 Research
 Open Access
On Zagier’s adele
 Pavel Guerzhoy^{1}Email author
https://doi.org/10.1186/2197984717
© Guerzhoy; licensee Springer. 2014
 Received: 16 January 2014
 Accepted: 3 February 2014
 Published: 18 August 2014
Abstract
Purpose
Don Zagier suggested a natural construction, which associates a real number and padic 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.
Methods
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 ζfunction is defined over $\mathbb{Z}[1/6]$.
MSC
11F37; 14H52; 14L05
Keywords
 Modular Form
 Elliptic Curve
 Fourier Coefficient
 Formal Power Series
 Primitive Form
Background
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 Γ_{0}(N), cf. [1], Section 4.6) of even integer weight k. Assume that all Fourier coefficients $b\left(n\right)\in \mathbb{Z}$ are rational integers.
associated with g.
be the canonical decomposition of M into its nonholomorphic 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 [3], implies the existence of $\alpha \in \mathbb{R}$
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 ${\lambda}_{p},{\mu}_{p}\in {\mathbb{Q}}_{p}$. It was Zagier (unpublished) who first considered the quantities λ_{ p } in the case when k=12 and g=Δ∈S_{12}(1): he claimed that there is an ‘optimal’ padic multiple of ${\mathcal{E}}_{g}$ to correct ${\mathcal{\mathcal{M}}}_{\alpha}$. In this case, Zagier observed the following phenomenon (see Proposition 7 in Section ‘Proof of Theorem 1’ for a proof).
Proposition 1
If g=Δ∈S_{12}(1), then for all but possibly finitely many primes p, we have that ${\lambda}_{p}\in {\mathbb{Z}}_{p}$.
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 [5] yields a systematic involvement of quantities μ_{ p }.
Theorem 1
Let g be a weight 2 primitive form. For all but possibly finitely many primes p, the quantities ${\lambda}_{p}\in {\mathbb{Z}}_{p}$ (and $p{\mu}_{p}\in {\mathbb{Z}}_{p}$) are padic integers.
Remark 1
Despite the obvious similarity between Proposition 1 and Theorem 1, our proofs of these two statements are based on completely different ideas.
Remark 2.
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 padic 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 ζfunction’, we relate the weak harmonic Maass form M of weight 2−k=0 which is good for g to a pullback of the Weierstrass ζfunction. Section ‘Onedimensional commutative formal group laws’ is devoted to several technical statements on onedimensional formal group laws. Finally, Section ‘The addition law for the Weierstrass ζfunction’ is devoted to (elementary) analysis of the addition law for the Weierstrass ζfunction which allows us to make a statement of global nature (i.e. ‘for all but possibly finitely many primes’).
Methods
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 $E=\mathbb{C}/\Lambda $ 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 $\psi :\phantom{\rule{1em}{0ex}}\mathbb{C}\to \mathbb{C}/\Lambda $ 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 ${\mathcal{E}}_{g}$ is defined here as a complexanalytic map, but it is known to be also a birational map between algebraic varieties. In particular, the values of the Jinvariant at the preimages of divison points are algebraic numbers. It is wellknown that since ${\mathcal{E}}_{g}=q+\dots \in \mathbb{Z}\mathrm{\u27e6q\u27e7}$, we have that ${\mathcal{E}}_{g}\left(\mathrm{i\infty}\right)=0\in \Lambda $, the origin of E, and the modular parametrization map ${\mathcal{E}}_{g}$ is unramified at i ∞ (see e.g. [7], Lemma 1).
In Section ‘Weak harmonic Maass forms and certain padic limits’, we compare the functions M and N, and obtain the following:
Proposition 2
for all but possibly finitely many primes p.
such that ${M}^{+}\alpha {\mathcal{E}}_{g}\in \mathbb{Q}\left(\phantom{\rule{0.3em}{0ex}}\right)q\phantom{\rule{0.3em}{0ex}})$. Moreover, it allows us to immediately reduce the proof of Theorem 1 to the proof of the following statement.
Theorem 2
be the associated Eichler integral, and let Λ=Λ_{ g } be the lattice in defined above.
is Λ^{2}periodic. This differential is exact if and only if h itself is Λperiodic. A standard addition formula for the Weierstrass ζfunction (see (22)) implies that δ(ζ) is Λ^{2}periodic. For both functions z and $\overline{z}$ we have that $\delta \left(z\right)=\delta \left(\overline{z}\right)=0$; therefore, these functions are Λ^{2}periodic trivially. We can now interpret both the Hodge decomposition (6) and the Λperiodicity of R as a decomposition of the meromorphic function Ψ(z)(=ζ(Λ,z)) such that δ(Ψ) is Λ^{2}periodic into a linear combination of z and $\overline{z}$ modulo Λperiodic functions. Note that Λperiodic functions are exactly those which pull back from E. We denote the linear space of meromorphic Λperiodic functions with poles outside Λ by . While ζ(Λ,z) has its poles in Λ, its shift by a Λperiodic function ζ(Λ,z)+℘^{′}(Λ,z)/2℘(Λ,z) has its poles outside Λ.
and ξ is called a coboundary if $\xi \left(X\right)\in {\mathbb{Z}}_{p}\mathrm{\u27e6X\u27e7}\otimes {\mathbb{Q}}_{p}$. The first de Rham cohomology ${H}_{\mathit{\text{dR}}}^{1}\left(F\right)$ is defined as the quotient of cocycles modulo coboundaries.
This definition introduces ${H}_{\mathit{\text{dR}}}^{1}\left(F\right)$ as a vector space over ${\mathbb{Q}}_{p}$ while Theorem 2 requires us to consider a related ${\mathbb{Z}}_{p}$module. In Section ‘Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζfunction’, we prove the following proposition which helps us to choose a natural normalization for that.
Proposition 3
Let F be a onedimensional formal group law over ${\mathbb{Z}}_{p}$, and let $\xi \in {H}_{\mathit{\text{dR}}}^{1}\left(F\right)$ be a cocycle. Then ${\xi}^{\prime}\left(X\right)\in {\mathbb{Z}}_{p}\mathrm{\u27e6X\u27e7}\otimes {\mathbb{Q}}_{p}$. □
This module was considered by N. Katz in [10]. In particular, by Theorem 5.3.3 of [10], $\mathbb{D}\left(F\right)$ is a free ${\mathbb{Z}}_{p}$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 ${\ell}_{F}\in \mathbb{D}\left(F\right)$ by Chapter IV, Proposition 5.5 of [11], and thus obviously spans $\mathbb{D}\left(F\right)$ if h=1. We will also need an explicit basis of $\mathbb{D}\left(F\right)$ if h=2.
Proposition 4
Let F be a onedimensional formal group law over ${\mathbb{Z}}_{p}$ such that its modulo p reduction F^{(p)} is of height h=1 or 2. Let ${\ell}_{F}\left(X\right)\in {\mathbb{Q}}_{p}\mathrm{\u27e6X\u27e7}$ be the logarithm of F.
When h=1, the onedimensional ${\mathbb{Z}}_{p}$module $\mathbb{D}\left(F\right)$ is spanned by ℓ_{ F }(X).
When h=2, the twodimensional ${\mathbb{Z}}_{p}$module $\mathbb{D}\left(F\right)$ is spanned by ℓ_{ F }(X) and p^{−1}ℓ_{ 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 ζfunction’ 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 [10] that in the supersingular case the Dieudonné module gives the whole ${H}_{\mathit{\text{crys}}}^{1}$ along with the wellknown fact that the Frobenius map is bijective on ${H}_{\mathit{\text{crys}}}^{1}$.
Theorem 2 would follow from (8) immediately if $\zeta (\Lambda ,{\mathcal{E}}_{g})$ was an element of $\mathbb{D}\left({D}_{g}\right)$. That however is not the case at least because $\zeta (\Lambda ,{\mathcal{E}}_{g})=1/q+\dots \in \mathbb{Q}\left(\phantom{\rule{0.3em}{0ex}}\right(q\left)\phantom{\rule{0.3em}{0ex}}\right)\u2ac5\u0338{\mathbb{Q}}_{p}\mathrm{\u27e6q\u27e7}$.
We now consider the onedimensional formal group law$\xca$ determined by Equation 5 (see Section ‘Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζfunction’ for a definition) with the formal group parameter X=−2x/y=−2℘(Λ,z)/℘^{′}(Λ,z). Since $X\in \mathrm{z\xe2\u201e\u0161\u27e6z\u27e7}$, one can write ζ(Λ,z)=1/X+C_{0}+Φ(X) with ${C}_{0}\in \mathbb{Q}$ and a formal power series $\Phi \left(X\right)\in \mathrm{X\xe2\u201e\u0161\u27e6X\u27e7}$.
It will be convenient for us to introduce the following notation. Throughout, we write for a subring of which coincides with $\mathbb{Z}[1/l]$ for some integer l. (For example, a power series $\varphi =\sum u\left(n\right){q}^{n}\in \mathcal{Z\u27e6q\u27e7}$ means that $u\left(n\right)\in {\mathbb{Z}}_{p}$ 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.
Results
Weak harmonic Maass forms and certain padic 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 $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma}_{0}\left(N\right)$ we have$M\left(\frac{\mathrm{a\tau}+b}{\mathrm{c\tau}+d}\right)={(\mathrm{c\tau}+d)}^{2k}M\left(\tau \right).$
 2.
We have that Δ _{2−k}(M)=0.
 3.
The function M(τ) has at most linear exponential growth at all cusps of Γ _{0}(N).
We denote the vector space of weak harmonic Maass forms by H_{2−k}(N).
takes weak harmonic Maass forms to weakly holomorphic (i.e. holomorphic on the upper halfplane 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 [2], we say that a harmonic weak Maass form M∈H_{2−k}(N) is good for g if the following conditions are satisfied:

The principal part of M at the cusp ∞ belongs to $\mathbb{Q}\left[\phantom{\rule{0.3em}{0ex}}{q}^{1}\right]$.

The principal parts of M at other cusps of Γ_{0}(N) are constant.

We have that ξ(M)=∥g∥^{−2}g, 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 $\parallel \varphi {\parallel}_{p}:={p}^{{\text{ord}}_{p}\left(\varphi \right)}$. We tacitly identify rational numbers with elements of ${\mathbb{Q}}_{p}$ under the natural embedding $\mathbb{Q}\hookrightarrow {\mathbb{Q}}_{p}$.
Proposition 5
Assume the notations and conventions above. If k>2, we assume additionally that p^{2}N for all primes pN.
The quantities ${\lambda}_{p},{\mu}_{p}\in {\mathbb{Q}}_{p}$ are defined uniquely if p^{2}∤N.
Proof.
The argument below is a refinement of similar arguments from the proof of Theorem 1.1 in [4] and the proof of Theorem 1.2 in [3] adapted for our current purposes.
We firstly consider the case when p^{2}∣N. It follows from Theorem 4.6.17 (3) of [1], that b(p)=0, thus ${\mathcal{\mathcal{M}}}_{\alpha}U\in {M}_{2k}^{!}(N)$; therefore, ${\text{ord}}_{p}\left({\mathcal{\mathcal{M}}}_{\alpha}\rightU)>\infty $. At the same time, by Bol’s identity, ${D}^{k1}\left({\mathcal{\mathcal{M}}}_{\alpha}\right)\in {M}_{k}^{!}\left(N\right)$, where D:=(2π i)^{−1}d/d τ, and we conclude that also ${\text{ord}}_{p}\left({\mathcal{\mathcal{M}}}_{\alpha}\right)>\infty $. It follows that λ_{ p }=μ_{ p }=0 does the job in this case.
Since now on, we assume that p^{2}∤N. The uniqueness clause of Proposition 5 follows immediately from ${\text{ord}}_{p}\left({\mathcal{E}}_{g}\right)=\infty $ (and therefore, ${\text{ord}}_{p}\left({\mathcal{E}}_{g}\rightV)=\infty $) in this case by ([1], 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 ([1], Theorem 4.6.17 (2)), we have b(p)^{2}=p^{k−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^{′}=p^{k−1}, thus ord_{ p }(β)+ord_{ p }(β^{′})=k−1>0, and therefore ord_{ p }(β^{′})>0.
We now consider separately pordinary (i.e. ord_{ p }(β)=0) and nonpordinary (i.e. ord_{ p }(β)>0) cases.
We begin with the pordinary case. In this case $\beta ,{\beta}^{\prime}\in {\mathbb{Q}}_{p}$ 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 ${lim}_{l\to \infty}{\beta {\prime}^{}}^{l}\mathcal{G}{U}^{l}$ exists in ${\mathbb{Q}}_{p}\mathrm{\u27e6q\u27e7}$, 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 ${\text{ord}}_{p}\left(\sum _{l\ge 0}{\beta {\prime}^{}}^{l}{R}_{p}{U}^{l}\right)>\infty $ as required.
and derive that ${\text{ord}}_{p}\left({\mathcal{\mathcal{M}}}_{\alpha}{\lambda}_{p}{\mathcal{E}}_{g}\right)>\infty $ as claimed.
We still have to show that both these quantities actually belong to ${\mathbb{Q}}_{p}$, not merely ${\mathbb{Q}}_{p}\left(\sqrt{p}\right)$. That follows from their definition along with the consideration of the action of the Galois group $\mathit{\text{Gal}}\left({\mathbb{Q}}_{p}\right(\sqrt{p})/{\mathbb{Q}}_{p})$ on (17).
The following proposition was proved by Zagier; this proposition motivates the claim about the adele.
Proposition 6
For all but possibly finitely many primes p such that g is pordinary, we have ord_{ p }(λ_{ p })≥0.
Proof.
The proposition follows from (19) combined with (20) and the ultrametric inequality.
The proof of Proposition 6 does not generalize to nonordinary primes, and in no case the set of nonordinary primes is known to be finite. However, in the case g=Δ∈S_{12}(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.
Proposition 7
If N=1 and k=12,16,18,20,22,26, then for all but possibly finitely many primes p we have that ${\lambda}_{p}\in {\mathbb{Z}}_{p}$, and ${p}^{k1}{\mu}_{p}\in {\mathbb{Z}}_{p}$.
Remark 3.
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 nonzero. 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.
Proof.
it follows from Theorem X.4.2 of [15], that all coefficients of ${\Delta}^{p}{p}^{k1}{R}_{p}\in \mathbb{Z}[1/6]\mathrm{\u27e6q\u27e7}$, and, therefore, of ${p}^{k1}{R}_{p}\in \mathbb{Z}[1/6]\mathrm{\u27e6q\u27e7}$ as required.
Weight zero weak harmonic Maass forms and the pullback of the Weierstrass ζfunction
is a meromorphic Γ_{0}(N)invariant function on $\overline{\mathfrak{H}}=\mathfrak{H}\cup $ 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 halfplane. However, since the map X_{0}(N)→E is algebraic, N may have only finitely many poles τ_{ i } of multiplicities κ_{ i } in the fundamental domain ${\Gamma}_{0}\left(N\right)\setminus \mathfrak{H}$, and the values $J\left({\tau}_{i}\right)\in \overline{\mathbb{Q}}$ of the Jinvariant at these poles are algebraic numbers. It follows that $(N\mathit{\text{CM}})\prod _{i}{\left(J\right(\tau )J({\tau}_{i}\left)\right)}^{{\kappa}_{i}}\in {M}_{0}^{!}\left(N\right)$ 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 [16], we have that the Fourier expansion at infinity $(N\mathit{\text{CM}})\prod _{i}{\left(J\right(\tau )J({\tau}_{i}\left)\right)}^{{\kappa}_{i}}\in \overline{\mathbb{Q}}\left(\phantom{\rule{0.3em}{0ex}}\right(q\left)\phantom{\rule{0.3em}{0ex}}\right)$, and dividing by back by $\prod _{i}{\left(J\right(\tau )J({\tau}_{i}\left)\right)}^{{\kappa}_{i}}$ we conclude that ${N}^{+}C{M}^{+}=N\mathit{\text{CM}}\in \overline{\mathbb{Q}}\left(\phantom{\rule{0.3em}{0ex}}\right(q\left)\phantom{\rule{0.3em}{0ex}}\right)$ 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 ${N}^{+}C{M}^{+}=N\mathit{\text{CM}}\in \mathbb{Q}\left(\phantom{\rule{0.3em}{0ex}}\right(q\left)\phantom{\rule{0.3em}{0ex}}\right)$, or, equivalently, that $\alpha +\mathbb{S}\left(\Lambda \right)/C\in \mathbb{Q}$. We note that ${\mathcal{\mathcal{M}}}_{\alpha}={M}^{+}\alpha {\mathcal{E}}_{g}\in \mathbb{Q}\left(\phantom{\rule{0.3em}{0ex}}\right(q\left)\phantom{\rule{0.3em}{0ex}}\right)$ and ${N}^{+}\frac{\mathbb{S}\left(\Lambda \right)}{C}{\mathcal{E}}_{g}\in \mathbb{Q}\left(\phantom{\rule{0.3em}{0ex}}\right(q\left)\phantom{\rule{0.3em}{0ex}}\right)$ by (2), take into the account that ${\text{ord}}_{p}\left({\mathcal{E}}_{g}\right)=\infty $ for almost all primes, and conclude that $\alpha +\mathbb{S}\left(\Lambda \right)/C\in \mathrm{\xe2\u201e\u0161.}$
Onedimensional commutative formal group laws
 (a)
F(X,Y)=X+Y+(terms of degree≥2)
 (b)
F(X,F(Y,Z))=F(F(X,Y),Z)
 (c)
F(X,Y)=F(Y,X).
 (d)
There is a unique power series $\iota \left(X\right)\in \mathcal{R\u27e6X\u27e7}$ such that F(X,ι(X))=0.
 (e)
F(X,0)=X and F(0,Y)=Y.
All formal group laws considered in this paper are onedimensional and commutative, and we will skip these adjectives.
A ring embedding ${\mathcal{R}}_{1}\hookrightarrow {\mathcal{R}}_{2}$ allows one to consider a formal group law over ${\mathcal{R}}_{1}$ as a formal group law over ${\mathcal{R}}_{2}$, and we do that tacitly using the ring embeddings $\mathbb{Z}\hookrightarrow \mathcal{Z}\hookrightarrow {\mathbb{Z}}_{p}$ for almost all primes p throughout.
We begin with Proposition 3.
Proof.
with t= max(a,b) and $b\in \mathbb{Z}$ such that ${p}^{b}{\xi}^{\prime}\left(0\right)\in {\mathbb{Z}}_{p}$ as required.
We record the outcome of the above argument in the special case a=b=t=0 as a separate proposition.
Proposition 8
Let F be a formal group law over ${\mathbb{Z}}_{p}$. If $\xi \in X{\mathbb{Q}}_{p}\mathrm{\u27e6X\u27e7}$ satisfies $\delta \left(\xi \right)(X,Y)\in {\mathbb{Z}}_{p}\mathrm{\u27e6X},\mathrm{Y\u27e7}$ and ord_{ p }(ξ^{′}(0))=0, then ${\xi}^{\prime}\left(X\right)\in {\mathbb{Z}}_{p}\mathrm{\u27e6X\u27e7}$.
Although the following proposition has a global nature, it follows immediately from Proposition 8.
Proposition 9
Let F be a onedimensional formal group over , and let $\xi \in \mathrm{X\xe2\u201e\u0161\u27e6X\u27e7}$ satisfy $\delta \left(\xi \right)(X,Y)\in \mathcal{Z\u27e6X},\mathrm{Y\u27e7}$. Then ${\xi}^{\prime}\left(X\right)\in \mathcal{Z\u27e6X\u27e7}$.
Proof.
Indeed, since ${\xi}^{\prime}\left(0\right)\in \mathbb{Q}$, we have that ord_{ p }(ξ^{′}(0))=0 for almost all primes p, and therefore, by Proposition 8, ${\xi}^{\prime}\left(X\right)\in {\mathbb{Z}}_{p}\mathrm{\u27e6X\u27e7}$ for almost all primes p, and our claim follows from that.
For example, for a formal group F over the ring $\mathbb{Z}/p\mathbb{Z}$ the power series F r_{ p }(X):=X^{ p } is an endomorphism of F, and it is called Frobenius endomorphism.
thus the logarithm series ℓ_{ F } determines the group law F. For formal group laws F and G over a algebra , the formal power series ${\ell}_{G}^{1}\left({\ell}_{F}\right(X\left)\right)\in \mathcal{R}\u27e6\mathcal{X}\u27e7$ gives an isomorphism F→G over .
We will need two formal group laws, D_{ g } and$\xca$, and an isomorphism between them.
which is a priori defined over , is in fact defined over (that is ${D}_{g}(X,Y)\in \mathbb{Z}\mathrm{\u27e6X},\mathrm{Y\u27e7}$).
where c_{2}=g_{2}(Λ)/20, c_{3}=g_{3}(Λ)/28, etc. be the Laurent series of the Weierstrass ℘function.
gives an isomorphism ${D}_{g}\to \xca$ over .
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 $\stackrel{~}{E}$ over using the addition law on E as in Chapter IV.2 of [11]. Then $\stackrel{~}{E}$ is isomorphic to our$\xca$ over . Honda [12] (see also [13]) proved that the formal group laws$\xca$ and D_{ g } are isomorphic over . Our Proposition 10 below is a simplified version of this statement adapted for our purposes.
Proposition 10
Proof.
We put q=e^{2π i τ}, and consider f as a function of τ which is the pullback of the rational function Z=−2x/y under ${\mathcal{E}}_{g}\phantom{\rule{1em}{0ex}}:\phantom{\rule{1em}{0ex}}{X}_{0}\left(N\right)\to E=\mathbb{C}/\Lambda $. Thus f(τ) is a meromorphic modular function on Γ_{0}(N). This function is bounded at infinity and has poles in the preimages τ_{ i } of $\frac{1}{2}\Lambda \setminus \Lambda $. Since the map ${\mathcal{E}}_{g}$ is an algebraic finite covering map between two algebraic varieties defined over $\overline{\mathbb{Q}}$, the function f(τ) may have only finitely many poles τ_{ i } with multiplicities κ_{ i } in the fundamental domain ${X}_{0}\left(N\right)=\overline{{\Gamma}_{0}\left(N\right)\setminus \mathfrak{H}}$, and the values of the Jinvariant $J\left({\tau}_{i}\right)\in \overline{\mathbb{Q}}$ at these points are algebraic numbers. The weakly holomorphic weight zero modular form $f\left(\tau \right)\prod _{i}{\left(J\right(\tau )J({\tau}_{i}\left)\right)}^{{\kappa}_{i}}\in {M}_{0}^{!}\left(N\right)$ has algebraic Fourier coefficients and a standard bounded denominators argument based on Theorem 3.52 of [16], implies that $f\in \mathcal{Z\u27e6q\u27e7}$ as required.
Let F be a formal group law defined over a ring ${\mathcal{R}}_{1}$. A ring homomorphism ${\mathcal{R}}_{1}\to {\mathcal{R}}_{2}$ allows one to consider F as a formal group law over ${\mathcal{R}}_{2}$ (taking the images under the ring homomorphism of all coefficients of the twovariable power series F).
Let us now prove Proposition 4.
Proof.
and that is exactly what we need.
The addition law for the Weierstrass ζfunction
with $\varphi \left(t\right)\in \mathbb{Z}[1/2,{g}_{2},{g}_{3}]\mathrm{\u27e6t\u27e7}\subset \mathcal{Z\u27e6t\u27e7}$.
Making use of the notations just introduced we firstly rewrite and analyze addition formula (21).
Proposition 11
with a formal power series $A({Z}_{1},{Z}_{2})\in \mathcal{Z\u27e6}{Z}_{1},{Z}_{2}\u27e7$ such that A(0,Z_{2})=A(Z_{1},0)=1.
Proof.
Note that (24) is obvious with a formal power series $A({Z}_{1},{Z}_{2})\in \mathrm{\xe2\u201e\u0161\u27e6}{Z}_{1},{Z}_{2}\u27e7$ satisfying A(Z_{1},Z_{2})=A(Z_{2},Z_{1}) and A(0,Z_{2})=A(Z_{1},0)=1, because ℘(z)=1/z^{2}+…. We thus only need to prove that in fact $A({Z}_{1},{Z}_{2})\in \mathcal{Z\u27e6}{Z}_{1},{Z}_{2}\u27e7$.
with $A({Z}_{1},{Z}_{2})\in \mathcal{Z\u27e6}{Z}_{1},{Z}_{2}\u27e7$ as required.
We are now ready to prove (9).
Proposition 12
Let $\Phi \left(Z\right)=\zeta \left({\ell}_{\xca}\right(Z\left)\right)1/Z\in \mathbb{Z}\mathrm{\u27e6Z\u27e7}$. Then $\Phi \left(\xca\right({Z}_{1},{Z}_{2}\left)\right)\Phi \left({Z}_{1}\right)\Phi \left({Z}_{2}\right)\in \mathcal{Z\u27e6}{Z}_{1},{Z}_{2}\u27e7$.
Proof.
For i=1,2 let x_{ i }=Z_{ i }/W_{ i } and y_{ i }=−2/W_{ i }.
We thus conclude that $\Phi \left(\xca\right({Z}_{1},{Z}_{2}\left)\right)\Phi \left({Z}_{1}\right)\Phi \left({Z}_{2}\right)\in \mathcal{Z\u27e6}{Z}_{1},{Z}_{2}\u27e7$ as required.
Conclusion
derived from (10) and (8) valid for all but finitely many primes p. Here $\mathit{\text{Fr}}\left({\mathcal{E}}_{g}\right)\left(q\right)={\mathcal{E}}_{g}\left({q}^{p}\right)$ is the action of Frobenius on ${H}_{\mathit{\text{dR}}}^{1}$ of the formal group law over ${\mathbb{Z}}_{p}$.
Declarations
Acknowledgements
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.
Authors’ Affiliations
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