SUPERSINGULAR REDUCTION OF DRINFELD MODULES

CHANTAL DAVID

Abstract. We present here an improvement of the results obtained by Brown in [1], on lower bounds for the number of supersingular primes of a rank 2 Drinfeld module ϕ over 𝔽_{q}(T). The improvement follows from a theorem giving a bound on certain supersingular primes of ϕ. This is similar to the case of elliptic curves, but the results obtained here for Drinfeld modules hold unconditionally, the Riemann hypothesis being known to hold over function fields.

1. Introduction.

Let R = 𝔽_{q}[T] and F = 𝔽_{q}(T) be, respectively, the ring of polynomials and the field of rational functions in one indeterminate over the finite field 𝔽_{q}, q a power of an odd prime.

Let ϕ be a rank 2 Drinfeld R-module defined over 𝔽_{q}(T). For p ∈ 𝔽_{q}[T] a prime of good reduction for ϕ, let ϕ_{p} denote the reduction of ϕ at p; ϕ_{p} is then a rank 2 Drinfeld R-module over the finite field

𝔽_{(p)} = 𝔽_{q}[T]/(p) ≃ 𝔽_{q^{deg p}}.

Depending on the structure of the endomorphism ring End(ϕ_{p}), the Drinfeld module ϕ_{p} is said to be ordinary or supersingular (see Section 2). Let

P (x) = { monic prime polynomials p ∈ 𝔽_{q}[T] : |p| = q^{deg p} ≤ x } .

In this paper, we investigate the asymptotic behavior of

π_{ϕ}(x) = # { p ∈ P (x) : ϕ has good supersingular reduction at p } .

A quadratic extension K/F is said to be imaginary if the prime at infinity does not split in K/F. We say that a Drinfeld module ϕ has complex multiplication by an order O of a quadratic imaginary extension K/F if O ≃ End(ϕ). In this case, by Deuring’s criterion (Theorem 2.2) and Čebotarev density theorem (Theorem 5.1), we get that

Partially supported by CICMA.

(i) when K ≠ F (\sqrt{u}), u a non-square in 𝔽_{q}^{∗},

π_{ϕ}(x) ∼ \frac{1}{2} \frac{\overline{x}}{log_{q} \overline{x}},

where \overline{x} is the largest power of q smaller than or equal to x; (ii) when K = F (\sqrt{u}), u a non-square in 𝔽_{q}^{∗},

π_{ϕ}(x) ∼ \frac{\overline{x}}{log_{q} \overline{x}}

where \overline{x} is the largest odd power of q smaller than or equal to x. When ϕ does not have complex multiplication, by analogy with the

case of elliptic curves over ℚ, it is natural to ask if

Conjecture 1.1 (Lang-Trotter conjecture for Drinfeld modules). There is a constant C_{ϕ} such that

π_{ϕ}(x) = C_{ϕ} \frac{x^{1/2}}{log_{q} x} + \underline{o} (\frac{x^{1/2}}{log_{q} x})

when x → ∞.

We show in [2] that, on average, this conjecture holds for each degree k.

Brown showed in [1] that if ϕ is a non-exceptional (see Section 3) rank 2 Drinfeld R-module, then

π_{ϕ}(x) ≫ log log log x  (1.1)

when x → ∞. We show in this paper that this bound can be improved to

π_{ϕ}(x) ≫ log log x  (1.2)

when x → ∞. This is similar to the case of elliptic curves, where conditional results, assuming the generalised Riemman hypothesis for L-series over number fields, were proved by Murty [15] and Elkies [7, 8]. Recently, some unconditional results were obtained by Fouvry and Murty [10]. It should be noted that the results obtained here are unconditional, since the Riemann hypothesis is true over function fields.

2. Supersingular Drinfeld modules.

For the basic definitions and properties of Drinfeld modules, see for example [11] or [13]. We just recall here some facts about supersingular reduction of Drinfeld modules.

Let ϕ = ϕ(a, b) be the rank 2 Drinfeld R-module of generic characteristic over 𝔽_{q}(T) with equation

ϕ(T) = T + af + bf^{2}, a, b ∈ 𝔽_{q}[T], b ≠ 0,  (2.1)

where f is the Frobenius automorphism of 𝔽_{q}

f : x ⟶ x^{q}.

The j-invariant of ϕ is then

j_{ϕ} = \frac{a^{q+1}}{b} .

We will consider only Drinfeld R-modules of rank 2, and we simply call them Drinfeld modules.

Let p be a monic prime polynomial of 𝔽_{q}[T]. If p |̸b, then p is a prime of good reduction of ϕ, and the reduction of ϕ at p is the Drinfeld module ϕ_{p} with equation

ϕ_{p}(T) = T + (a mod p) f + (b mod p) f^{2} = T + a′ f + b′f^{2},

where

a′, b′ ∈ 𝔽_{(p)} = 𝔽_{q}[T]/(p) ≃ 𝔽_{q^{deg p}}, and b′ ≠ 0.

Definition 2.1. A Drinfeld module ψ defined over 𝔽_{(p)} = 𝔽_{q}[T]/(p) is supersingular if the following equivalent conditions hold:

(i) ψ(p) is purely inseparable, i.e. ψ(p) = f^{h} for some integer h; (ii) There are no p-torsion points, i.e. Ker ψ(p) = 0; (iii) End\overline{𝔽_{(p)}}(ψ) is a non-commutative ring; (iv) End\overline{𝔽_{(p)}}(ψ) is a maximal order in the unique quaternion algebra over F which is ramified exactly at p and ∞.

The equivalence of these properties is proved in [11, Thm 5.3] and [12, Proposition 4.1]. A Drinfeld module which is not supersingular is said to be ordinary. For p a prime of good reduction of ϕ, we say p is a supersingular (respectively ordinary) prime for ϕ if ϕ_{p} is a supersingular (respectively ordinary) Drinfeld module over 𝔽_{(p)}.

The following theorem is the statement of the criterion of Deuring [3], which allows the detection of supersingular primes of elliptic curves, in the context of Drinfeld modules.

Theorem 2.2 (Deuring’s Criterion). [1, Lemma 2.9.3] Let ϕ(a, b) be a Drinfeld module as defined by (2.1). Let p be a prime of good reduction of ϕ. Then ϕ has supersingular reduction at p if and only if there exists an order O of a quadratic imaginary extension K/F such that

(i) P_{O}(j_{ϕ}) ≡ 0 mod p;

(ii) p is inert or ramified in K.

In the previous theorem, P_{O}(x) denotes the monic polynomial whose roots are exactly the j-invariants of Drinfeld modules defined over C with complex multiplication by the order O of the quadratic imaginary extension K/F ; C is the completion at the infinite prime of the algebraic closure \overline{F}_{∞}, and plays the role of the complex plane for Drinfeld modules over 𝔽_{q}(T). Let K = F (\sqrt{d}), O = R[\sqrt{d}] for some square-free d ∈ 𝔽_{q}[T]. We then denote P_{O}(x) by P_{d}(x), and the roots of P_{d}(x) are then called the singular moduli associated to d. In [6], Drinfeld proved the existence of an analytic parametrisation for Drinfeld modules over C, given by the analytic function j(z). Using this, we can write

P_{d}(x) = ∏_{R[\sqrt{d}]⊆End(ϕ)}(x − j_{ϕ}) = ∏_{disc(z)=d^{[z]}}(x − j(z))  (2.2)

where the product ∏_{[z]} runs over equivalence classes [z] of discriminant d. P_{d}(x) is then a polynomial of degree h(d), the class number of F (\sqrt{d}), over R.

3. Supersingular primes of ϕ.

All the material of this section can be found in [1]. Fix some Drinfeld module ϕ as defined by (2.1). We also suppose that ϕ is a nonexceptional Drinfeld module. Let F_{∞} be the completion of F at the infinite prime. To simplify the discussion, we will consider only nonexceptional Drinfeld modules ϕ which satisfy one of the following

(i) q ≡ 3(4) and j_{ϕ}T is not a square in F_{∞} ,

(ii) q ≡ 1(4) and j_{ϕ} is not a square in F_{∞} ,

but our proof applies to all non-exceptional Drinfeld modules, as defined in [1, Definition 1.1.5] . Let P = {p_{1}, p_{2}, …, p_{n}} be a set of distinct monic prime polynomials of 𝔽_{q}[T], where each p_{i} is a either a prime of bad reduction or a supersingular prime of ϕ. We also suppose that P contains all primes of bad reduction.

Any f ∈ F^{∗} can be written uniquely in F_{∞} as f = ω(f) u T^{deg f}, where ω(f) ∈ 𝔽_{q}^{∗}, and u is an element of the valuation ring of F_{∞} such that u ≡ 1 mod ∞. We also define

sgn f = \{ \begin{matrix}1 & if ω(f) is a square in 𝔽_{q}^{∗} \\ −1 & if ω(f) is not a square in 𝔽_{q}^{∗}. \end{matrix}

Choose β ∈ 𝔽_{q}^{∗} such that

(−β)^{(q−1)/2} = −ω(j_{ϕ})^{(q−1)/2}.  (3.1)

Let π be a prime polynomial of 𝔽_{q}[T] such that (i) ω(π) = β; (ii) deg(π) is odd; (iii) (\frac{π}{p_{i}} ) = 1, 1 ≤ i ≤ n.

We say that π is an admissible prime associated to P and β. Let N_{π} be the numerator of P_{π}(j_{ϕ}), i.e.

N_{π} = P_{π}(j_{ϕ})b^{h(π)},  (3.2)

where j_{ϕ} = \frac{a}{b} and h(π) is the class number of F (\sqrt{π}). We will see in Section 5 that there is an infinite number of such primes. Then, since ϕ is non-exceptional and (3.1) holds, one can show that [1, (4.2.10)]

(\frac{π}{N_{π}} ) = −1.

There is then a prime p(π) ∈ 𝔽_{q}[T] dividing N_{π} such that

(\frac{π}{p(π)}) = −1.

By Deuring’s criterion (Theorem 2.2), p(π) is a supersingular prime for ϕ, and p(π) cannot be one of the p_{i}’s, since π is a quadratic nonresidue mod p(π). This shows that there exist an infinite number of supersingular primes for ϕ.

Finally, it is clear that

|p(π)| ≤ |N_{π}|.  (3.3)
4. A bound for N_{d}.

We prove in this section that

Theorem 4.1. Let d ∈ 𝔽_{q}[T]^{∗} be a square-free polynomial such that F (\sqrt{d})/F is a quadratic imaginary extension, and let ϕ be a Drinfeld module as given by (2.1). Let N_{d} be the numerator P_{d}(j_{ϕ}). Then

N_{d} ≪ exp (C |d|^{1/2} log^{2} |d|),

where the C and the O-constant depend only on q and ϕ.

As in the classical case of elliptic curves over ℚ, we can write the singular moduli associated to d as j(z_{i}), where for 1 ≤ i ≤ h(d)

z_{i} = \frac{−b_{i} ± \sqrt{b_{i}^{2} − 4a_{i}c_{i}}}{2a_{i}} , b_{i}^{2} − 4a_{i}c_{i} = d  (4.1)

with

a_{i}, b_{i}, c_{i} ∈ 𝔽_{q}[T], and |b_{i}| < |a_{i}| ≤ |c_{i}|.  (4.2)

It can also be shown that

|j(z_{i})| ≪ exp (C_{q}|z_{i}|) = exp (C_{q} \frac{|d|^{1/2}}{|a_{i}|} )

where C_{q} depends only on q ([1, Lemma 2.8.2]). Choose

C ≥ max (C_{q}, log |j_{ϕ}|).

Then

P_{d}(j_{ϕ}) = ∏_{i=1}^{h(d)} (j_{ϕ} − j(z_{i})) ≪ 2^{h(d)} exp (C|d|^{1/2} ∑_{i=1}^{h(d)} \frac{1}{|a_{i}|}),

so we have to bound h(d) and ∑_{i=1}^{h(d)} \frac{1}{|a_{i}|}.

Lemma 4.2.

h(d) ≪ |d|^{1/2} log |d|

where the O-constant depends only on q.

Proof: Define the L-series

L(s, χ_{d}) = ∑_{a} χ_{d}(a) |a|^{−s},

where

χ_{d}(a) = (\frac{d}{a})

and where the sum is taken over monic polynomials a ∈ 𝔽_{q}[T]. By quadratic reciprocity, one shows that L(s, χ_{d}) is the finite sum

L(s, χ_{d}) = ∑_{deg a<deg d} (\frac{d}{a}) |a|^{−s},

which gives

L(1, χ_{d}) ≪ log |d|.  (4.3)

We now use the class number formula for function fields. Since F (\sqrt{d}) is a quadratic imaginary extension of F, we only have to consider the following cases

(i) when deg d is odd, L(1, χ_{d}) = \begin{matrix}√ & \overline{q} \\ √ & \overline{|d|} \end{matrix} h(d);

(ii) when deg d is even, and sgn d = −1, L(1, χ_{d}) = \frac{q+1}{2\sqrt{|d|}} h(d).

With (4.3), this completes the proof of Lemma 4.2.

Lemma 4.3.

∑_{i=1}^{h(d)} \frac{1}{|a_{i}|} ≪ log^{2} |d|

where the O-constant depends only on q.

Proof: The number of z_{i} with a_{i} = a, where a is some fixed value, is smaller than the number of solutions to x^{2} ≡ d mod a. Also, for any 1 ≤ i ≤ h(d), |a_{i}| ≤ |d|^{1/2} by (4.2). Then

∑_{i=1}^{h(d)} \frac{1}{|a_{i}|} ≤ ∑_{|a|≤|d|^{1/2}} \frac{2^{δ(a)}}{|a|} ,

where the sum runs over monic polynomials a ∈ A, and δ(a) is the number of monic prime divisors of a. But

∑_{|a|≤|d|^{1/2}} \frac{2^{δ(a)}}{|a|} ≤ ∑_{|a|≤|d|^{1/2}} ∑_{|b|≤|d|^{1/2}} \frac{1}{|a||b|} ≪ log^{2} |d|.

This completes the proof of Theorem 4.1, since

N_{d} = Num P_{d}(j_{ϕ}) = b^{h(d)} P_{d}(j_{ϕ})

where j_{ϕ} = \frac{a}{b} with a, b ∈ 𝔽_{q}[T]. Remark: This improves the bound given in [1], which is N_{d} ≪ exp (|d|^{4}).

5. Distribution of primes.

Let E be a finite Galois extension of F = 𝔽_{q}(T), with Galois group G = Gal(E/F), and genus g_{E}. Let L be the algebraic closure of 𝔽_{q} in E, and

N = [L : 𝔽_{q}];
M = [E : LF].

For C a conjugacy class in G, we denote

P_{k}(F) = {p monic prime ∈ 𝔽_{q}[T] : deg p = k}
P_{k}(E/F; C) = \{p ∈ P_{k}(F) : p is unramified in E/F and (\frac{E/F}{p} ) = C\} .

Then, since the Riemann hypothesis is true over function fields, we get

Theorem 5.1 (Čebotarev Density Theorem). ([9, Proposition 5.16]) Let f be the Frobenius element of 𝔽_{q}, i.e.

f : x ⟶ x^{q}.

Let a be a positive integer such that

res_{L} τ = res_{L} f^{a} for any τ ∈ C.

Then, for k ≢ a mod N,

|P_{k}(E/F; C)| = 0

and for k ≡ a mod N

| |P_{k}(E/F; C)| − \frac{|C|}{M} \frac{q^{k}}{k} | ≪ |C|q^{k/2} (1 + \frac{g_{E}}{kM} ) ,

where the O-constant is absolute.

Remark: This is in fact stronger than the statement of Proposition 5.16 of [9], since we are in the particular case of a Galois extension over 𝔽_{q}(T).

We now apply the last theorem to count the number of admissible primes π, as defined above. Let P = {p_{1}, …, p_{n}} be a set of monic prime polynomials of 𝔽_{q}[T], and β ∈ 𝔽_{q}^{∗}. Let

K_{n} = F (\sqrt{α}, \sqrt{p_{1}}, …, \sqrt{p_{n}}) ,

with α a non-square of 𝔽_{q}^{∗}. Then the algebraic closure of 𝔽_{q} in K_{n} is L = 𝔽_{q^{2}} and

N = [𝔽_{q^{2}} : 𝔽_{q}] = 2
M = [K_{n} : 𝔽_{q^{2}}(T)] = 2^{n}
Gal(K_{n}/F) ≃ (ℤ/2ℤ)^{n+1}.

Let π_{P,β}^{k} be the number of prime polynomials π of 𝔽_{q}[T] such that (i) π is unramified in K_{n}/F; (ii) π is an admissible prime associated to P and β; (iii) deg π = k.

Lemma 5.2. There exists a unique g(P, β) ∈ Gal (K_{n}/F) such that for a prime polynomial π ∈ 𝔽_{q}[T], unramified in K_{n} with ω(π) = β, the following are equivalent:

(i) deg π is odd, and for 1 ≤ i ≤ n, (\frac{π}{p_{i}} ) = 1;
(ii) (\frac{K_{n}/F}{π} ) = g(P, β).

Proof: This is a consequence of quadratic reciprocity over function fields, which reads as

(\frac{α}{π}) = (−1)^{deg π}

for α a non-square of 𝔽_{q}[T], and

(\frac{p_{i}}{π} ) = (\frac{π}{p_{i}} ) (−1)^{\frac{q−1}{2} deg p_{i} deg π}(sgn π)^{deg p_{i}}(sgn p_{i})^{deg π}

for p_{i} a polynomial of 𝔽_{q}[T] with deg p_{i} ≥ 1.

Applying the Čebotarev density theorem to the extension K_{n}/F, with C = {g(P, β)}, we obtain

π_{P,β}^{k} = π_{C}(K_{n}/F; k) = 0 when k is even,

and

π_{P,β}^{k} = π_{C}(K_{n}/F; k) = \frac{1}{2^{n}} \frac{q^{k}}{k} + O (\frac{q^{k/2}}{k} (k + ∑_{i=1}^{n} deg p_{i})) when k is odd ,

(5.1)

with an absolute O-constant.

We then show

Proposition 5.3. There is a sufficiently large absolute constant C such that for any odd k such that

q^{k} ≥ C 4^{n} (∑_{i=1}^{n} deg p_{i})^{2} ,  (5.2)

we have

π_{P,β}^{k} ≫ \frac{1}{2^{n}} \frac{q^{k}}{k} .

Proof: For k odd such that (5.2) holds, we can choose C big enough such that the error term in (5.1) is bounded by a fraction of the main term. The result follows.

We also define

π_{P,β}(x) = ∑_{q^{k}≤x} π_{P,β}^{k} ,

and rewrite last proposition as

Proposition 5.4. There is a sufficiently large absolute constant C such that for

x ≥ C q^{2} 4^{n} (∑_{i=1}^{n} deg p_{i})^{2} ,  (5.3)

we have

π_{P,β}(x) ≫_{q} \frac{1}{2^{n}} \frac{x}{log x}.

Proof: Let l be the unique odd integer such that q^{l} ≤ x < q^{l+2}. Then, by the last proposition,

π_{P,β}(x) ≥ π_{P,β}^{l} ≫ \frac{1}{2^{n}} \frac{q^{l}}{l} ≥ \frac{log q}{q^{2}} \frac{1}{2^{n}} \frac{x}{log x}

when q^{l} ≥ C 4^{n} (∑_{i=1}^{n} deg p_{i})^{2}, which is insured by (5.3).

Remark: With Proposition 5.4 and Theorem 4.1, it follows by induction that

π_{ϕ}(x) ≫ \sqrt{log log x},

which is a first improvement of (1.1).

6. Supersingular primes of Drinfeld modules with two complex multiplications.

Theorem 6.1. Let ϕ be a Drinfeld module defined over 𝔽_{q}(T). Let d_{1}, d_{2} ∈ 𝔽_{q}[T] be two distinct fundamental discriminants, and let p be a prime of good reduction of ϕ such that

p | P_{d_{1}}(j_{ϕ}) and p | P_{d_{2}}(j_{ϕ}).

Then deg p ≤ deg d_{1}d_{2}.

Proof: For at least one of d_{1} or d_{2} of even degree, the result was proved by Dorman (see [4] and [5]), as a corollary of the explicit factorization of

J(d_{1}, d_{2}) = ∏_{disc(τ_{2})=d_{2}^{disc(τ_{1})=d_{1}}^{[τ_{1}] , [τ_{2}]}} (j(τ_{1}) − j(τ_{2}))^{(q^{2}−1/ω_{1}ω_{2})} ,

where ω_{1} and ω_{2} are the number of roots of unity in the quadratic orders of discriminant d_{1} and d_{2} respectively. This is not sufficient here, since all admissible primes π as chosen in Section 3 are odd degree polynomials.

In general, one can prove Theorem 6.1 following the ideas of Kaneko [14] who proved a similar result for elliptic curves. By hypothesis, ϕ has good supersingular reduction ϕ_{p} at p. Then, End(ϕ_{p}) is a maximal order in H, the unique quaternion algebra over F which ramifies exactly at p and ∞, and

R [\sqrt{d_{1}}] ⊆ End(ϕ_{p})
R [\sqrt{d_{2}}] ⊆ End(ϕ_{p}).

Let α_{i} be the image of \sqrt{d_{i}} in End(ϕ_{p}), i = 1, 2, and consider the R-submodule of End(ϕ_{p})

L = R + Rα_{1} + Rα_{2} + Rα_{1}α_{2}.

This is an order in H; its reduced discriminant d(L) is given by

d(L)^{2} = R ( det tr(μ_{i}μ_{j}) ) ,

where

μ_{1} = 1, μ_{2} = α_{1}, μ_{3} = α_{3}, μ_{4} = α_{1}α_{2}

[16, Chapter 1, Lemma 4.7]. One computes

det tr(μ_{i}μ_{j}) = − (4d_{1}d_{2} − s^{2})^{2} ,

where s = tr(α_{1}α_{2}). Then, since L is contained in the maximal order End(ϕ_{p})

p^{2} | − (4d_{1}d_{2} − s^{2})^{2} ,

which gives

|p| ≤ max (|d_{1}d_{2}|, |s^{2}|).

If

|s^{2}| > |d_{1}d_{2}|,

h = tr^{2}(α_{1}α_{2}) − 4n(α_{1}α_{2}) is a polynomial of even degree with leading coefficient in (𝔽_{q}^{∗})^{2}, i.e. ∞ splits in the quadratic extension F (\sqrt{h})/F. Since F (\sqrt{h}) ⊆ H, this is impossible. Then

|s^{2}| ≤ |d_{1}d_{2}|

and |p| ≤ |d_{1}d_{2}| ⇔ deg p ≤ deg d_{1}d_{2}.

7. Lower bound for π_{ϕ}(x).

Given P = {p_{1}, …, p_{n}}, we choose admissible primes π associated to p_{1}, …, p_{n}, as in Section 3. For each of these admissible π, we get a new supersingular prime p(π) | N_{π}, and by construction p(π) ∉ P = {p_{1}, …, p_{n}}. If we can also insure that they are all distinct, we can use Proposition 5.4 to count the new supersingular primes p(π). Theorem 6.1 gives the criterion which insures that the primes p(π) are distinct.

Theorem 7.1. Let ϕ a non-exceptional Drinfeld module, as given by (2.1). Then, there exists an infinite sequence x_{1}, x_{2}, … with x_{m} → ∞ and a positive constant K such that

π_{ϕ}(x_{m}) ≥ K log x_{m}.

Furthermore, log x_{m+1} ≪ x_{m}^{1/4} log^{2} x_{m}.

Proof: Let K be any positive constant such that

K < \frac{1}{2 log 4}.  (7.1)

It suffices to show that for x large enough with

π_{ϕ}(x) < K log x,

there exists x′ > x with π_{ϕ}(x′) ≥ K log x′, and log x′ ≪ x^{1/4} log^{2} x.

Then, let x be such that

n = π_{ϕ}(x) < K log x,

and let p_{1}, …, p_{n} be the n supersingular primes of ϕ with norm smaller than or equal to x. Then,

4^{n} (∑_{i=1}^{n} deg p_{i})^{2} ≤ 4^{n}n^{2}(log_{q} x)^{2} < x^{K log 4}K^{2} log^{2} x log_{q}^{2} x,

and by (7.1), we obtain

\sqrt{x} ≥ Cq^{2}4^{n} (∑_{i=1}^{n} deg p_{i})^{2}

for x large enough. Then, by Proposition 5.4,

π_{P,β}(\sqrt{x}) ≫ \frac{1}{2^{n}} \begin{matrix}√ & \overline{x} \\ log & \sqrt{x} \end{matrix}.

For each of those admissible π with norm smaller than or equal to \sqrt{x}, we get a new supersingular prime p(π). Suppose that p = p(π_{1}) = p(π_{2}) for some π_{1} ≠ π_{2}. Then by Theorem 6.1, |p| ≤ x, which is impossible since p(π) ∉ P = {p_{1}, …, p_{n}} . Let x′ = max |p(π)|. Then by Theorem 4.1,

log x′ ≪ max (C |π|^{1/2} log^{2} |π|) ≪ x^{1/4} log^{2} x.

Finally, since n < K log x,

π_{ϕ}(x′) ≥ π_{P,β}(\sqrt{x}) ≫ \frac{1}{2^{n}} \begin{matrix}√ & \overline{x} \\ log & \sqrt{x} \end{matrix} > \frac{x^{\frac{1}{2} −K log 2}}{log \sqrt{x}} .

Then for x large enough, using 7.1, we get

π_{ϕ}(x′) > K log x′.

This completes the proof of the Theorem.

Corollary 7.2.

π_{ϕ}(x) = Ω (log x)

and

π_{ϕ}(x) ≫ log log x.

Proof: Corollary 7.2 follows applying Theorem 7.1 to find x_{m} ≤ x ≤ x_{m+1}. Then

π_{ϕ}(x) ≥ K log x_{m} ≫ (log log x_{m+1}) ≥ (log log x).

Acknowledgements. I would like to thank Ram Murty who suggested me this problem, and David Dorman for helpful discussions related to this work.

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CICMA, Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montréal, Québec, H3G 1M8, CANADA

E-mail address: chantal@abacus.concordia.ca