PERIOD RATIOS OF MODULAR FORMS

NEIL DUMMIGAN

1. Introduction

The ultimate generalisation of Dirichlet’s class number formula and the Birch and Swinnerton-Dyer conjecture is the Bloch-Kato conjecture on L-functions and Tamagawa numbers of motives [BK]. The main purpose of this paper is to point out that the Bloch-Kato conjecture provides a natural explanation for certain observations concerning the prime factorisations of period ratios of modular forms. One could also say we are offering a little evidence in support of the conjecture, in a case which has not been tested previously. We predict the existence of non-trivial elements in certain Shafarevich-Tate groups.

Let f be a normalised Hecke eigenform, cuspidal of weight k > 2. For the basics on modular forms see, for instance, [Se1]. Two assumptions will be in force for most of the paper, though the second one is mainly for convenience, and we shall drop it in a later section.

1. f has level one, i.e. it is a modular form for the whole of SL_{2}(ℤ).

2. The coefficients in the Fourier expansion of f are rational.

These are restrictive assumptions, in fact the only known examples are those where f is the unique normalised cusp form of (level one and) weight k. This happens precisely when k = 12, 16, 18, 20, 22 or 26. In the case k = 12, f is the discriminant function Δ = ∑ τ(n)q^{n}, where q = e^{2πiz} and τ is Ramanujan’s function.

Generally, suppose f = ∑_{n=1}^{∞} a_{n}q^{n}, with a_{1} = 1 and all the a_{n} are rational integers. Associated with f is the L-function defined by the Dirichlet series L_{f}(s) = ∑ a_{n}n^{−s} for ℜ(s) > (k+1)/2, but continuable to a holomorphic function on the whole complex plane by means of the integral formula

∫_{0}^{∞} f(iy)y^{s−1}dy = (2π)^{−s}Γ(s)L_{f}(s).

Date: January 11th, 2000.

1991 Mathematics Subject Classification. 11F67, 11G40, 11F33.

Key words and phrases. modular form, Bloch-Kato conjecture, Shafarevich-Tate group, p-adic cohomology.

For 0 ≤ j ≤ k − 2 define the jth period of f,

r_{j}(f) = ∫_{0}^{i∞} f(z)z^{j}dz,

where the integral is taken along the positive imaginary axis. Clearly then

r_{j}(f) = j!i^{j+1}(2π)^{−(j+1)}L_{f}(j + 1)  (1)

Note that the period r_{j}(f) is either purely imaginary or real, according as j is even or odd, respectively.

The points s = j + 1 for 1 ≤ j + 1 ≤ k − 1, at which the L-function is evaluated, are critical points in the sense of [De2]. It is shown in [Ma] that for even j with 0 < j ≤ k − 2, the period ratio r_{j}/r_{0} is a rational number, as is the ratio r_{j}/r_{1} for any odd j with 1 < j < k − 2. More generally, if we drop the restriction that the Fourier coefficients are rational, the number field they generate contains the period ratios. But the beautiful proof of this fact (see [Ma]) offers little clue to the nature of the prime factorisations of these ratios.

Anybody who has examined the tables of period ratios in [Ma] and [Ko] (see Section 2 below) must have been struck by the following fact, at least for the forms of weights k = 12, 16, 18 and 20. The primes occurring in the numerators and denominators of the “odd” period ratios are all less than the weight k. The same is true of the even period ratios, except that certain “Eisenstein” primes dividing the numerator of the kth Bernoulli number appear in all the numerators. This latter fact is actually a consequence of the formula which is Theorem 1 of [KZ], at least in those cases where the dimension of the space of cusp forms is one. It also follows (for k ≥ 16) from Zagier’s generalisation of an identity of Rankin. See Remarks 2 and 3 following the corollary to Proposition 6 in §5 of [Za]. This latter approach depends on the fact that irregular prime divisors of the numerator of B_{k} divide the denominators of coefficients in the normalised Eisenstein series E_{k}. This fact also leads to the simplest proof of Ramanujan’s congruence σ_{11}(n) ≡ τ(n)(mod 691) and its generalisations.

Manin uses the observed occurrence of these Eisenstein primes, together with his formula for the Fourier coefficients of f in terms of the periods (the Coefficients Theorem, 1.3 of [Ma]), to give an alternative proof of the Ramanujan-style congruences.

Conversely, starting from a Ramanujan-style congruence, one can deduce the divisibility by p (the Eisenstein prime concerned) of the even period ratios, if one assumes that the system of linear equations for the periods, due to Eichler, Shimura and Manin, (the Coefficients Theorem again) is invertible modulo p. We shall see, in Section 6, that this confirms a prediction of the Bloch-Kato conjecture, which can be viewed as giving yet another explanation for the occurrence of the Eisenstein primes in the even period ratios. We shall need Serre’s interpretation of Ramanujan congruences in terms of Galois representations [Se2].

Deligne [De1] constructed l-adic Galois representations for Hecke eigenforms and used them to prove Ramanujan’s conjecture |τ(p)| < 2p^{11/2} (p prime), as a consequence of the Weil conjectures, whose proof he completed a few years later. Scholl [Sc] proved that these Galois representations come from a Grothendieck motive, specifically from a projector on a desingularised product of universal elliptic curves over a modular curve. They live in the (k −1)st cohomology of the variety. This is reviewed in Section 3.

In Section 7, using the injectivity of the q-expansion map, and slightly modified results of Ulmer [U], we show that certain Hodge cohomology groups are torsion-free. This allows us, in Section 8, to apply Faltings’s comparison theorem for crystalline and p-adic étale cohomologies [Fa]. We deduce that certain factors c_{p} appearing in the Bloch-Kato conjecture (which, thanks to our restriction to level one, are at worst powers of p) are trivial for p > k. This then explains the observation about non-Eisenstein primes in the period ratios being less than k. Faltings’s comparison theorem is a refinement and generalisation of the one originally proved by Fontaine and Messing [FM]. The Fontaine-Messing theorem is insufficient for our purposes (it works over ℚ_{p} instead of ℤ_{p}), though what we need would follow from Fontaine’s conjecture “C2” [Fo]. Note that we do not need the “deRham conjecture”, and see 0.13 and the final footnote of [Ne2] concerning the status of Faltings’s proof of the “crystalline conjecture”, which is what we use.

In the course of our “explanations” we have to make one or two assumptions, even allowing the Bloch-Kato conjecture. When p is an Eisenstein prime we cannot be certain which of two globally Galois-invariant lattices is the integral p-adic étale cohomology. Also, we assume that for weights k = 12, 16, 18 and 20 the order of the Shafarevich-Tate group is not divisible by large primes. However, for weights k = 22 and 26, which we have excluded from consideration until now, there are “extra” primes occurring in the numerators of some of the period ratios which, assuming Bloch-Kato, must be the orders of elements in certain Shafarevich-Tate groups. The orders of these elements are 41 (when k = 22, occurring in one of the odd period ratios) and 29, 43 and 97 (when k = 26, occurring in certain of the even period ratios).

The absence of large prime factors in the large denominators of some of the period ratios is remarkable. If Ω(n) is defined to be the total number of primes (allowing repeats) in the factorisation of n, then the “normal order” of Ω(n) is log log(n). Hence it is very unusual for a large integer to be “round” or “smooth”, i.e. composed of a large number of relatively small primes, see 22.12 of [HW]. For example, if n = 2^{5}.3^{4}.5^{3}.7^{2}.11.13.23 (the denominator of one of the period ratios) then Ω(n) = 17 whereas log log(n) ≈ 3.2. Moreover, if one looks at all the non-Eisenstein, non-extra primes occurring altogether in the period ratios for a given f, one obtains (with the exception of p = 11 when k = 22) all the primes less than k −2, so the significance of the weight, rather than just the general occurrence of small primes, is very striking.

The formula of Kohnen and Zagier [KZ] appears to be no help in explaining this phenomenon.

In another paper [Du] we look instead at ratios of critical values of symmetric square L-functions for modular forms, and find certain occurrences of Eisenstein and other large primes. In this case we can give conditional constructions for some of the elements of Shafarevich-Tate groups demanded by the Bloch-Kato conjecture.

In Section 10 of this paper, we drop the assumption that f has rational Fourier coefficients, and find that it makes little difference to the validity of our arguments. We look at the even period ratios for the normalised cuspidal eigenforms of weight 24 and level one. In Section 11 we find that it is more difficult to drop the level one assumption, but look at the periods of a couple of quadratic twists of forms of level one.

I am grateful to K. Buzzard for helpful comments on an earlier version of this paper.

2. The forms and their period ratios

Let Δ = ∑ τ(n)q^{n} = q ∏_{n=1}^{∞}(1 − q^{n})^{24}, Q = E_{4} = 1 + 240 ∑ σ_{3}(n)q^{n} and R = E_{6} = 1 − 504 ∑ σ_{5}(n)q^{n}. The unique normalised cusp forms of level one and weights k = 12, 16, 18, 20, 22 and 26 are respectively Δ, QΔ, RΔ, Q^{2}Δ, QRΔ, Q^{2}RΔ.

It follows from the fact that f(−1/z) = z^{k}f(z) that r_{j}(f) = (−1)^{j+1}r_{k−2−j}(f). Hence we need only tabulate the period ratios for 0 ≤ j ≤ (k/2) − 1. Furthermore, when (k/2) − 1 is even (i.e. when k = 18, 22 or 26) r_{(k/2)−1} = 0. In these cases, the L-function vanishes at the central point s = k/2.

The even period ratios, taken from [Ma], are set out in the following table.

k  r_{2}/r_{0}  r_{4}/r_{0}  r_{6}/r_{0}  r_{8}/r_{0}  r_{10}/r_{0}
12  \frac{−691}{2^{2}.3^{4}.5}  \frac{691}{2^{3}.3^{2}.5.7}
16 \frac{−3617}{2^{2}.3^{2}.5.7.13}  \frac{3617}{2^{3}.3^{5}.5.11.13}  \frac{−3617}{2^{3}.3^{3}.5.7.13}
18  \frac{−43867}{2^{4}.3^{3}.5^{4}}  \frac{43867}{2^{6}.3^{2}.5^{2}.7.13}  \frac{−43867}{2^{6}.3^{2}.5^{3}.7.11}
20 \frac{−283.617}{2^{2}.3^{4}.5.7^{2}.17}  \frac{283.617}{2^{3}.3^{5}.5.7^{2}.17}  \frac{−283.617}{2^{3}.3^{4}.5.7^{2}.13.17}  \frac{283.617}{2^{3}.3^{5}.5.7^{2}.11.17}
22  \frac{−131.593}{3^{2}.5^{4}.7.19}  \frac{131.593}{2^{4}.3^{2}.5^{3}.17.19}  \frac{−131.593}{2^{4}.3^{3}.5^{4}.7.19}  \frac{131.593}{2^{5}.3^{3}.5^{3}.7.13.19}
26 \frac{−43.657931}{2^{3}.3^{3}.5^{3}.7^{2}.13.23} \frac{97.657931}{2^{5}.3^{4}.5^{3}.7^{2}.11.13.23} \frac{−29.657931}{2^{5}.3^{3}.5^{3}.7^{2}.13.19.23} \frac{657931}{2^{4}.3^{4}.5^{2}.7^{2}.13.17.23} \frac{−657931}{2^{5}.3^{4}.5^{3}.7^{2}.11.13.23}

The relevant Bernoulli numbers are

B_{12} = −691/2730, B_{16} = −3617/510, B_{18} = 43867/798,

B_{20} = −283.617/330, B_{22} = 11.131.593/138, B_{26} = 13.657931/6.

Notice that the irregular (≥ k+3) prime divisors of the numerators of these Bernoulli numbers appear in the numerators of the even period ratios.

Kohnen and Zagier [KZ] consider cusp forms R_{n} of weight k and level one, such that for any cusp form f of weight k and level one, the period r_{n}(f) is given by the Petersson inner product (f, R_{n}). Their Theorem 1 is a formula for the periods r_{m}(R_{n}), where m and n have different parity. If, as is the case for us, the dimension of the space of cusp forms is one, then (for even 0 < m < k − 2, say), the ratio r_{m}(f)/r_{0}(f) is the same as the ratio r_{m}(R_{1})/r_{0}(R_{1}). But if p is an irregular prime divisor of the numerator of B_{k} then the formula shows that r_{0}(R_{1}) has a pole at p while r_{m}(R_{1}) does not, thus explaining the appearance of p in the numerator of r_{m}(f)/r_{0}(f).

As an example we consider the case k = 22. Using the formula of Kohnen and Zagier, we find

r_{18}(R_{1}) = 18!(−19B_{2} + B_{0} + 19B_{18}) = 18!(−19/6 + 1 + 19.43867/798)

= 18!(831744/798) = 18!(2^{7}.3.19/7)

and

r_{20}(R_{1}) = 19!(B_{0} − 10B_{2} − B_{20} + 11B_{2}B_{20}/B_{22})

= 19!(1 − 5/3 + 174611/330 − 4016053/25635390)

= 19!(41040000/77683) = 2^{7}.3^{3}.5^{4}.19/131.593.

We recover r_{18}/r_{20} = 131.593/3^{2}.5^{4}.7.19. Notice how miraculously the smoothness occurs.

As already noted, Manin deduces the Ramanujan-style congruences a_{n} ≡ σ_{k−1}(n)(mod p) from the divisibility by p of the even period ratios, via his Coefficients Theorem.

The odd period ratios, taken from [Ko], are set out in the following table.

k r_{3}/r_{1} r_{5}/r_{1} r_{7}/r_{1} r_{9}/r_{1}  r_{11}/r_{1}
12 \frac{−5^{2}}{2^{4}.3}  \frac{5}{2^{2}.3}
16 \frac{−5.7^{2}}{2^{3}.3^{2}.13}  \frac{7^{2}}{2^{2}.3^{2}.13}  \frac{−5.7}{2^{2}.3^{2}.13}
18 \frac{−11}{2^{2}.3.5}  \frac{1}{2^{3}.3}  \frac{−1}{2^{3}.3.5}
20 \frac{−5.11.13}{2^{5}.3^{2}.17} \frac{11^{2}}{2^{2}.3^{2}.7.17} \frac{−5}{2^{2}.3^{2}.17}  \frac{1}{2^{2}.3.17}
22 \frac{−7.13^{2}}{2^{2}.3^{3}.5.19} \frac{7.13}{2^{5}.3^{2}.19} \frac{−41}{2^{4}.3^{2}.5.19} \frac{7}{2^{4}.3^{2}.5.19}
26 \frac{−17.19}{2^{2}.3^{2}.5.23} \frac{5.17}{2^{3}.3^{2}.7.23} \frac{−7}{2^{3}.3^{2}.5.23} \frac{1}{2^{3}.3^{2}.5.23} \frac{−1}{2^{3}.3^{3}.5.7.23}

3. The underlying motives

Let f be one of the normalised eigenforms already introduced. Since f is a Hecke eigenform, the L-function has an Euler product.

L_{f}(s) = ∏_{p}(1 − a_{p}p^{−s} + p^{k−1−2s})^{−1},

the product taken over all prime numbers. The product converges for ℜ(s) > (k + 1)/2. A special case of a theorem of Deligne [De1] implies the existence, for each prime l, of a representation

ρ_{l} : Gal(\overline{ℚ}/ℚ) → Aut(V_{l})

(V_{l} is a two-dimensional vector space over ℚ_{l}), such that 1. ρ_{l} is unramified at p for all primes p ≠ l

2. If Frob_{p} is an arithmetic Frobenius element at p then the characteristic polynomial of Frob_{p}^{−1} acting on V_{l} is x^{2} − a_{p}x + p^{k−1}.

For N ≥ 3 let M_{N} be the modular scheme over ℤ[1/N] parametrising elliptic curves, each with a given basis for the group of points of order N. Let X_{N} be the universal elliptic curve over M_{N}. Let \overline{M_{N}} be the compactification of M_{N} and let \overline{X_{N}} be the universal generalised elliptic curve over \overline{M_{N}}. Let \overline{X_{N}}^{k−2} be the (k −

2)-fold fibre product of \overline{X_{N}} over \overline{M_{N}} and let X = \overline{\overline{X_{N}}}^{k−2} be Deligne’s canonical desingularisation. The scheme X is proper and smooth over ℤ[1/N] (though not geometrically irreducible).

In [De1], V_{l} is constructed as a subquotient of the étale l-adic cohomology H_{et}^{k−1}(X⊗ \overline{ℚ}, ℚ_{l}). In [Sc] it is shown that V_{l} is actually the l-adic realisation of a Grothendieck motive M_{f}. Firstly one constructs a certain projector Π_{ϵ} in the group ring of the automorphism group of X, with coefficients in ℤ[1/(2N.(k − 2)!)]. The image of this projector is the level N parabolic cohomology group with l-adic coefficients. Then to obtain V_{l} we just apply another projector to take GL_{2}(ℤ/Nℤ)-invariants, restricting from level N to level one. In our case the projector involving Hecke correspondences is unnecessary, since the spaces of cusp forms are one-dimensional. Let P_{f} be the composite of the two projectors, so that V_{l} = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℚ_{l}).

In fact, the results of Sections 7 and 8 below (and their proofs) remain valid even when the space of cusp forms is not one-dimensional, if we simply agree to concern ourselves with the motive attached to the whole space of cusp forms. See Section 10 below for more on the case k = 24.

We shall now fix N = 3. Noting that #GL_{2}(ℤ/3ℤ) = 2^{4}.3 and that k ≥ 12, we see that the coefficients of P_{f} do not require the inversion of any primes greater than k − 2. The projector P_{f} is, of course, independent of l. For each prime l we now have an l-adic realisation V_{l} of the motive M_{f}. We also need the Betti and deRham realisations V_{B} = P_{f}H_{B}^{k−1}(X ⊗ ℂ, ℚ) and V_{dR} = P_{f}H_{dR}^{k−1}(X ⊗ ℚ). Here, H_{B} and H_{dR} are singular cohomology and algebraic deRham cohomology, respectively.

H_{dR}^{k−1}(X ⊗ ℚ) has its Hodge filtration, which induces a filtration

V_{dR} = F^{0} ⊃ F^{1} = ⋯ = F^{k−1} ⊃ {0}.

It also induces a filtration on V_{B} ⊗ ℂ via its natural isomorphism with V_{dR} ⊗ ℂ. The two non-trivial subquotients are one-dimensional.

It is really the twists M_{f}(j), for 1 ≤ j ≤ k − 1, which we shall be using. In terms of realisations, V_{B}(j) = P_{f}H_{B}^{k−1}(X ⊗ ℂ, ℚ(2πi)^{j}) and V_{l}(j) = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℚ_{l}(j)), the Tate twist as a Galois representation. (In particular, for p ≠ l, the action of Frob_{p} on V_{l}(j) is p^{j} times its action on V_{l}.) Strictly speaking, the numbering in the Hodge filtration of the deRham realisation should get shifted by j, but it is convenient for us not to do this.

4. The Bloch-Kato conjecture

For details of the Bloch-Kato conjecture on special values of the L-functions of motives, one should of course consult [BK]. We only describe our special case here. A concise account of a different special case, and its consequences for the degrees of modular parametrisations of elliptic curves, may be found in [Fl].

Inside each of the realisations of the motive M_{f}, we choose a natural lattice. Inside V_{dR} we choose any ℤ-lattice T_{dR} such that T_{dR} ⊗ ℤ[1/3] = P_{f}H_{dR}^{k−1}(X/ℤ[1/3]). Thus, for any prime p ≠ 3, T_{dR} ⊗ ℤ_{p} = P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}). For each prime l let T_{l}(j) = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℤ_{l}(j))/tors. Let A_{l}(j) = V_{l}(j)/T_{l}(j). Finally, let T_{B}(j) = P_{f}H_{B}^{k−1}(X ⊗ ℂ, ℤ(2πi)^{j})/tors.

For each finite prime p define the set of local points

Γ_{p}(j) = ⊕_{l}H_{f}^{1}(ℚ_{p}, T_{l}(j)).

Here the sum is over all primes l and we are taking Galois cohomology with respect to a decomposition subgroup at p. H_{f}^{1}(ℚ_{p}, T_{l}(j)) is the inverse-image of H_{f}^{1}(ℚ_{p}, V_{l}(j)) under the natural map.

H_{f}^{1}(ℚ_{p}, V_{l}(j)) = \begin{cases}Ker(H^{1}(ℚ_{p}, V_{l}(j)) → H^{1}(ℚ_{p}^{ur}, V_{l}(j))) & p ≠ l \\ Ker(H^{1}(ℚ_{p}, V_{l}(j)) → H^{1}(ℚ_{p}, V_{l}(j) ⊗ B_{cris})) & p = l. \end{cases}

Apart from at l = p, all the summands are finite, in fact only finitely many of them are non-trivial (see below for their evaluation in most cases). Actually, it follows from the fact that ρ_{l} is unramified at all p ≠ l, that for such p, H_{f}^{1}(ℚ_{p}, V_{l}(j)) is trivial. See the comment following Proposition 9.1 of [Du] for something similar.

At ∞ we define the set of local points Γ_{∞}(j) = (V_{B}(j) ⊗ ℝ)^{+}, where the + means we take the part fixed under the action of complex conjugation. This action is simultaneously on the manifold X ⊗ ℂ and on the coefficients ℝ(2πi)^{j}.

It follows from the fact that the points s = j are critical that integration over cycles induces an isomorphism

(V_{dR}/F^{j}) ⊗ ℝ ≃ Γ_{∞}(j).

There is a measure on the right hand side such that the image under projection of the lattice T_{B}(j) has covolume one. We then define vol_{∞}(j) to be the covolume of the image of T_{dR}. Notice that for different j of the same parity, these volumes differ only by a factor of a power of 2πi. This follows from the fact that the F^{j} are all the same, for 1 ≤ j ≤ k − 1.

For finite p, the exponential map of Bloch and Kato gives an isomorphism

(V_{dR}/F^{j}) ⊗ ℚ_{p} ≃ H_{f}^{1}(ℚ_{p}, V_{p}(j)).

For this, we really ought to assume that j ≥ k/2, which is no loss, thanks to the symmetry about this central point. There is a measure on the right hand side such that the image of T_{dR} ⊗ ℤ_{p} has volume one. We then define vol_{p}(j) to be the volume of Γ_{p}(j) (for the l ≠ p summands, just multiply by the order).

Define the global points Γ_{ℚ}(j) = ⊕_{l}H^{0}(ℚ, A_{l}(j)). (In the elliptic curve case, this would (when j = 1) give only the torsion subgroup of the global points.) Now the conjecture of Bloch and Kato predicts that

( ∏_{p≤∞} vol_{p}(j))/#Γ_{ℚ}(j) = #Γ_{ℚ}(k − j)/#\underline{III}(j).

For the precise definition of the Shafarevich-Tate groups \underline{III}(j), which are only assumed to be finite, we refer to p. 27 of [Fl] or 5.13 of [BK]. Strictly speaking we should have j > k/2 for the product on the left to converge, but in the next section we shall describe the interpretation in terms of values of the L-function, so s = k/2 will make sense again. But we must ignore the point s = k/2 when k/2 is odd and the L-function vanishes. Whenever we consider primes dividing the numerator or denominator of a period ratio, we are assuming that this ratio is non-zero. For work on the conjectured relation between the behaviour of L_{f}(s) at the central point and the height pairing on algebraic cycles, see [Zh] and [Ne1].

5. Local measures, Euler factors and fudge factors

For a finite prime p let P_{p}(x) = 1 − a_{p}x + p^{k−1}x^{2}. Hence

L_{f}(s) = ∏_{p}(P_{p}(p^{−s}))^{−1}.

For j ≥ k/2 and all but finitely many p, vol_{p}(j) = P_{p}(p^{−j}), by Theorem 4.1 of [BK], so for j > k/2 we can write

L_{f}(j) = ∏_{p}(c_{p}(j)/vol_{p}(j)),

where the product is over all finite primes, c_{p}(j) = vol_{p}(j)/P_{p}(p^{−j}), and c_{p}(j) = 1 for all but finitely many p.

Now the Bloch-Kato conjecture may be written as

L_{f}(j) = (∏_{p} c_{p}(j))vol_{∞}(j)#\underline{III}(j)/(#Γ_{ℚ}(j)#Γ_{ℚ}(k − j)).  (2)

Henceforth we only insist that k/2 ≤ j ≤ k − 1. Since V_{p}(j) is unramified at l for all primes l ≠ p, it follows exactly as on p. 30 of [Fl] that |c_{p}(j)|_{l} = 1 for all such l.

Recall that r_{j−1}(f) = (j − 1)!i^{j}(2π)^{−j}L_{f}(j), that the vol_{∞}(j) for j of equal parity differ only by multiplication by a power of 2πi, and observe that the prime factors of (j − 1)! are certainly less than k (in fact less than k − 2). In Sections 7 and 8 we shall show that |c_{p}(j)|_{p} = 1 for all primes p > k, so that c_{p}(j) = 1 for all primes p > k. It then follows that, if the Bloch-Kato conjecture is true, any primes p > k occurring in the factorisation of a period ratio r_{j−1}/r_{0} (when j − 1 is even) or r_{j−1}/r_{1} (when j − 1 is odd), must be accounted for by #\underline{III}(j), #Γ_{ℚ}(j) or #Γ_{ℚ}(k − j). In the next section we deal with the possible prime divisors of the latter two quantities. It would be nice to be able to show that c_{p} = 1 for p > k − 2, not just for p > k.

6. Eisenstein primes and Ramanujan congruences

The factor #Γ_{ℚ}(j) is divisible by the prime l iff the 𝔽_{l}[Gal(\overline{ℚ}/ℚ)]-module T_{l}(j)/lT_{l}(j) has a trivial one-dimensional submodule. Untwisting, and noting that the determinant of the Galois representation associated to f is the (k − 1)st power of the cyclotomic character, it follows that, for all primes p ≠ l, Frob_{p}^{−1} can be represented by a

matrix of the form (\begin{matrix}p^{j} & ∗ \\ 0 & p^{k−1−j} \end{matrix}). Hence

a_{p} ≡ p^{j} + p^{k−1−j}(mod l).

It is proved in [SwD] that the only primes l for which this can possibly happen are l < k (which, remember, do not really concern us) and l dividing the numerator of the Bernoulli number B_{k}. On p. 32 of [SwD] there is a table listing exactly which congruences occur for the six possible f. Those primes dividing the numerator of B_{k} for which there is a congruence, we call “Eisenstein primes”. It follows from [SwD] that the only possible congruences as above, where l is an Eisenstein prime, are of the shape a_{p} ≡ 1+p^{k−1}( mod l). Also, this implies that either T_{l}/lT_{l} or T_{l}(k−1)/lT_{l}(k−1) has a Galois-fixed line. We have no way of proving which actually happens, but conjecture that it is always the latter for the following reason. When j = 1, #Γ_{ℚ}(k−j) will then be divisible by l. This factor of l will then appear in the denominator of the Bloch-Kato formula for L_{f}(1), explaining why it appears in the numerators of all the even period ratios r_{m}/r_{0}. (Recall that, up to a factorial and a power of 2πi, r_{0} is the same as L_{f}(1).)

7. The Hodge cohomology groups

Proposition 7.1. P_{f}H^{s}(X ⊗ ℤ_{p}, Ω^{r}) is torsion-free, for r + s = k − 1 and p > k − 2.

Proof. It suffices to prove that Π_{ϵ}H^{s}(X ⊗ ℤ_{p}, Ω^{r}) is torsion-free, for r + s = k − 1, where X and Π_{ϵ} are as in Section 3.

For 0 ≤ r ≤ k − 1 let a_{r} be the ℤ_{p}-rank of Π_{ϵ}H^{k−1−r}(X ⊗ ℤ_{p}, Ω^{r}) and let b_{r} be the dimension of the 𝔽_{p}-vector space Π_{ϵ}H^{k−1−r}(X ⊗ 𝔽_{p}, Ω^{r}). It follows from Corollary 9.4 of Chapter III of [Ha] that a_{r} is also the dimension of the ℚ_{p}-vector space Π_{ϵ}H^{k−1−r}(X⊗ℚ_{p}, Ω^{r}), and that b_{r} ≥ a_{r}, with equality iff Π_{ϵ}H^{k−1−r}(X⊗ℤ_{p}, Ω^{r}) is torsion-free.

Let π : \overline{X_{3}} → \overline{M_{3}} be the natural map and let ω^{−1} = R^{1}π_{∗}O\overline{X_{3}}. We shall sloppily use the same notation for a sheaf before and after base change. Slightly modifying the proof of Theorem 5.5 of [U] to cover our simpler situation (see also Section 7 of [U]), we find that

Π_{ϵ}H^{0}(X ⊗ 𝔽_{p}, Ω^{k−1}) ≃ H^{0}(\overline{M_{3}} ⊗ 𝔽_{p}, Ω^{1} ⊗ ω^{k−2}) ≃ H^{0}(\overline{M_{3}} ⊗ 𝔽_{p}, ω^{k}(−cusps)),

which is a subspace of H^{0}(\overline{M_{3}} ⊗ 𝔽_{p}, ω^{k}),

Π_{ϵ}H^{k−1}(X ⊗ 𝔽_{p}, O_{X⊗𝔽_{p}}) ≃ H^{1}(\overline{M_{3}} ⊗ 𝔽_{p}, ω^{2−k})

and that Π_{ϵ}H^{s}(X⊗𝔽_{p}, Ω^{r}) is trivial for all other (r, s) (whether or not r+s = k−1). In fact, by Serre duality these two non-trivial 𝔽_{p}-vector spaces have the same dimension, λ say, and λ is the dimension of “the” space of cusp forms of weight k for Γ_{1}(3). We are using here the fact that if R is a ℤ[1/3]-algebra (such as 𝔽_{p} or ℚ_{p}) then H^{0}(\overline{M_{3}} ⊗ R, ω^{k}) ≃ H^{0}(\overline{M_{3}}, ω^{k}) ⊗ R. See Prop. 2.5 of [Gr] or 1.7.1 of [Ka]. Actually, for the R we use, Proposition 9.3 and Corollary 9.4 of Chapter III of [Ha] would suffice. We are also using the fact that H^{0}(\overline{M_{3}}, ω^{k}) is a torsion-free ℤ[1/3]-module, which follows from the injectivity of the map from modular forms to q-expansions, see Prop. 2.7 of [Gr] or 1.6 of [Ka].

Since the Hodge-deRham spectral sequence in characteristic zero degenerates at E_{1}, we have ∑ a_{r} = dim Π_{ϵ}H_{dR}^{k−1}(X ⊗ℚ_{p}), which is known to be 2λ [Sc]. But we have just seen that ∑ b_{r} = 2λ. Hence a_{r} = b_{r} for all r, and the Π_{ϵ}H^{k−1−r}(X ⊗ ℤ_{p}, Ω^{r}) are indeed torsion-free. ■

In the following corollaries, p > k − 2.

Corollary 7.2. P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}) is torsion-free.

This follows easily by consideration of the Hodge-deRham spectral sequence.

Corollary 7.3. P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}) ≃ Hom_{ℤ_{p}}(P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}), ℤ_{p}) via Poincaré duality.

This follows from Corollary 7.2 and the exact sequence 3.5.4 of [I], and from the fact that f is equal to its own complex conjugate.

Corollary 7.4. P_{f}H_{dR}^{k−1}(X ⊗ (ℤ/p^{n}ℤ)) ≃ P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}) ⊗ (ℤ/p^{n}ℤ), for all n ≥ 1.

For r + s = k − 1, we know that P_{f}H^{s}(X ⊗ ℤ_{p}, Ω^{r}) is torsion-free, by Proposition 7.1, and that P_{f}H^{s}(X ⊗ (ℤ/p^{n}ℤ), Ω^{r}) ≃ P_{f}H^{s}(X ⊗ ℤ_{p}, Ω^{r}) ⊗ (ℤ/p^{n}ℤ), by Corollary 9.4 of Chapter III of [Ha]. In the universal coefficients exact sequence

0 → P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}) ⊗ (ℤ/p^{n}ℤ) → P_{f}H_{dR}^{k−1}(X ⊗ (ℤ/p^{n}ℤ))
→ Tor_{1}^{ℤ_{p}}(P_{f}H_{dR}^{k}(X ⊗ ℤ_{p}), ℤ/p^{n}ℤ) → 0

the Tor group on the right must be trivial, otherwise the middle term would be bigger than what is allowed by the Hodge-deRham spectral sequence and our knowledge of the Hodge cohomology groups. Alternatively, we could just use the fact that P_{f}H_{dR}^{k}(X ⊗ ℤ_{p}) = 0.

Corollary 7.5. For all n ≥ 1, Corollary 7.3 still holds if ℤ_{p} is replaced by ℤ/p^{n}ℤ.

This follows directly from Corollary 7.4.

8. Filtered modules and associated representations

Proposition 8.1. The Bloch-Kato fudge factors c_{p} are trivial for p > k.

Proof. It remains to show that |c_{p}|_{p} = 1 for p > k. Let O_{\overline{ℚ_{p}},n} be the reduction modulo p^{n} of the ring of integers of an algebraic closure of ℚ_{p}.

For each n there is a homomorphism θ_{n} : W_{n}(O_{\overline{ℚ_{p}},1}) → O_{\overline{ℚ_{p}},n} given by

θ_{n}(a_{0}, …, a_{n−1}) = â_{0}^{p^{n}} + ... + p^{n−1}â_{n−1}^{p}.

(The right hand side is independent of the choices of liftings â_{i}.) This homomorphism extends to the divided power envelope W_{n}^{DP} (O_{\overline{ℚ_{p}},1}) of the ideal ker(θ_{n}). Let J_{n} be the kernel of this extended homomorphism. Let B_{∞} = \underleftarrow{lim} W_{n}^{DP} (O_{\overline{ℚ_{p}},1}), the inverse limit being taken with respect to the maps (a_{0}, …, a_{n}) ↦ (a_{0}^{p}, …, a_{n−1}^{p}). The inverse limits of the divided powers of the J_{n} provide a natural filtration of the ring B_{∞}. See [BK] or [FM] (Chapter 1 in both cases) for more details.

Consider T_{dR} = P_{f}H_{dR}^{k−1}(X ⊗ ℤ_{p}) ≃ P_{f}H_{cris}^{k−1}(X ⊗ 𝔽_{p}/ℤ_{p}) with its Hodge filtration. We let ϕ denote the natural p^{th}-power Frobenius map on T_{dR} or on B_{∞}. There is also an action of G = Gal(\overline{ℚ_{p}}/ℚ_{p}) on B_{∞}. Define a ℤ_{p}G-module

T (k − 1) := ker(ϕ − p^{k−1} : ∑_{r+s=k−1} Fil^{r}B_{∞} ⊗ Fil^{s}T_{dR} → B_{∞} ⊗ T_{dR}).

Now, for any integer j, define T (j) := (T (k − 1))(j − (k − 1)) (a Tate twist).

If we assume that p > k − 1 (the dimension of X) then, by the results of [FM], T_{dR} is a “strongly divisible lattice” in the “admissible filtered module” T_{dR} ⊗ ℚ_{p}, and T (j) is a ℤ_{p}-lattice in P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ_{p}}, ℚ_{p}(j)). Note that the condition p > k − 1 is not only concerned with the length of the Hodge filtration, but also excludes primes dividing the denominator of the projector P_{f}, which might have stopped T_{dR} from being a strongly divisible lattice. Now Theorem 4.1 (iii) of [BK] implies that the volume of H_{f}^{1}(ℚ_{p}, T (j)), with respect to the measure induced by the exponential map, is |P_{p}(p^{−j})|_{p}^{−1}.

To show that |c_{p}(j)|_{p} = 1 for p > k, it then suffices to show that T (j) = T_{p}(j) = P_{f}H_{et}^{k−1}(X ⊗ \overline{ℚ}, ℤ_{p}(j))/tors. But this is a consequence of Faltings’s comparison theorem (Theorem 5.3 of [Fa]), given the results of the previous section. We briefly indicate how, referring to [Fa] for notation, definitions and the statement of the theorem.

In the last paragraph of 2(h) (of [Fa]), M = P_{f}H_{cris}^{k−1}(X⊗𝔽_{p}/ℤ_{p}) ≃ P_{f}H_{dR}^{k−1}(X⊗ℤ_{p}) and M/p^{r}M = P_{f}H_{cris}^{k−1}(X ⊗ 𝔽_{p}/(ℤ/p^{r}ℤ)) ≃ P_{f}H_{dR}^{k−1}(X ⊗ (ℤ/p^{r}ℤ)) (by Corollary 7.4 above, and the comparison theorem between deRham and crystalline cohomologies). M/p^{r}M is called “M” in the first paragraph of 2(h), where M ≃ M^{t} by Corollary 7.5 above.

In Theorem 5.3 (of [Fa]) we have a = k − 1 so require p > k. E = O_{X}/p^{r}O_{X}, 𝕃 = ℤ/p^{r}ℤ, and we take an inverse limit. The application of the projector P_{f} is justified by the functoriality of the comparison isomorphism. ■

9. Conjecturally non-trivial Shafarevich-Tate groups

As noted in the introduction, any non-Eisenstein primes p > k appearing in the period ratios must come from non-trivial Shafarevich-Tate groups, at least if the Bloch-Kato conjecture is true. Recalling the symmetry between j and k − j, we just list the cases for j ≤ k/2.

When k = 22, \underline{III}(8) should have an element of order 41. When k = 26, \underline{III}(3), \underline{III}(5) and \underline{III}(7) should have elements of orders 43, 97 and 29, respectively. To prove the existence of such elements without assuming any conjecture seems like a very difficult problem. (As already mentioned in the introduction, we have much better luck with this when we look instead at critical values of the symmetric square L-function.)

It is reassuring that the “extra” primes all occur in the numerators of period ratios. If any occurred in the denominators (without occurring in the denominators of all the even or odd period ratios for a given f) then the Bloch-Kato conjecture would have to be false. It is also worth pointing out that the contributions of primes p < k to the period ratios cannot be accounted for entirely by factorials and by congruences with Eisenstein series. There has to be something coming from non-trivial fudge factors c_{p}, or from Shafarevich-Tate groups, or both.

10. Level one, non-rational coefficients

In this section we allow any weight k, not just k = 12, 16, 18, 20, 22, 26, but continue to insist that the level is one. Using a projector as in Section 3, we get a motive M with rational coefficients, but in general it is a direct sum of motives (with non-rational coefficients) corresponding to all the different normalised, cuspidal eigenforms, f_{1}, ..., f_{n} of weight k and level one.

The L-function of M is the product of those for the f_{i}. For M, the proof that the fudge factors c_{p} are powers of p, trivial for p > k, goes through unchanged. Also, it follows from Ribet’s generalisation of Swinnerton-Dyer’s work that for M, #Γ_{ℚ}(j) (1 ≤ j ≤ k − 1) is divisible at most by primes l ≤ k + 1 or l dividing the numerator of B_{k}/2k. See the proof of Lemma 5.2 of [R].

According to a conjecture of Maeda, the f_{i} always form a single orbit under the action of Gal(\overline{ℚ}/ℚ), so c_{p} will be the norm of an ideal c_{p,i}, a corresponding fudge factor for the motive attached to f_{i}. See [Bu] and [FJ] for more on Maeda’s conjecture, including its verification for all weights k ≤ 2000.

According to the Bloch-Kato conjecture, any large, non-Eisenstein prime occurring in the product of a given period ratio taken over all the f_{i}, must be the order of an element in an appropriate Shafarevich-Tate group.

When k = 24 there are two normalised cuspidal eigenforms, f_{1} and its Galois conjugate f_{2}. According to [R], if we let A = Δ^{2} and B = ΔE_{6}^{2} then f_{1} = B + (1572 + 12β)A, where β := \sqrt{144169}.

f_{1} = q + (540 + 12β)q^{2} + (169740 − 576β)q^{3} + (12663328 + 12960β)q^{4}
+(36534510 − 180480β)q^{5} + (−904830480 + 1725840β)q^{6} + ...

For even m with 2 ≤ m ≤ 10 let R_{m} = r_{m}(f_{1})/r_{0}(f_{1}). We calculated the R_{m} by solving the linear equations of Eichler, Shimura and Manin. For safety, we got Maple to solve eight equations in the five unknowns, and they were consistent. To get a rational number, the product of period ratios for f_{1} and f_{2}, we let P_{m} be the norm from ℚ(β) to ℚ of R_{m}. Here are the results:

P_{2} = 103.2294797/3^{4}.5^{3}.7^{4}.11^{3}; P_{4} = 73.103.2294797/2^{5}.3^{6}.5^{4}.7^{3}.11^{2}.19^{2}; P_{6} = 103.179.2294797/2^{9}.3^{7}.5^{3}.7^{3}.11^{2}.17^{2}.19; P_{8} = 103.2294797/2^{10}.3^{6}.5^{4}.7^{3}.11^{2}.17.19; P_{10} = 103.2294797/2^{9}.3^{6}.5^{3}.7^{3}.11^{3}.13.17.19.

Notice again the preponderance of primes less than k − 2. B_{24} = −103.2294797/2.3.5.7.13, so this time the extra primes are 73 and 179. In this situation, where the cuspidal eigenforms form a single Galois orbit, it is easy to check that the occurrence of the Eisenstein primes in the even period ratios may still be accounted for using Zagier’s generalisation of Rankin’s identity (see the reference in the introduction to this paper). Strictly speaking, we are also using the fact that, in this particular case, the Eisenstein primes concerned do not divide B_{r} for any r < k.

I am grateful to William Stein for permission to direct the reader to his web page, which may be accessed via Number Theory Web (http://www.maths.uq.edu.au/ krm/ web aust.html). Here there is a modular forms database with data on special values for cusp forms of level one with weights ≤ 82.

11. Higher level: some quadratic twists

Once the level is increased there are primes of bad reduction. If p is such a bad prime, dividing the level, then in general it seems difficult to say anything about the possible prime divisors of the fudge factor c_{p}.

However, if the form happens to be the twist by some Dirichlet character of a form of level one then things are much simpler. In fact it is not difficult to check that the primes which could possibly divide c_{p} are p itself and primes dividing the order of the Dirichlet character. These latter are those primes l for which T_{l}/(1 − τ)T_{l} can be non-trivial, where τ is a topological generator of the tame inertia group at p, c.f. p.31 of [Fl]. For a twist by a quadratic character of conductor m (this will be a form for the congruence subgroup Γ_{0}(m^{2})), we need only worry about 2 and divisors of m. It is particularly convenient to twist one of the six forms of level one with rational coefficients by a quadratic character, since then we will be dealing with a form with rational coefficients.

Using modular symbols (see for instance [MTT]) and Maple, we computed the periods for f_{χ}, the twist by the quadratic character χ of conductor 3 of the form f of weight k = 12 or 20. Letting R_{j} = L(f_{χ}, j + 1)j!(−3)^{j+1}(2πi)^{−(j+1)}(i\sqrt{3})^{−1}, we express the R_{j} (for 0 ≤ j ≤ k − 2) in terms of the periods r_{0} and r_{1}. Actually, R_{k−2−j} = (−1)^{j}R_{j}, so it is only necessary to go as far as j = k/2 − 1.

k = 12 Even periods R_{0} = 2^{6}.3.5^{3}.7r_{1} R_{2} = −2^{4}.3.5^{2}.7r_{1} R_{4} = 2^{3}.3.5^{2}r_{1} Odd periods R_{1} = −2^{5}.691r_{0} R_{3} = 2.691r_{0} R_{5} = 0

k = 20 Even periods R_{0} = 2^{6}.3^{2}.5^{2}.7^{2}.11.13.19r_{1} R_{2} = −(2^{4}.3.5.11.13.13577/17)r_{1} R_{4} = (2^{3}.3.5.11.13.491/17)r_{1} R_{6} = −(2^{3}.3^{2}.5.7^{2}.11/17)r_{1} R_{8} = (2^{3}.3.5.43/17)r_{1} Odd periods R_{1} = −(2^{6}.439.283.617/3^{2}.7)r_{0} R_{3} = (2.1277.283.617/3.7.17)r_{0} R_{5} = −(2^{3}.137.283.617/3^{2}.7^{2}.17)r_{0} R_{7} = (2^{2}.283.617/3^{2}.7.17)r_{0} R_{9} = 0

As a check on our calculations, these periods are real or purely imaginary in exactly the expected way. (In each case, some linear combination of period ratios of f, representing the coefficient of either r_{0} or r_{1}, miraculously comes out to zero.) In both cases R_{k/2−1} vanishes just as it should. For further safety when k = 20, we checked that R_{10} = R_{8} and R_{16} = R_{2}.

The congruences necessary for Γ_{ℚ}(j + 1) to be non-trivial are ruled out by [SwD], so everything in the period ratios should be accounted for by fudge factors and orders of Shafarevich-Tate groups. True to the lack of congruences, all the Eisenstein primes in the above expressions cancel out when we look at ratios of periods, R_{j}/R_{1} for odd j.

Again we notice a preponderance of small prime factors in the period ratios. When k = 12, the primes involved in the denominators of the even period ratios are all those less than k−2, whereas when k = 20 we get all primes less than k, including k−1 = 19. When k = 12 there are no “extra” primes, whereas for k = 20 there are some surprisingly large ones. If the Bloch-Kato conjecture is correct then \underline{III}(2), \underline{III}(3), \underline{III}(4), \underline{III}(5), \underline{III}(6) and \underline{III}(9) must possess elements of orders 439, 13577, 1277, 491, 137 and 43, respectively.

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Merton College, Oxford, OX1 4JD, U.K.

  E-mail address: dummigan@maths.ox.ac.uk