{a(n)^{∗} \atop (r^{2}−af^{2},n)=1}} (\frac{a}{n});

2. c_{f}^{r}(n) is a multiplicative function of n;

3. for any prime l, c_{f}^{r}(l^{α}) = c_{(f,l)}^{r}(l^{α});

4. if α ≥ 1, then c_{1}^{r}(2^{α}) = (−2)^{α}/2;

5. if l is an odd prime, then \frac{c_{1}^{r}(l^{α})}{l^{α−1}} = \{ \begin{matrix}l − 1 − (\frac{r^{2}}{l} ) & if α is even \\ − (\frac{r^{2}}{l} ) & if α is odd; \end{matrix}

6. if l is an odd prime (l |̸r), then \frac{c_{l}^{r}(l^{α})}{l^{α−1}} = \{ \begin{matrix}0 & if α is odd \\ l − 1 & if α is even; \end{matrix}

7. for all n, |c_{f}^{r}(n)| ≤ \frac{n}{κ(n)}.

Proof: 1. By definition,

c_{f}^{r}(n) = ∑_{{a(4n)^{∗},a≡1(4) \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n}) + ∑_{{a(4n)^{∗},a≡3(4) \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n}) .

The second sum is empty since r^{2} + f^{2} ≡ 2 (4). If a ≡ 1 (4), then (r^{2} − af^{2}, 4n) = 4 if and only if (r^{2} − af^{2}, n) = 1. This gives

c_{f}^{r}(n) = ∑_{{a(4n)^{∗},a≡1(4) \atop (r^{2}−af^{2},n)=1}} (\frac{a}{n}) .

As n is odd, there is a bijection between the invertible residues modulo 4n which are congruent to 1 modulo 4 and the residues modulo n. We then use the property (6) of the Kronecker symbols to deduce the claim;

2. Clearly, c_{f}^{r}(1) = 1. Let n = n_{1} n_{2} with (n_{1}, n_{2}) = 1. We can suppose without loss of generality that 2 |̸n_{1}. By 1., we have

c_{f}^{r}(n_{1}) c_{f}^{r}(n_{2}) = ∑_{{a_{1}(n_{1})^{∗} \atop (r^{2}−a_{1}f^{2},n_{1})=1}} ∑_{{a_{2}(4n_{2})^{∗} \atop (r^{2}−a_{2}f^{2},4n_{2})=4}} (\frac{a_{1}}{n_{1}} ) (\frac{a_{2}}{n_{2}} ) .  (17)

Now for any a_{1} and a_{2} in the above sums, let a be the unique integer such that 1 ≤ a ≤ 4n, (a, 4n) = 1, a = a_{1} + k_{1}n_{1} = a_{2} + k_{2}4n_{2} for some integers k_{1} and k_{2}.

It is easy to see that (r^{2} − a_{1}f^{2}, n_{1}) = 1 and (r^{2} − a_{2}f^{2}, 4n_{2}) = 4 if and only if (r^{2} − af^{2}, 4n) = 4. Therefore we can write the right hand side of (17) as

∑_{{a_{1}(n_{1})^{∗}, a_{2}(4n_{2})^{∗} \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n_{1}} ) (\frac{a}{n_{2}} ) = ∑_{{a_{1}(n_{1})^{∗}, a_{2}(4n_{2})^{∗} \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n}) .

The statement now follows by the Chinese Remainder Theorem;

3. If (f, l) = 1 then the two sets {a | a(mod4l^{α}), (a, 4l^{α}) = 1} and {af^{2} | a(mod4l^{α}), (a, 4l^{α}) = 1} are equal. So, if l is odd,

c_{f}^{r}(l^{α}) = ∑_{{a(l^{α})^{∗} \atop (r^{2}−af^{2},l)=1}} (\frac{a}{l})^{α} = ∑_{{b(l^{α})^{∗} \atop (r^{2}−b,l)=1}} (\frac{b}{l})^{α} (\frac{f^{∗}}{l} )^{2α} = c_{1}^{r}(l^{α})  (18)

where f^{∗}f ≡ 1 (l). The proof is similar for l = 2. If (f, l) = l, then l is odd and since (r, f) = 1, we have that (r^{2} − af^{2}, l) = 1 for all a invertible residues modulo l^{α}. Therefore,

c_{f}^{r}(l^{α}) = ∑_{a(l^{α})^{∗}} (\frac{a}{l})^{α} = c_{l}^{r}(l^{α});  (19)

4. Since (\frac{a_{1}}{2} ) = (\frac{a_{2}}{2} ) when a_{1} ≡ a_{2} (8), we have

c_{1}^{r}(2^{α}) = 2^{α−1} ∑_{{a(8)^{∗} \atop (r^{2}−a,8)=4}} (\frac{a}{2})^{α} .

Now r^{2} ≡ 1 (8), so (r^{2} − a, 8) = 4 if and only if a ≡ 5 (8). Therefore

c_{1}^{r}(2^{α}) = 2^{α−1} (\frac{5}{2})^{α} = (−2)^{α}/2;

5. Using (18), we write

c_{1}^{r}(l^{α}) = l^{α−1} ∑_{(r^{2}−a,l)=1^{a(l)^{∗}}} (\frac{a}{l})^{α} = l^{α−1} ∑_{a(l)^{∗}} (\frac{a}{l})^{α} − l^{α−1} (\frac{r^{2}}{l} )

and the claim is deduced from the orthogonality relations of the Legendre symbols;

6. is a consequence of (19), by the orthogonality relations of the Legendre symbols;

7. is a consequence of 2., 4., 5. and 6. □

Lemma 3.4 Let c = ∏_{l prime} (1 + \frac{1}{l(\sqrt{l}−1)}) . Then

∑_{n>U} \frac{1}{κ(n)φ(n)} ∼ \frac{c}{\sqrt{U}} .

In particular, ∑_{n=1}^{∞} \frac{1}{κ(n)φ(n)} converges. Proof: Let

C(t) = ∑_{n≤t} \frac{n^{3/2}}{κ(n)φ(n)}.

Using the partial summation formula [12, Exercise 1.1], we immediately deduce that

∑_{n>U} \frac{1}{κ(n)φ(n)} = \frac{3}{2} ∫_{U}^{∞} \frac{C(t)}{t^{5/2}} + lim_{N→∞} \frac{C(N)}{N^{3/2}} − \frac{C(U)}{U^{3/2}}.  (20)

We claim that the following asymptotic formula holds:

C(t) ∼ \frac{c}{2} t.  (21)

The lemma then follows easily by substituting (21) in (20).

To prove (21), we consider the Dirichlet series

K(s) = ∑_{n=1}^{∞} \frac{n^{3/2}}{κ(n)φ(n)} n^{−s},

which clearly converges for ℜ(s) > 5/2. Since both κ and φ are multiplicative functions, a straightforward computation gives the Euler product expansion

K(s) = ∏_{l prime} (1 + \frac{l(l^{s−3/2} + 1)}{(l − 1)(l^{2s−1} − 1)}) .

This shows that K(s) converges for ℜ(s) > 1.

By computing the product

K(s) · \frac{1}{ζ(2s − 1)} = ∏_{l prime} (1 + \frac{1 + l^{s−1/2}}{l^{2s−1}(l − 1)})

which converges for ℜ(s) > 1/2, we deduce that K(s) admits a meromorphic continuation in the half plane ℜ(s) > 1/2, with only a simple pole at s = 1 and residue

\frac{1}{2} ∏_{l prime} (1 + \frac{1}{l(\sqrt{l} − 1)}) .

Since K(s) is regular on the vertical line ℜ(s) = 1 (s ≠ 1), we apply the Wiener-Ikehara Tauberian Theorem (see for example [12] Theorem 1.1) to K(s) to deduce (21). This proves the lemma. □

Theorem 3.1 now follows easily from Lemmas 3.3 and 3.4, as

x ∑_{(2r,f)=1^{n≤U,f≤V}} \frac{c_{f}^{r}(n)}{fnφ(nf^{2})} = x ∑_{(2r,f)=1^{f≤V}} ∑_{n=1}^{∞} \frac{c_{f}^{r}(n)}{fnφ(nf^{2})} + O (x ∑_{f≤V} \frac{1}{f} ∑_{n>U} \frac{1}{κ(n)φ(nf^{2})})

from Lemma 3.3(7). But φ(nf^{2}) ≥ φ(n)φ(f^{2}), which gives

O (x ∑_{f≤V} \frac{1}{f} ∑_{n>U} \frac{1}{κ(n)φ(nf^{2})}) = O (\frac{x}{U^{1/2}} )

from Lemma 3.4. Finally,

x ∑_{(2r,f)=1^{n≤U,f≤V}} \frac{c_{f}^{r}(n)}{fnφ(nf^{2})} = x ∑_{(2r,f)=1^{f=1}}^{∞} ∑_{n=1}^{∞} \frac{c_{f}^{r}(n)}{fnφ(nf^{2})}

+O (x ∑_{f>V} \frac{1}{fφ(f^{2})} ∑_{n=1}^{∞} \frac{1}{κ(n)φ(n)}) + O (\frac{x}{U^{1/2}} )

= K_{r}x + O (\frac{x}{V^{2}} ) + O (\frac{x}{U^{1/2}} )  (22)

This completes the proof of the theorem. Indeed, from (16) and (22)

∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p≤x}^{∗}L(1, χ_{d}) log p = K_{r}x + O (\frac{x}{V^{2}} + \frac{x}{U^{1/2}} )

+O (U log V + \frac{x log V (log U)^{1/2}}{log^{c+2} x} + \frac{x^{3/2} log x}{U} + \frac{x log x log U}{V^{2}} )

for any U, V satisfying (15). Choosing

U = \sqrt{x} log^{c+1} x
V = (log x)^{1/2(c+2)} ,

we deduce the result. □

4 Proof of Theorem 1.2.

For r ≤ 2\sqrt{p}, the number of 𝔽_{p}-isomorphism classes of elliptic curves over 𝔽_{p} with p + 1 − r points is the total number of ideal classes of the ring ℤ [(D + \sqrt{D}) /2] where D = r^{2} −4p is a negative integer which is congruent to 0 or 1 modulo 4. This total number of ideal classes is the Kronecker class number

H(r^{2} − 4p) = 2 ∑_{{f^{2}|r^{2}−4p \atop d≡0,1(4)}} \frac{h(d)}{w(d)},  (23)

where the sum ranges over positive integers f such that f^{2} divides r^{2}−4p, and d = (r^{2}−4p)/f^{2} is congruent to 0 or 1 modulo 4. As usual, h(d) and w(d) denote the class number and the number of units respectively, of the order of discriminant d.

Suppose that p ≠ 2, 3. Then, any elliptic curve over 𝔽_{p} has a model

E : Y^{2} = X^{3} + aX + b

with a, b ∈ 𝔽_{p}. The elliptic curves E′(a′, b′) over 𝔽_{p} which are 𝔽_{p}-isomorphic to E are given by all the choices

a′ = μ^{4}a and b′ = μ^{6}b

with μ ∈ 𝔽_{p}^{∗}. The number of such E′ is

(p − 1)/6 when a = 0 and p ≡ 1(3);
(p − 1)/4 when b = 0 and p ≡ 1(4);
(p − 1)/2 otherwise.

Then, the number of curves E(a, b) with a, b ∈ ℤ, 0 ≤ a, b < p and a_{p}(E(a, b)) = r is

H(r^{2} − 4p) (\frac{p − 1}{2}) + O(p) = \frac{p H(r^{2} − 4p)}{2} + O(p)  (24)

This result can be found in Birch [1].

We then write

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} π_{E(a,b)}^{r}(x) = \frac{1}{4AB} ∑_{p≤x} # \{|a| ≤ A, |b| ≤ B : a_{p} (E(a, b)) = r\}

as

\frac{1}{4AB} ∑_{B(r)<p≤x} (\frac{2A}{p} + O(1)) (\frac{2B}{p} + O(1)) (\frac{p H(r^{2} − 4p)}{2} + O(p)) .

This last equation can be rewritten as

\frac{1}{2} ∑_{B(r)<p≤x} \frac{H(r^{2} − 4p)}{p} + O ( ∑_{B(r)<p≤x} H(r^{2} − 4p) (\frac{1}{A} + \frac{1}{B} + \frac{p}{AB} )) + O (log log x) . (25)

Using

H(r^{2} − 4p) ≤ ∑_{d≡0,1(4)^{f^{2}|r^{2}−4p}} h(d) ≪ \sqrt{p} log p ∑_{f^{2}|r^{2}−4p} \frac{1}{f} ,

and the Brun-Titchmarsh Theorem (see for example [8]), we have the estimates

∑_{p≤x} H(r^{2} − 4p) ≪ \sqrt{x} log x ∑_{p≤x} ∑_{f^{2}|r^{2}−4p} \frac{1}{f}

≪ \sqrt{x} log x ∑_{f≤2\sqrt{x}} \frac{1}{fφ(f)} \frac{x}{log x} ≪ x^{3/2}  (26)
∑_{p≤x} pH(r^{2} − 4p) ≪ x^{5/2}.  (27)

Finally, replacing these estimates in (25), we have

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} π_{E(a,b)}^{(r)}(x) = \frac{1}{2} ∑_{B(r)<p≤x} \frac{H(r^{2} − 4p)}{p}

+ O ((\frac{1}{A} + \frac{1}{B} ) x^{3/2} + \frac{x^{5/2}}{AB} + log log x)  (28)

We now use Theorem 3.1 to evaluate the main term of (28). Using (23), we write

\frac{1}{2} ∑_{B(r)<p≤x} \frac{H(r^{2} − 4p)}{p} = ∑_{f≤2\sqrt{x}} ∑_{p∈S_{f} (x)} \frac{1}{p} \frac{h(d)}{w(d)},  (29)

where

S_{f}(x) = \{B(r) < p ≤ x | 4p − r^{2} ≡ 0 (f^{2}), and d = \frac{r^{2} − 4p}{f^{2}} ≡ 0, 1 (4)\}.

As d = (r^{2} − 4p)/f^{2} is a negative integer, the class number formula reads as

h(d) = \frac{w(d)|d|^{1/2}}{2π} L(1, χ_{d}),  (30)

where χ_{d} is the Kronecker symbol defined in Section 3. Therefore, replacing (30) in (29), we have

\frac{1}{2} ∑_{B(r)<p≤x} \frac{H(r^{2} − 4p)}{p} = \frac{1}{2π} ∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p∈S_{f} (x)} \frac{\sqrt{4p − r^{2}}}{p} L(1, χ_{d})

= \frac{1}{π} ∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p∈S_{f} (x)} \frac{L(1, χ_{d})}{\sqrt{p}} + O (log^{2} x) .

With partial summation, we write

∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p∈S_{f} (x)} \frac{L(1, χ_{d})}{\sqrt{p}} = \frac{1}{\sqrt{x} log x} ∑_{f≤2\sqrt{x}} \frac{1}{f} ∑_{p∈S_{f} (x)} L(1, χ_{d}) log p

− ∫_{2}^{x} ∑_{f≤2\sqrt{t}} (\frac{1}{f} ∑_{p∈S_{f} (t)} L(1, χ_{d}) log p) \frac{d}{dt} (\frac{1}{\sqrt{t} log t}) dt,  (31)

since S_{f}(t) = ∅ for f > 2\sqrt{t}. Using Theorem 3.1, (31) can be rewritten as

K_{r} ∫_{2}^{x} \frac{dt}{2\sqrt{t} log t} + K_{r} (\frac{\sqrt{x}}{log x} + ∫_{2}^{x} \frac{dt}{\sqrt{t} log^{2} t})

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

which gives

\frac{1}{2} ∑_{B(r)<p≤x} \frac{H(r^{2} − 4p)}{p} = \frac{2K_{r}}{π} π_{1/2}(x) + O (\frac{\sqrt{x}}{log^{c} x}) .  (32)

Replacing (32) in (28), the only thing left to show is that C_{r} = \frac{2}{π} K_{r} has the correct Euler product expansion.

Lemma 4.1 Suppose that r is an odd integer. Let

K_{r} = ∑_{(2r,f)=1^{f=1}}^{∞} ∑_{n=1}^{∞} \frac{c_{f}^{r}(n)}{nfφ (nf^{2})} with c_{f}^{r}(n) = ∑_{{a(4n)^{∗} \atop (r^{2}−af^{2},4n)=4}} (\frac{a}{n}) ,

where ∑_{a(n)^{∗}} is the sum over a complete set of invertible residues mod n. Then

K_{r} = ∏_{l|r} (1 − \frac{1}{l^{2}} )^{−1} ∏_{l |̸ r} \frac{l(l^{2} − l − 1)}{(l − 1)(l^{2} − 1)}.

Proof of Lemma 4.1: Since c_{f}^{r}(n) is a multiplicative function of n (2. of Lemma 3.3), we have that

K_{r} = ∑_{{f=1 \atop (f,2r)=1}}^{∞} \frac{1}{fφ(f^{2})} ∏_{l} (∑_{α≥0} \frac{c_{f}^{r}(l^{α})}{l^{α}φ(l^{α})} · \frac{φ((f^{2}, l^{α}))}{(f^{2}, l^{α})}) .  (33)

Using 3. of Lemma 3.3, we rewrite the above product as

∏_{l} (∑_{α≥0} \frac{c_{f}^{r}(l^{α})}{l^{α}φ(l^{α})} · \frac{φ((f^{2}, l^{α}))}{(f^{2}, l^{α})}) = ∏_{l |̸ f} (∑_{α≥0} \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})}) ∏_{l|f} (∑_{α≥0} \frac{c_{f}^{r}(l^{α})}{l^{α}φ(l^{α})} \frac{φ((f^{2}, l^{α}))}{(f^{2}, l^{α})})

= ∏_{l} (∑_{α≥0} \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})}) ∏_{l|f} (\begin{matrix}∑ & α≥0 & \frac{c_{f}^{r} (l^{α})}{l^{α}φ(l^{α})} & \frac{φ((f^{2},l^{α}))}{(f^{2},l^{α})} \\ ∑ & α≥0 & \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})} \end{matrix} ) .

Replacing inside (33), and using the multiplicativity of the functions in the outer sum of (33), we obtain

K_{r} = ∏_{l} (∑_{α≥0} \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})}) ∏_{l |̸ 2r} (1 + ∑_{β≥1} \frac{1}{l^{β}φ(l^{2β})} · \frac{∑_{α≥0} \frac{c_{l}^{r}(l^{α})}{l^{α}φ(l^{α})} · \frac{φ((l^{2β},l^{α}))}{(l^{2β},l^{α})}}{∑α≥0 \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})}} )

= ∏_{l|2r} (∑_{α≥0} \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})}) ∏_{l |̸ 2r} (1 + ∑_{α≥1} \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})} + \frac{1}{l^{3} − 1} (\frac{l}{l − 1} + ∑_{α≥1} \frac{c_{l}^{r}(l^{α})}{l^{α}φ(l^{α})}))

With Lemma 3.3, we compute

∑_{α≥1} \frac{c_{1}^{r}(l^{α})}{l^{α}φ(l^{α})} = \{ \begin{matrix}\frac{2}{3} & if l = 2 \\ \frac{1}{l^{2} − 1} & if l|r \\ \frac{−2}{(l − 1)(l^{2} − 1)} & if l |̸2r \end{matrix}

and for l |̸2r

∑_{α≥1} \frac{c_{l}^{r}(l^{α})}{l^{α}φ(l^{α})} = \frac{1}{l^{2} − 1}.

Replacing in the expression for K_{r} above, this gives

K_{r} = \frac{2}{3} ∏_{l|r} (1 − \frac{1}{l^{2}} )^{−1} ∏_{l |̸ 2r} (1 − \frac{2}{(l − 1)(l^{2} − 1)} + \frac{1}{(l − 1)(l^{2} − 1)})

= ∏_{l|r} (1 − \frac{1}{l^{2}} )^{−1} ∏_{l |̸ r} \frac{l(l^{2} − l − 1)}{(l − 1)(l^{2} − 1)}.

This completes the proof of Theorem 1.2 for r odd. □

5 Proof of Theorem 1.4.

We prove in this section Theorem 1.4. As in the proof of Theorem 1.2, the main ingredient is Theorem 3.1.

Let

μ = \frac{1}{4AB} ∑_{|a|≤A, |b|≤B} π_{E(a,b)}^{r}(x).

Fix any c > 0. Then, for A, B > x^{1+ϵ}, Theorem 1.2 gives

μ = C_{r}π_{1/2}(x) + O (\frac{\sqrt{x}}{log^{c} x}) .  (34)

Thus, by the triangle inequality, the left hand side of (4) is

≪ \frac{1}{4AB} ∑_{|a|≤A, |b|≤B} |π_{E(a,b)}^{r}(x) − μ|^{2} + O (\frac{x}{log^{2c} x}) .  (35)

Now in general, if μ = \frac{1}{N} ∑_{n=1}^{N} λ_{n}, then \frac{1}{N} ∑_{n=1}^{N} (λ_{n} − μ)^{2} = \frac{1}{N} ∑_{n=1}^{N} λ_{n}^{2} − μ^{2}. Therefore, the left hand side of (4) is

≪ \frac{1}{4AB} ∑_{|a|≤A, |b|≤B} (π_{E(a,b)}^{r}(x))^{2} − μ^{2} + O (\frac{x}{log^{2c} x}) .  (36)

We then write

(π_{E(a,b)}^{r}(x))^{2} = π_{E(a,b)}^{r}(x) + # {p, q ≤ x | p ≠ q, a_{p}(E(a, b)) = a_{q}(E(a, b)) = r}

where the pairs p, q and q, p are both counted.

Proceeding as in the proof of Theorem 1.2 and using the Chinese Remainder theorem, we obtain from the last line that

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} (π_{E(a,b)}^{r}(x))^{2} = μ + \frac{1}{4} ∑_{B(r)<p,q≤x, p≠q} \frac{H(r^{2} − 4p)}{p} \frac{H(r^{2} − 4q)}{q} + E(x, A, B)

where

E(x, A, B) ≪ ∑_{p,q≤x} \frac{H(r^{2} − 4p) + H(r^{2} − 4q)}{pq} + ∑_{p,q≤x} H(r^{2}−4p) H(r^{2}−4q) (\frac{1}{A} + \frac{1}{B} + \frac{pq}{AB} ) .

Therefore, using the estimates (26) and (27), we obtain

E(x, A, B) ≪ \frac{\sqrt{x} log log x}{log x} + (\frac{1}{A} + \frac{1}{B} ) x^{3} + \frac{x^{5}}{AB} .  (37)

Then

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} (π_{E(a,b)}^{r}(x))^{2} = μ+(\frac{1}{2} ∑_{B(r)<p≤x} \frac{H(r^{2} − 4p)}{p})^{2}−\frac{1}{4} ∑_{p≤x} \frac{H(r^{2} − 4p)^{2}}{p^{2}} +E(x, A, B),

and using (32) and (37), this gives

\frac{1}{4AB} ∑_{|a|≤A, |b|≤B} (π_{E(a,b)}^{r}(x))^{2} = (C_{r}π_{1/2}(x))^{2} + O (\frac{x}{log^{c} x} + (\frac{1}{A} + \frac{1}{B} ) x^{3} + \frac{x^{5}}{AB} ) .

Replacing in (36), and using (34), this completes the proof of Theorem 1.4 □

Acknowledgments: This work was done when the second author was visiting the Centre Interuniversitaire en Calcul Mathématique Algébrique (CICMA). He thanks CICMA for its hospitality and financial support. Both authors would like to thank A. Granville for helpful discussions, and for mentioning to us the result of Theorem 1.4. We also thank W. D. Banks and H. Kisilevsky for helpful discussions.

References

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

[2] H. Davenport, Multiplicative Number Theory, Springer-Verlag, 1980.

[3] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.

[4] W. Duke, Rational elliptic curves with no exceptional primes, C.R. Acad. Sci. Paris 325 (1997), 813–818.

[5] N. D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over ℚ, Invent. Math. 89 (1987), 561-567.

[6] E. Fouvry and M. R. Murty, On the distribution of supersingular primes, Can. J. Math. 48 (1996), 81-104.

[7] E. Fouvry and E. Ullmo, personal communication.

[8] H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, London, 1974.

[9] G. H. Hardy and J. E. Littlewood, Some problems of partitio numenorum III, Acta. Math. 44 (1923), 1-70.

[10] L.-K. Hua, Introduction to number theory, Springer-Verlag, 1982.

[11] S. Lang and H. Trotter, Frobenius distributions in GL_{2}-extensions, Springer-Verlag Lecture Notes in Mathematics 504, 1976.

[12] M. R. Murty and V. M. Murty, Non-vanishing of L-functions and applications, Birkhauser, 1997.

[13] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Inventiones Math., 15 (1972), 259-331.

Concordia University, Department of Mathematics, 1455 de Maisonneuve Blvd. West, Montréal, Quebec H3G 1M8, CANADA E–mail: chantal@discrete.concordia.ca Università degli Studi Roma Tre, Dipartimento di Matematica, Largo S. L. Murialdo, 1 , 00146, Roma, ITALIA E–mail: pappa@mat.uniroma3.it