On Zagier’s adele
© Guerzhoy; licensee Springer. 2014
Received: 16 January 2014
Accepted: 3 February 2014
Published: 18 August 2014
Don Zagier suggested a natural construction, which associates a real number and p-adic numbers for all primes p to the cusp form g=Δ of weight 12. He claimed that these quantities constitute a rational adele. In this paper we prove this statement, and, more importantly, a similar statement when g is a weight 2 primitive form with rational integer Fourier coefficients.
While a simple modular argument suffices for the proof of Zagier’s original claim, consideration of the case when g is of weight 2 involves Hodge decomposition for the formal group law of the rational elliptic curve associated with g.
Results and Conclusions
While in the weight 12 setting considered by Zagier the claim under consideration depends on a specific choice of a mock modular form which is good for g, in the case when g is of weight 2, the statement has a global nature, and depends on the fact that the classical addition law for the Weierstrass ζ-function is defined over .
11F37; 14H52; 14L05
with q= exp(2π i τ) and I(τ)>0 be a primitive form of conductor N (i.e., a new normalized cusp Hecke eigenform on Γ0(N), cf. , Section 4.6) of even integer weight k. Assume that all Fourier coefficients are rational integers.
associated with g.
be the canonical decomposition of M into its non-holomorphic part M− and a mock modular form M+. Although the mock modular form M+ does not typically have rational Fourier coefficients, Theorem 1.1 in , implies the existence of
Here and throughout, ϕ∈K((q)) means that q t ϕ∈K ⟦ q ⟧ for some positive integer t.
and we will suppress the index p when that does not lead to confusion.
are bounded. The mock modular form M+ is defined modulo an addition of a weakly holomorphic modular form h∈M 2−k!(N) which is bounded at all cusps except infinity and has rational Fourier coefficients at infinity. However, since the Fourier coefficients of h must have bounded denominators, the choice of M+ does not affect the quantities . It was Zagier (unpublished) who first considered the quantities λ p in the case when k=12 and g=Δ∈S12(1): he claimed that there is an ‘optimal’ p-adic multiple of to correct . In this case, Zagier observed the following phenomenon (see Proposition 7 in Section ‘Proof of Theorem 1’ for a proof).
If g=Δ∈S12(1), then for all but possibly finitely many primes p, we have that .
to a primitive form g with rational Fourier coefficients.
In this paper (see Theorem 1 below), we prove (1) in the case when g is of weight 2. There are advantages to this case. Firstly, there is an abundance of examples since there are infinitely many primitive forms with rational integer Fourier coefficients. Secondly, the infinitude of supersingular primes for such a form proved by Elkies  yields a systematic involvement of quantities μ p .
Let g be a weight 2 primitive form. For all but possibly finitely many primes p, the quantities (and ) are p-adic integers.
Despite the obvious similarity between Proposition 1 and Theorem 1, our proofs of these two statements are based on completely different ideas.
It is not difficult to prove the assertion of Theorem 1 for ordinary primes (Proposition 6 below). However, the argument used in that proof of Proposition 6 does not generalize to other primes.
We derive Theorem 1 from Theorem 2 below. The latter, in our opinion, is an elegant statement of independent interest.
Section ‘Proof of Theorem 1’ of the paper is devoted to principal ideas involved into the proof of Theorem 1. The proofs of several propositions formulated in Section ‘Proof of Theorem 1’ are postponed to further sections. Specifically, in Section ‘Weak harmonic Maass forms and certain p-adic limits’ we recall some facts and definitions related to weak harmonic Maass forms and prove the initial version of Zagier’s claim (see Propositions 1, 7). In Section ‘Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζ-function’, we relate the weak harmonic Maass form M of weight 2−k=0 which is good for g to a pullback of the Weierstrass ζ-function. Section ‘One-dimensional commutative formal group laws’ is devoted to several technical statements on one-dimensional formal group laws. Finally, Section ‘The addition law for the Weierstrass ζ-function’ is devoted to (elementary) analysis of the addition law for the Weierstrass ζ-function which allows us to make a statement of global nature (i.e. ‘for all but possibly finitely many primes’).
Proof of Theorem 1
In order to clarify the ideas of our proof of Theorem 1, we begin with a relation between the theory of weak harmonic Maass forms of weight zero and classical Weierstrass theory of elliptic functions. This relation allows us, in particular, to obtain an interpretation (3) for the quantity α. The assumption k=2 is crucial for this discussion.
are rational numbers. The quotient is thus an elliptic curve over , and we denote by ω the nowhere vanishing differential on E normalized such that its pullback with respect to the covering map is dz.
such that ω pulls back to −2π i g(τ) d τ. Note that we do not consider the minimal model of E here, and thus, in particular, our considerations do not depend on Manin’s constant. The map is defined here as a complex-analytic map, but it is known to be also a birational map between algebraic varieties. In particular, the values of the J-invariant at the preimages of divison points are algebraic numbers. It is well-known that since , we have that , the origin of E, and the modular parametrization map is unramified at i ∞ (see e.g. , Lemma 1).
In Section ‘Weak harmonic Maass forms and certain p-adic limits’, we compare the functions M and N, and obtain the following:
for all but possibly finitely many primes p.
such that . Moreover, it allows us to immediately reduce the proof of Theorem 1 to the proof of the following statement.
is Λ2-periodic. This differential is exact if and only if h itself is Λ-periodic. A standard addition formula for the Weierstrass ζ-function (see (22)) implies that δ(ζ) is Λ2-periodic. For both functions z and we have that ; therefore, these functions are Λ2-periodic trivially. We can now interpret both the Hodge decomposition (6) and the Λ-periodicity of R as a decomposition of the meromorphic function Ψ(z)(=ζ(Λ,z)) such that δ(Ψ) is Λ2-periodic into a linear combination of z and modulo Λ-periodic functions. Note that Λ-periodic functions are exactly those which pull back from E. We denote the linear space of meromorphic Λ-periodic functions with poles outside Λ by . While ζ(Λ,z) has its poles in Λ, its shift by a Λ-periodic function ζ(Λ,z)+℘′(Λ,z)/2℘(Λ,z) has its poles outside Λ.
and ξ is called a coboundary if . The first de Rham cohomology is defined as the quotient of cocycles modulo coboundaries.
This definition introduces as a vector space over while Theorem 2 requires us to consider a related -module. In Section ‘Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζ-function’, we prove the following proposition which helps us to choose a natural normalization for that.
Let F be a one-dimensional formal group law over , and let be a cocycle. Then . □
This module was considered by N. Katz in . In particular, by Theorem 5.3.3 of , is a free -module of rank h=height(F(p)), where F(p) is the reduction of F modulo p, assuming that h<∞. As an example, for such a formal group law F, its logarithm by Chapter IV, Proposition 5.5 of , and thus obviously spans if h=1. We will also need an explicit basis of if h=2.
Let F be a one-dimensional formal group law over such that its modulo p reduction F(p) is of height h=1 or 2. Let be the logarithm of F.
When h=1, the one-dimensional -module is spanned by ℓ F (X).
When h=2, the two-dimensional -module is spanned by ℓ F (X) and p−1ℓ F (X p ).
We provide an elementary proof of Proposition 4 for h=2 in Section ‘Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζ-function’ of the paper. This proof generalizes to any h<∞, though we do not need and do not prove any generalization here. This proposition does not seem to be new. For instance, it was pointed out by the referee that Proposition 4 should follow from the fact proved by N. Katz in  that in the supersingular case the Dieudonné module gives the whole along with the well-known fact that the Frobenius map is bijective on .
Theorem 2 would follow from (8) immediately if was an element of . That however is not the case at least because .
We now consider the one-dimensional formal group law determined by Equation 5 (see Section ‘Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζ-function’ for a definition) with the formal group parameter X=−2x/y=−2℘(Λ,z)/℘′(Λ,z). Since , one can write ζ(Λ,z)=1/X+C0+Φ(X) with and a formal power series .
It will be convenient for us to introduce the following notation. Throughout, we write for a subring of which coincides with for some integer l. (For example, a power series means that for all integers n for all but possibly finitely many primes p.)
This observation combined with (10) and (8) accomplishes our proof of Theorem 2.
As it was mentioned above, Theorem 1 follows from Theorem 2.
Weak harmonic Maass forms and certain p-adic limits
In this section, g∈S k (N) is a primitive form with rational integral Fourier coefficients of arbitrary even integer weight k≥2. In a moment, we will pay special attention to the cases when N=1, and dimS k (1)=1, namely k=12,16,18,20,22,26. In particular, Zagier’s initial claim, namely Proposition 1 which motivated this project and served as its starting point, is a special case of Proposition 7 which is proved in this section.
- 1.For all we have
We have that Δ 2−k(M)=0.
The function M(τ) has at most linear exponential growth at all cusps of Γ 0(N).
takes weak harmonic Maass forms to weakly holomorphic (i.e. holomorphic on the upper half-plane with possible poles at cusps) modular forms of weight k and level N.
We now restrict our attention to the subspace of weak harmonic Maass forms which map to cusp forms in S k (N) under the ξ operator. Following , we say that a harmonic weak Maass form M∈H2−k(N) is good for g if the following conditions are satisfied:
The principal part of M at the cusp ∞ belongs to .
The principal parts of M at other cusps of Γ0(N) are constant.
We have that ξ(M)=∥g∥−2g, where ∥·∥ is the usual Petersson norm.
and introduce the metrics on the set of formal power series ϕ such that ord p (ϕ)>−∞ by putting . We tacitly identify rational numbers with elements of under the natural embedding .
Assume the notations and conventions above. If k>2, we assume additionally that p2|N for all primes p|N.
The quantities are defined uniquely if p2∤N.
We firstly consider the case when p2∣N. It follows from Theorem 4.6.17 (3) of , that b(p)=0, thus ; therefore, . At the same time, by Bol’s identity, , where D:=(2π i)−1d/d τ, and we conclude that also . It follows that λ p =μ p =0 does the job in this case.
Since now on, we assume that p2∤N. The uniqueness clause of Proposition 5 follows immediately from (and therefore, ) in this case by (, Theorem 4.6.17). We thus only need to prove the existence clause. Moreover, we can and will assume that k=2 if p∣N.
We now introduce certain quantities β and β′. Our definition depends on whether p∣N or p∤N.
If p∣N then by (, Theorem 4.6.17 (2)), we have b(p)2=pk−2, and we assumed that k=2; therefore, b(p)=±1. In this case, we put β=b(p) and β′=p β.
ordered such that ord p (β)≤ord p (β′).
Note that in both cases β β′=pk−1, thus ord p (β)+ord p (β′)=k−1>0, and therefore ord p (β′)>0.
We now consider separately p-ordinary (i.e. ord p (β)=0) and non-p-ordinary (i.e. ord p (β)>0) cases.
We begin with the p-ordinary case. In this case by Hensel’s lemma if p∤N, and by the definition above if p∣N.
Since R p ∈M 2−k!(N), we have that ord p (R p )>−∞. Thus, the limit exists in , since ord p (β′)=k−1>0.
We now consider the cases when p∤N and p∣N separately. We claim that in either case, λ p =u and μ p =0 do the job.
Since ord p (R p )>−∞, we conclude that as required.
and derive that as claimed.
We still have to show that both these quantities actually belong to , not merely . That follows from their definition along with the consideration of the action of the Galois group on (17).
The following proposition was proved by Zagier; this proposition motivates the claim about the adele.
For all but possibly finitely many primes p such that g is p-ordinary, we have ord p (λ p )≥0.
The proposition follows from (19) combined with (20) and the ultrametric inequality.
The proof of Proposition 6 does not generalize to non-ordinary primes, and in no case the set of non-ordinary primes is known to be finite. However, in the case g=Δ∈S12(1) (and in some similar cases), there is an easy argument independent on whether a prime is ordinary or not. Specifically, we now prove the following proposition which mildly generalizes Proposition 1.
If N=1 and k=12,16,18,20,22,26, then for all but possibly finitely many primes p we have that , and .
The case N=1 and g=Δ of weight k=12 was considered by Zagier. It was his observation that (−α,(λ p )) is an adele of . Our statement about μ p here does not seriously enhance Proposition 7. Firstly, we do not know whether there are infinitely many primes p such that p∣b(p) and therefore μ p may be non-zero. Secondly, the quantities μ p are obviously independent on the choice of α, therefore we do not see any natural way to speak about an adele class determined by these quantities. Although our proof of Proposition 7 may be generalized to some other cases, we do not know any interesting and natural infinite series of examples which may be treated using this kind of argument.
it follows from Theorem X.4.2 of , that all coefficients of , and, therefore, of as required.
Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζ-function
is a meromorphic Γ0(N)-invariant function on cusps. Note that while M+ has its only pole at infinity, N+ (and thus N+−C M+) may have poles both at other cusps and in the interior of the upper half-plane. However, since the map X0(N)→E is algebraic, N may have only finitely many poles τ i of multiplicities κ i in the fundamental domain , and the values of the J-invariant at these poles are algebraic numbers. It follows that is a weakly holomorphic modular form with algebraic coefficients of the principal parts of its its Fourier expansion at all cusps. Therefore, by Chapter 6.2 in , we have that the Fourier expansion at infinity , and dividing by back by we conclude that with ord p (N+−C M+)>−∞ for all but possibly finitely many primes p. That will imply the second claim of Proposition 2 if we show that , or, equivalently, that . We note that and by (2), take into the account that for almost all primes, and conclude that
One-dimensional commutative formal group laws
F(X,Y)=X+Y+(terms of degree≥2)
There is a unique power series such that F(X,ι(X))=0.
F(X,0)=X and F(0,Y)=Y.
All formal group laws considered in this paper are one-dimensional and commutative, and we will skip these adjectives.
A ring embedding allows one to consider a formal group law over as a formal group law over , and we do that tacitly using the ring embeddings for almost all primes p throughout.
We begin with Proposition 3.
with t= max(a,b) and such that as required.
We record the outcome of the above argument in the special case a=b=t=0 as a separate proposition.
Let F be a formal group law over . If satisfies and ord p (ξ′(0))=0, then .
Although the following proposition has a global nature, it follows immediately from Proposition 8.
Indeed, since , we have that ord p (ξ′(0))=0 for almost all primes p, and therefore, by Proposition 8, for almost all primes p, and our claim follows from that.
For example, for a formal group F over the ring the power series F r p (X):=X p is an endomorphism of F, and it is called Frobenius endomorphism.
We will need two formal group laws, D g and, and an isomorphism between them.
where c2=g2(Λ)/20, c3=g3(Λ)/28, etc. be the Laurent series of the Weierstrass ℘-function.
The short form of the Weierstrass equation typically is not minimal. One may use the minimal Néron model of E, and produce out of it a formal group law over using the addition law on E as in Chapter IV.2 of . Then is isomorphic to our over . Honda  (see also ) proved that the formal group laws and D g are isomorphic over . Our Proposition 10 below is a simplified version of this statement adapted for our purposes.
We put q=e2π i τ, and consider f as a function of τ which is the pullback of the rational function Z=−2x/y under . Thus f(τ) is a meromorphic modular function on Γ0(N). This function is bounded at infinity and has poles in the preimages τ i of . Since the map is an algebraic finite covering map between two algebraic varieties defined over , the function f(τ) may have only finitely many poles τ i with multiplicities κ i in the fundamental domain , and the values of the J-invariant at these points are algebraic numbers. The weakly holomorphic weight zero modular form has algebraic Fourier coefficients and a standard bounded denominators argument based on Theorem 3.52 of , implies that as required.
Let F be a formal group law defined over a ring . A ring homomorphism allows one to consider F as a formal group law over (taking the images under the ring homomorphism of all coefficients of the two-variable power series F).
Let us now prove Proposition 4.
and that is exactly what we need.
The addition law for the Weierstrass ζ-function
Making use of the notations just introduced we firstly rewrite and analyze addition formula (21).
with a formal power series such that A(0,Z2)=A(Z1,0)=1.
Note that (24) is obvious with a formal power series satisfying A(Z1,Z2)=A(Z2,Z1) and A(0,Z2)=A(Z1,0)=1, because ℘(z)=1/z2+…. We thus only need to prove that in fact .
with as required.
We are now ready to prove (9).
Let . Then .
For i=1,2 let x i =Z i /W i and y i =−2/W i .
We thus conclude that as required.
derived from (10) and (8) valid for all but finitely many primes p. Here is the action of Frobenius on of the formal group law over .
The author is very grateful to Don Zagier for sharing his (unpublished) idea about the adele. The author thanks the referee for a big amount of remarks and suggestions which helped the author to improve the presentation significantly. This research is supported by Simons Foundation Collaboration Grant.
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