On the 2^{n} divisibility of the Fourier coefficients of J_{q} functions and the Atkin conjecture for p = 2

Shigeki Akiyama

College of General Education, Niigata University Niigata, 950-21, JAPAN

§1. Introduction

Let f be the holomorphic modular form of weight 2k, which is a normalized common eigenform with respect to Hecke operators. Then it is well known that the Fourier coefficient τ(n) of f satisfies the equation

τ(np) − τ(n)τ(p) + p^{2k−1}τ(n/p) = 0,  (1)

for any prime p and any positive integer n. Here τ(n/p) is defined to be zero when n/p is not an integer. In [2] and [3], Atkin made a similar conjecture for a modular \underline{function}:

Conjecture (Atkin).

Let j(z) be the modular invariant:

j(z) = ∑_{n≥−1} c(n)x_{3}^{n} = x_{3}^{−1} + 744 + 196884x_{3} + · · · · ·· ,

where x_{3} = exp(2π\sqrt{−1}z). Let p ≤ 23 be a fixed prime and l be a prime other than p. For any positive integer α, put a_{α}(n) = c(np^{α})/c(p^{α}). Then the following congruences hold

a_{α}(nl) − a_{α}(n)a_{α}(l) + l^{−1}a_{α}(n/l) ≡ 0 (mod p^{α}),  (2)
a_{α}(np) − a_{α}(n)a_{α}(p) ≡ 0 (mod p^{α}).  (3)

Remark 1.

Atkin asserted in [2] that a_{α}(n) are in ℚ ∩ ℤ_{p}. The author knows the proof of this fact only for the case p = 2, 3 and 13. Atkin also announced in [2], that he had proved the conjecture for p = 2, 3, 5, 7 and 13. But we cannot find his proofs in the literature. In [4], Atkin and O’Brien showed the congruence (3) for p = 13. Koike showed the congruence (2) for p = 13 in [8] and completed the proof for p = 13. Koike’s work suggests us that, at least, Atkin had been in the right direction to the proof for p = 13. There seems no published proof for other primes.

Remark 2.

This conjecture is a p-adic version of (1). Thus the conjecture gives us the starting point of the vast theory of p-adic modular forms and p-adic Hecke operators (see Katz [7], Dwork [5], Serre [11]).

In this article, we will prove the Atkin conjecture for p = 2 in more precise form:

Theorem 1.

Let α be a positive integer and a_{α}(n) = c(2^{α}n)/c(2^{α}). Then we have, for any odd prime l,

a_{α}(nl) − a_{α}(n)a_{α}(l) + l^{−1}a_{α}(n/l) ≡ 0 (mod 2^{4α+4}),  (4)
a_{α}(2n) − a_{α}(n)a_{α}(2) ≡ 0 (mod 2^{4α+7}).  (5)

Remark 3.

Atkin already noticed in [2], that when p ≤ 5, the exponent of the prime p of the congruence (2) and (3) is not best possible. As for this exponent of (4) and (5), see the last part of §4 and Remark 7 in §5.

The original purpose of the study is to prove the 2^{n} divisibility property of Hecke’s absolute invariant such as the conjecture (A) in §2. Fortunately, the author found that his argument was very close to the Atkin conjecture for p = 2. First, we will introduce Hecke’s absolute invariants.

§2. Hecke’s absolute invariants.

Let G_{q} be the Hecke group, which is a discontinuous subgroup of P SL_{2}(ℝ) generated by

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and ( \begin{matrix}1 & λ_{q} \\ 0 & 1 \end{matrix} ) ,

where λ_{q} = 2 cos(π/q) and q = 3, 4, · · · · ·, ∞. The standard fundamental domain of G_{q}, as a transformation group of a complex upper half plane ℍ, is given by

F_{q} = {z ∈ ℍ : |z| ≥ 1, |Re(z)| ≤ λ_{q}/2}.

Let J_{q} be the bijective conformal mapping from ” the half of F_{q}”, that is,

\{z ∈ ℍ : |z| ≥ 1, −λ_{q}/2 ≤ Re(z) ≤ 0\} ∪ \{\sqrt{−1}∞\},

to ℍ. This mapping is uniquely determined by the conditions:

J_{q}(− exp(−π\sqrt{−1}/q)) = 0, J_{q}(\sqrt{−1}) = 1, J_{q}(\sqrt{−1}∞) = ∞.

Using reflection principle repeatedly, we can define the value of J_{q} on ℍ and consider J_{q} as a mapping from ℍ to ℂ. From this construction we see that

J_{q}(γz) = J_{q}(z)

for any γ ∈ G_{q}. The automorphic function field of G_{q} is nothing but a rational function field generated by J_{q} over ℂ. This function J_{q} is called Hecke’s absolute invariant with respect to G_{q}. Since J_{q} is invariant under the transformation z → z+λ_{q}, we have the Fourier expansion of J_{q} at \sqrt{−1}∞:

J_{q}(z) = ∑_{n≥−1} A_{q}(n)x_{q}^{n},

where x_{q} = exp(2π\sqrt{−1}z/λ_{q}). J. Raleigh [10] showed

A_{q}(n) = r_{q}^{n}B_{q}(n),

where r_{q} ∈ ℝ, B_{q}(n) ∈ ℚ. The value r_{q} is determined up to rational multiples, so we put r_{q}^{−1} = A_{q}(−1) and B_{q}(−1) = 1. The actual value is

r_{q} = exp (2\frac{Γ′(1)}{Γ(1)} − \frac{Γ′(1/4 + 1/2q)}{Γ(1/4 + 1/2q)} − \frac{Γ′(1/4 − 1/2q)}{Γ(1/4 − 1/2q)} − \frac{1}{cos(π/q)}) ,

where Γ(s) is the gamma function. Further, J. Wolfart [12] showed that r_{q} is algebraic if and only if q = 3, 4, 6, ∞. So we treat only the case q = 3, 4, 6, ∞ in the following. Let j_{q}(z) = r_{q}J_{q}(z) then j_{q}(z) is contained in ℤ[x_{q}, x_{q}^{−1}]. Put

j_{q}(z) = ∑_{n≥−1} c_{q}(n)x_{q}^{n},

where c_{q}(n) ∈ ℤ and c_{q}(−1) = 1. Consider the case q = 3. Then G_{3} = P SL(2, ℤ) and j_{3}(z) coincides with the modular invariant j(z) appeared in the introduction. From now on, write c(n) instead of c_{3}(n). The first few values of c_{q}(n) are found in Table 1.

The author proposed conjectures concerning c_{q}(n) as a rational function of q in [1]. As a special case, we see:

Conjecture.

For all integer n, we have

ord_{2}(c(n)) = ord_{2}(c_{4}(n)) = ord_{2}(c_{∞}(n)),

ord_{3}(c(n)) = ord_{3}(c_{6}(n)).

For the later convenience, we call the statement

ord_{2}(c(n)) = ord_{2}(c_{∞}(n))

to be conjecture (A). In the next section, we will take a close look at this conjecture (A).

§3. O.Kolberg’s results and the conjecture (A).

Note that G_{∞} is a subgroup of index 3 of G_{3}, this is the reason why we first treat the conjecture (A) among others. ( The group G_{∞} is called theta group.) Thus there exist an algebraic relation between j and j_{∞}:

j(z) = (j_{∞}(z) − 2^{4})^{3}/j_{∞}(z).

Considering x_{∞}^{2} = x_{3}, we easily see that

c(n) ≡ c_{∞}(n) (mod 2),  (6)

which is our first knowledge about the conjecture (A). Using the famous λ-invariant, which is a generator of the automorphic function field with respect to principal congruence subgroup of level two, we can express j_{∞} as

j_{∞}(z) = −\frac{16}{λ(z)(λ(z) − 1)}.

Moreover, employing the expression of λ(z) by theta null series, we have

j_{∞}(z) = x_{∞}^{−1} ∏_{n≥1}(1 + x_{∞}^{2n−1})^{24}.

From this infinite product representation, we see

j_{∞} (\frac{z − 1}{2}) j_{∞} (\frac{z + 1}{2}) = j_{∞}(z).  (7)

And we also have

j_{∞} (\frac{z − 1}{2}) + j_{∞} (\frac{z + 1}{2}) = 48 − \frac{2^{12}}{j_{∞}(z)}.  (8)

Note that the left hand side of (8) is the result of the action of the double coset G_{∞}( \begin{matrix}1 & 1 \\ 0 & 2 \end{matrix} )G_{∞} as a Hecke operator on j_{∞}.

These two relations (7) and (8) seem to be fundamental. Define the action V by

f(z)|V = f (\frac{z − 1}{2}) + f (\frac{z + 1}{2}) .

Remark that j(2z) + j(z/2) is invariant under G_{∞} and

j(z)|T (2) = j(2z) + j(z/2) + j (\frac{z + 1}{2}) .

Here T (n) is the Hecke operator of degree n with respect to G_{3}. This shows that j (\frac{z+1}{2}) is contained in ℂ(j_{∞}). A precise calculation shows

j (\frac{z + 1}{2}) = −j_{∞}(z)^{−2}(j_{∞}(z) − 2^{8})^{3}  (9)
= −j_{∞}(z) + 2^{5} · 3 · 7 + j_{∞}|V^{2}.  (10)

Here the symbol j_{∞}|V^{2} means (j_{∞}|V)|V. The last formula (10) is verified by the repeated use of (7) and (8). Comparing coefficients of (10), we have, for n ≥ 1,

c(n) = (−1)^{n−1}c_{∞}(n) + 4c_{∞}(4n).  (11)

This formula, together with the product ( or theta ) representation of j_{∞}, gives us an easy alternative way of calculating c(n). The author does not know that someone had mentioned this formula (11) before. Using (11) , by the aid of computer calculations, our conjecture (A) is reduced to the following:

Conjecture (B).

For any positive integer n, we have ord_{2}(c_{∞}(2n)) ≥ 3 + ord_{2}(c_{∞}(n)).

Conjecture (B’)

For any positive integer n, we have ord_{2}(c(2n)) ≥ 3 + ord_{2}(c(n)).

Note that the conjecture (B) and the conjecture (B’) are equivalent, which is easily seen by (11). Numerical calculations suggest that the equality holds in both (B) and (B’) when n is even. Concerning the conjecture (B’), O. Kolberg [9] showed:

Proposition 1 (O.Kolberg).

For any positive integer α and odd integer n, we have

c(2^{α}n) ≡ −2^{3α+8}3^{α−1}σ_{7}(n) (mod 2^{3α+13}).  (12)

For any positive integer n,

c(8n + 1) ≡ 20σ_{7}(8n + 1) (mod 2^{7}),
c(8n + 3) ≡ σ_{1}(8n + 3)/2 (mod 2^{3}),
c(8n + 5) ≡ −12σ_{7}(8n + 5) (mod 2^{8}),

where σ_{s}(n) = ∑_{d|n} d^{s}. The value of c(8n + 7) becomes both even and odd infinitely often.

This proposition implies the validity of the conjecture (B) and (A) for a certain type of n. For example, if

n = 2^{β}m, m ≡ 1 (mod 8), σ_{7}(m) ≢ 0 (mod 2^{5}),  (13)

where β is any non negative integer. Then

ord_{2}(c(2n)) ≥ 3 + ord_{2}(c(n))

holds. Using (11), we see that

ord_{2}(c_{∞}(2n)) ≥ 3 + ord_{2}(c_{∞}(n)).

Thus again by (11),

ord_{2}(c(n)) = ord_{2}(c_{∞}(n))

holds for every n of type (13). To prove Proposition 1, O. Kolberg explicitly calculated j_{∞}^{−n}|V^{k} for each positive integer n and k. And this calculation is crucial in proving the Atkin conjecture for p = 2 in §4.

Remark 4.

Define the operator U(2) by

∑ a(n)x_{3}^{n}|U(2) = ∑ a(2n)x_{3}^{n}.

By using the results of Koike [8], there exists a unique modular cusp form F of weight 2^{t−1} such that

j(z)|U(2)^{m} − 744 ≡ F (z) (mod 2^{t}).  (14)

It is well known that the space of modular cusp forms is decomposed into common eigenspaces with respect to Hecke operators. Thus

F (z) = ∑_{i} F_{i}(z),

and each F_{i} is a common eigenfunction of eigenvalue λ_{i}. If t ≥ 3 then the action of Hecke operator of degree two and of weight 2^{t−1} coincides with that of U(2) modulo 2^{t}. K. Hatada showed in [6],

λ_{i} ≡ 0 (mod 8).

Thus we have

j(z)|U(2)^{m+1} − 744 ≡ ∑ λ_{i}F_{i}(z) (mod 2^{t}).  (15)

Comparing (14) and (15), we see that the conjecture (B’) seems to be reasonable.

§4. Proof of the Atkin conjecture for p = 2.

In this section, we prove Theorem 1 cited in the introduction in a slightly stronger form. Our discussion is almost the same as in the proof of Koike [8]. So the precise description will be omitted if not necessary. We also use the idea of Atkin-O’Brien [4] and the results of O. Kolberg [9].

Let S_{k} be the space of modular cusp forms of weight k and S(α, λ) be the ℤ submodule of S_{λ+2^{α−1}} whose elements have integer Fourier coefficients in the expansion at the cusp \sqrt{−1}∞. Denote by d(α, λ) the dimension of S_{λ+2^{α−1}}. Then S(α, λ) has rank d(α, λ). Let α′ > α ≥ 3 be two positive integers. Then for each f ∈ S(α, λ), there exists f′ ∈ S(α′, λ) such that f′ ≡ f (mod 2^{α}), where the symbol ≡ means that the corresponding Fourier coefficients are congruent modulo 2^{α}. Thus there exists a system of free basis {f_{α,i}^{(λ)}}_{i=1}^{d(α,λ)} of S(α, λ) such that

f_{α,i}^{(λ)} ≡ f_{α′,i}^{(λ)} (mod 2^{α})

for any α′ > α ≥ 3. Let f̃_{i}^{(λ)} be the 2-adic limit of f_{α,i}^{(λ)} when α tends to infinity. Define by S(λ) the set consisting of all elements ∑ a_{i}f̃_{i}^{(λ)} such that a_{i} ∈ ℚ_{2} and there are only finitely many a_{i}’s for which ord_{2}(a_{i}) < t for any positive integer t. This space is called 2-adic Banach space admitting orthonormal basis {f̃_{i}^{(λ)}}_{i} over ℚ_{2}. Itis known that S(0) admits orthonormal basis {j(z)^{−i}}_{i=1}^{∞}. Let l be an odd prime. The 2-adic Hecke operator Ũ_{λ}(2) and T̃_{λ}(l) acting on S(λ) is defined by

f̃|Ũ_{λ}(2) = ∑ A(2n)x_{3}^{n},
f̃|T̃_{λ}(l) = ∑{A(nl) + l^{λ−1}A(n/l)}x_{3}^{n},

for f̃ = ∑ A(n)x_{3}^{n} . We define A(n/l) to be zero when n is not a multiple of l.

Proposition 2.

The space S(0) admits orthonormal basis {j_{∞}(2z + 1)^{−i}}_{i=1}^{∞}.

Proof) Let j^{(0)}(z) = j(z) − 744 and j_{∞}^{(0)}(z) = j_{∞}(z) − 24. Then by (11),

j^{(0)}(z) = −j_{∞}^{(0)}(2z + 1) + 4j_{∞}^{(0)}(2z + 1)|U(2)^{2}.

Thus j_{∞}^{(0)}(2z + 1) is 2-adically approximated by − ∑_{i=0}^{∞} 2^{2i}j^{(0)}(z)|U(2)^{2i}. By using Theorem 1 of Koike [8], j^{(0)}(z)|U(2)^{i} belongs to S(0) when i ≥ 1. From (8),

j_{∞}(2z + 1)^{−1} = 2^{−12}{48 − 2j_{∞}(2z + 1)|U(2)}.

This shows j_{∞}(2z+1)^{−1}, so j_{∞}(2z+1)^{−i} for i ≥ 1, belongs to S(0). By the congruence (6), we have

j_{∞}^{−i}(2z + 1) ≡ j(z)^{−i} (mod 2).

Recalling S(0) admits an orthonormal basis {j(z)^{−i}}_{i=1}^{∞}, we see the assertion.

Note that

ξ = −j_{∞}(2z + 1)^{−1} = x_{3} ∏_{n≥1}(1 + x_{3}^{n})^{24}.

In [9], O. Kolberg showed, for any positive integer k

ξ^{2k−1}|U(2) = ∑_{j=0}^{3k−2} 2^{8j+3} \frac{6k − 3}{2j + 1} ( \begin{matrix}3k + j − 2 \\ 2j \end{matrix} ) ξ^{k+j},  (16)

ξ^{2k}|U(2) = ∑_{j=0}^{3k} 2^{8j} \frac{3k}{3k + j} ( \begin{matrix}3k + j \\ 2j \end{matrix} ) ξ^{k+j}.  (17)

He derived these formulas by the elementary argument of trigonometric function. Koike’s proof of the Atkin conjecture for p = 13 essentially needs the same type of calculation due to Atkin-O’Brien [4].

Let F be the ℤ_{2} submodule of S(0) consisting of all elements:

∑_{r≥1} a_{r}ξ^{r},

where a_{r} ∈ ℤ_{2} and ord_{2}(a_{r}) ≥ 8(r − 1). We define the operator U′(2) = 2^{−3}U(2) on S(0). Then by (16) and (17), we see

2^{8r−8}ξ^{r}|U′(2) = ∑_{j≥r/2}^{2r} 2^{8j−8}c_{r,j}ξ^{j},  (18)

where c_{r,j} are integers for which ord_{2}(c_{r,j}) ≥ 4r − 4 and c_{1,1} is odd. This shows that U′(2) acts on F. Moreover, also by (18), the eigenfunction of Ũ′(2) = 2^{−3}Ũ_{0}(2) on F whose eigenvalue is a unit of ℤ_{2} exists uniquely up to ℚ_{2}^{×} multiples. For abbreviation, we call an eigenfunction with a unit eigenvalue to be a unit eigenfunction.

Remark 5.

For the case p = 13, the action of Ũ_{0}(13) and the uniqueness of the unit eigenfunction were considered on the \underline{whole} space. But in our case, we must restrict the action to F and consider Ũ′(2) instead of Ũ(2) to separate a unique unit eigenfunction.

Let M be the ℤ_{2} module generated by { f|U′(2)^{n}: f ∈ ℤ_{2}[j(z)], n ≥ 0 }. Then we have

Proposition 3.

For any f ∈ M, there exist a unique h ∈ ℤ_{2}[j(z)] and g ∈ F such that f = h+2^{8}g.

Proof) By the Theorem 1 of Koike [8], there exist a unique h ∈ ℤ_{2}[j(z)] and g ∈ S(0) such that f = h + g. Thus we have to show g ∈ 2^{8}F. As the operator U′(2) acts on F, it suffices to show the assertion on j(z)^{k}|U′(2) for k ≥ 1. From (9), we have

j(z) = −j_{∞}(2z + 1) + 3 · 2^{8} − 3 · 2^{16}j_{∞}(2z + 1)^{−1} + 2^{24}j_{∞}(2z + 1)^{−2}.

By the repeated use of this formula, it is sufficient to show that j_{∞}(2z + 1)^{k}|U′(2) can be decomposed into h_{1} ∈ ℤ_{2}[j_{∞}(2z + 1)] and g_{1} ∈ 2^{8}F. We proceed this proof by induction. By (8), we see

j_{∞}(2z + 1)|U′(2) − 3 = −2^{8}j_{∞}(2z + 1)^{−1} ∈ 2^{8}F.

So it is true for k = 1. Note that by (7),

j_{∞}(2z + 1)^{k+1}|U′(2) =

2^{4} (j_{∞}(2z + 1)^{k}|U′(2)) (j_{∞}(2z + 1)|U′(2)) − j_{∞}(2z + 1) (j_{∞}(2z + 1)^{k−1}|U′(2)) .

We easily complete the proof from this formula.

Proposition 4.

Let 2^{8}f ∈ M such that f = ∑_{n≥1} a(n)x_{3}^{n} with a(1) ≢ 0 (mod 2). Then there exists a constant k_{α} ∈ ℤ_{2}^{×} such that

f|Ũ′(2)^{α+1} ≡ k_{α}f|Ũ′(2)^{α} (mod 2^{4α+8}),

for each non negative integer α.

Proof) The idea of the proof is due to Atkin-O’Brien [4]. So precise calculations will be omitted. By Proposition 3, we see f|Ũ′(2)^{α} ∈ F. Thus f|Ũ′(2)^{α} is written in the form

∑_{j≥1} 2^{8(j−1)}d_{j}(α)ξ^{j},  (19)

where d_{j}(α) ∈ ℤ_{2}. Then by (18),

d_{j}(α + 1) = ∑_{r≥j/2}^{2j} d_{r}(α)c_{r,j},

and ord_{2}(c_{r,j}) ≥ 4r − 4. Put

γ_{ij}(α) = d_{j}(α + 1)d_{i}(α) − d_{j}(α)d_{i}(α + 1).

The key of the proof is the relation;

γ_{ij}(α + 1) = ∑_{k,l} γ_{kl}(α)c_{k,i}c_{l,j},

where integers k, l are taken over i/2 ≤ k ≤ 2i and j/2 ≤ l ≤ 2j. By induction, we have

ord_{2}(γ_{ij}(α)) ≥ 4α + 4 max{0, [(i + j − 5)/2]},

where [x] stands for the greatest integer not exceeding x. Especially we see

d_{j}(α + 1)d_{1}(α) ≡ d_{j}(α)d_{1}(α + 1) (mod 2^{4α}).

Our assumption implies d_{1}(α) ≢ 0 (mod 2). So we put k_{α} = d_{1}(α + 1)/d_{1}(α). Thus

d_{j}(α + 1) ≡ k_{α}d_{j}(α) (mod 2^{4α}).

Substitute this congruence into (19), we get

f|Ũ′(2)^{α+1} = ∑_{j≥1} 2^{8j−8}d_{j}(α + 1)ξ^{j}

≡ k_{α} ∑_{j≥1} 2^{8j−8}d_{j}(α)ξ^{j} = k_{α}f|Ũ′(2)^{α} (mod 2^{4α+8}).

Here we use the fact d_{1}(α + 1) = k_{α}d_{1}(α) to make bigger the exponent of 2 of the congruence. This completes the proof.

Remark 6.

Put a(n) = 2^{−11}c(2n), then f = ∑_{n≥1} a(n)x_{3}^{n} satisfies the assumption of Proposition 4. So we have

2^{−3α−11}c(2^{α+1}n) ≡ k_{α−1}2^{−3α−8}c(2^{α}n) (mod 2^{4(α−1)+8}),

for any positive integer n and α. Using this, we have

c(2^{α+1}n)c(2^{α}) ≡ c(2^{α}n)c(2^{α+1}) (mod 2^{10α+23}).

This implies the congruence (5) in Theorem 1, because

ord_{2}(c(2^{α})) = 3α + 8,

which is seen by (12).

Now we state our main theorem.

Theorem 2.

Let l be an odd prime. Let 2^{8}f be an element of M expanded as f = ∑_{n≥1} a(n)x_{3}^{n} with a(1) ≢ 0 (mod 2). Then we have, for any positive integer n,

b_{α}(nl) − b_{α}(n)b_{α}(l) + l^{−1}b_{α}(n/l) ≡ 0 (mod 2^{4α+8}),

where b_{α}(n) = a(2^{α}n)/a(2^{α}) and α is any non negative integer.

Proof) By Proposition 4, we have

f|Ũ′(2)^{α} = 2^{−3α} ∑_{n≥1} a(2^{α}n)x_{3}^{n} ≡ k_{α−1}k_{α−2} · · · · · k_{0}f (mod 2^{8}).

Especially, this shows ord_{2}(a(2^{α}n)) ≥ 3α and equality holds when n = 1. Thus b_{α}(n) ∈ ℤ_{2}. Put f_{α} = ∑ b_{α}(n)x_{3}^{n} then, by Proposition 4,

f_{α+1} ≡ f_{α} (mod 2^{4α+8}).

Let f′ be the 2-adic limit of {f_{α}}. Then, by definition

f|Ũ′(2) = κf,

where κ ∈ ℤ_{2}^{×} . Recalling that Ũ′(2) and T̃_{0}(l) are commutative, f′|T̃_{0}(l) is also an eigenfunction of Ũ′(2). By the uniqueness of the unit eigenfunction of Ũ′(2), we see f′|T̃_{0}(l) = b(l)f′ for some b(l). Considering this equality modulo 2^{4α+8}, we get the result.

From this theorem, we can show Theorem 1 as a corollary. To see this we have to specialize a(n) = 2^{−11}c(2n), as in Remark 6.

The exponent 4α + 8 of Theorem 2 can be replaced by 4α + 9. So we can also show that (4) holds modulo 2^{4α+5}. This little improvement follows from the fact that b(l) ≡ 0 (mod 2) in the above proof, which is shown by the precise argument similar to the proof of (12). We expect that the exponent of Theorem 2 will be improved to 4α + 15, as in Remark 7.

§5. Further conjectures.

By the aid of computer calculations, we will propose a more precise conjecture. To describe this, define

Ξ(n, α, p) = ( \begin{matrix}ord_{2}(a_{α}(np) − a_{α}(n)a_{α}(p) + p^{−1}a_{α}(n/p)), & for odd prime p, \\ ord_{2}(a_{α}(2n) − a_{α}(2)a_{α}(n)), & for p = 2, \end{matrix}

for any positive integer α and n ≥ 2. Recall that a_{α}(n) = c(2^{α}n)/c(2^{α}). Then we have

Conjecture (C).

There exists non negative integer valued function γ from the set of integers greater than 1 such that

Ξ(n, α, p) = 4α + 7 + γ(p) + γ(n).

For odd n, we have

γ(n) ≥ (1 + (2/n))/2 + (1 + (−1/n)) + 4,

where (·/n) is the Jacobi symbol. Equality holds when (2/n) = −1 and n is an odd prime. For even integers, we have

γ(2^{β}m) = 3(β − 1) + ord_{2}(σ_{1}(m)),

for any odd integer m and positive integer β.

Remark 7.

In special cases, the above conjecture (C) says that the exponent 4α + 7 of (5) is best possible and that of (4) can be replaced by 4α + 11, which is best possible. To see this, consider the case p = 2 or p ≡ 3 (mod 8) and n = 2m^{2} with an odd integer m.

The conjecture (C) gives us an impression that something interesting remain unrecognized in the Atkin conjecture for p = 2. We give first 100 values of γ(n) in Table 2.

Table 1.

\begin{matrix}n & q = 3 \\ 0 & 2^{3} · 3 · 31 \\ 1 & 2^{2} · 3^{3} · 1823 \\ 2 & 2^{11} · 5 · 2099 \\ 3 & 2 · 3^{5} · 5 · 355679 \\ 4 & 2^{14} · 3^{3} · 45767 \\ 5 & 2^{3} · 5^{2} · 2143 · 777421 \\ 6 & 2^{13} · 3^{6} · 11 · 13^{2} · 383 \\ 7 & 3^{3} · 5 · 7 · 271 · 174376673 \\ 8 & 2^{17} · 3 · 5^{3} · 199 · 41047 \\ 9 & 2^{2} · 3^{7} · 5 · 4723 · 15376021 \\ 10 & 2^{12} · 3^{5} · 5^{2} · 13^{2} · 5366467 \\ 11 & 2 · 3 · 11 · 13^{3} · 1008344102147 \\ 12 & 2^{16} · 3^{5} · 5 · 10980221089 \\ 13 & 2^{3} · 3^{3} · 5 · 23 · 112291 · 1746673133 \\ 14 & 2^{14} · 7 · 281 · 96457 · 8202479 \\ 15 & 3^{6} · 5^{2} · 7 · 1483 · 666739430527 \end{matrix}

\begin{matrix}n & q = 4 & q = 6 & q = ∞ \\ 0 & 2^{3} · 13 & 2 · 3 · 7 & 2^{3} · 3 \\ 1 & 2^{2} · 1093 & 3^{3} · 29 & 2^{2} · 3 · 23 \\ 2 & 2^{11} · 47 & 2^{5} · 271 & 2^{11} \\ 3 & 2 · 3^{3} · 22963 & 3^{5} · 269 & 2 · 3 · 1867 \\ 4 & 2^{14} · 653 & 2^{6} · 3^{3} · 5 · 43 & 2^{14} · 3 \\ 5 & 2^{3} · 5 · 13 · 41 · 3491 & 5 · 163 · 2137 & 2^{3} · 23003 \\ 6 & 2^{13} · 3^{3} · 1951 & 2^{5} · 3^{6} · 307 & 2^{13} · 3 · 5^{2} \\ 7 & 3^{4} · 7 · 1801 · 2161 & 2 · 3^{3} · 53 · 9283 & 3 · 337 · 1861 \\ 8 & 2^{17} · 77191 & 2^{7} · 3 · 19^{2} · 653 & 2^{17} · 41 \\ 9 & 2^{2} · 3^{5} · 59 · 743129 & 3^{7} · 157 · 839 & 2^{2} · 3 · 5 · 241303 \\ 10 & 2^{12} · 5 · 7 · 1063 · 1093 & 2^{6} · 3^{5} · 5 · 7^{2} · 227 & 2^{12} · 3^{2} · 19 · 53 \\ 11 & 2 · 23 · 281 · 523 · 90499 & 2 · 3 · 17 · 97 · 103 · 2423 & 2 · 5^{2} · 53 · 173 · 199 \\ 12 & 2^{16} · 3^{3} · 17^{2} · 4157 & 2^{6} · 3^{5} · 433931 & 2^{16} · 3 · 7 · 157 \\ 13 & 2^{3} · 5 · 491 · 953 · 376153 & 3^{3} · 613 · 1072231 & 2^{3} · 3 · 11 · 1875943 \\ 14 & 2^{14} · 3^{3} · 7 · 7210349 & 2^{10} · 5 · 37 · 238001 & 2^{14} · 3 · 11 · 2039 \\ 15 & 3^{4} · 5 · 7 · 24033246929 & 3^{6} · 5 · 31 · 43 · 22859 & 3^{2} · 15913 · 16691 \end{matrix}

n γ(n) n γ(n) n γ(n) n γ(n)  n γ(n)
1  21  8 41  8 61  6 81  7
2  0 22  2 42  5 62  5 82  1
3  4 23  5 43  4 63  5 83  4
4  3 24  8 44  5 64 15 84  8
5  6 25  8 45  6 65  8 85  7
6  2 26  1 46  3 66  4 86  2
7  5 27  5 47  6 67  4 87  5
8  6 28  6 48  11 68  4 88  8
9  7 29  6 49  9 69  8 89  7
10  1 30  3 50  0 70  4 90  1
11  4 31  7 51  5 71  5 91  6
12  5 32  12 52  4 72  6 92  6
13  6 33  7 53  6 73  7 93  10
14  3 34  1 54  3 74  1 94  4
15  5 35  6 55  5 75  4 95  5
16  9 36  3 56  9 76  5 96  14
17  7 37  6 57  7 77  8 97  7
18  0 38  2 58  1 78  3 98  0
19  4 39  5 59  4 79  6 99  4
20  4 40  7 60  6 80 10 100  3

References

1. S. Akiyama, A note on Hecke’s absolute invariants, J. Ramanujan Math. Soc., 7 (1992), 65-81.

2. A.O.L. Atkin, Congruence for modular forms, Proc. IBM Conf. on computers mathematical research, Blaricum 1966, (North Holland).

3. A.O.L. Atkin, Congruence Hecke operators, Proc. Symp. Pure Math., vol. 12, 33-40.

4. A.O.L. Atkin and J.N. O’Brien, Some properties of p(n) and c(n) modulo powers of 13, Trans. Amer. Math. Soc., 126 (1967), 442-459.

5. B. Dwork, The U_{p} operator of Atkin on modular functions of level 2 with growth conditions, Lecture Notes in Math., 350 (1973), 57-67

6. K. Hatada, Eigenvalues of Hecke Operators on SL(2, ℤ), Math. Ann.,239 (1979), 75-96.

7. N. Katz, p-adic properties of modular schemes and modular forms, Lecture Notes in Math., 350 (1973), 69-190 8. M. Koike, Congruences between modular forms and functions and applications to the conjecture of Atkin, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20 (1973), 129-169.

9. O. Kolberg, Congruences for the coefficients of the modular invariant j(τ) modulo powers of 2, Årbok Univ. Bergen, 16 (1961), 1-9.

10. J. Raleigh, On the Fourier coefficients of triangle groups, Acta Arith., 8 (1962), 107-111.

11. J-P. Serre, Formes modulaires et fonctions zêta p-adiques, Lecture Notes in Math., 350 (1973), 191-268.

12. J. Wolfart, Transzendente Zahlen als Fourierkoeffizienten von Heckes Modulformen, Acta Arith., 39 (1981), 193-205.