# Explicit t-expansions for the elliptic curve [y.sup.2] = 4([x.sup.3] + Ax + b).

1. Introduction and the main result. Let R be a commutative ring with unit on which 2 is invertible. Let [??] be an elliptic curve over Spec R whose affine form is given by the equation [y.sup.2] = 4([x.sup.3] + Ax + B) for some A, B [member of] R satisfying 4[A.sub.3] + 27[B.sup.2] [member of] [R.sup.x]. Let [??] be the completion of E at the origin. We set t = -2x/y. Then [??] is canonically isomorphic to the formal spectrum of R[[t]].

In this paper, we give an explicit description of the pullbacks to [??] of some important functions and 1-forms on E. Our main result is the following:

Theorem 1. Let [??] [member of] R[[t]]dt denote the pull back of the invariant differential w = dx/y to [??]. Then for any integer k, the formal power series [x.sup.k][??]/dt [member of] R((t)) is equal to the sum

(1) [[infinity].summation over [(m,n=0)] (m + 2n - k + 1).sub.mn]/m!n!] [A.sup.m] [B.sup.n] [t.sup.4m+6n-2k].

Here [(m + 2n - k + 1).sub.m+n] denotes the Pochhammer symbol

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we understand [(m + 2n - k + 1).sub.m+n] = 1 when m = n = 0.

The proof of Theorem 1 will be given in Section 2. We give some other results in Section 3.

Remark 2. According to [6, p. 924, Remark], the formula (1) for k = 0 was already obtained by Beukers [3]. According to [7, p. 273], a generalization of the formula for k = 0 to the case of an elliptic curvegiven by a more general Weierstra[beta] equation was also obtained by Beukers [3], and recently perhaps independently by Sadek [5].

Remark 3. When A and B vary, the sum (1) is a formal power series of three variables A, B, and t with coefficients in Z.Ifweset A' = [At.sup.4] and B' = [Bt.sup.6], then Theorem 1 for k [less than or equal to] 0 is rewritten as

[t.sup.2k][x.sup.k][??]/dt = F((1 - k,k), -A', B'),

where the right hand side is the hypergeometric series of two variables in the sense of [2, Definition 3.1], associated to the set {2[v.sub.1] + 3[v.sub.2], - [v.sub.1] - 2[v.sub.2]} of linear forms. Similarly we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for k [greater than or equal to] 1. Here the right hand side is the coefficient-wise limit in R.

Now we give several consequences of Theorem

1. All of them follow immediately from Theorem 1 or from the argument of its proof, and the proofs are omitted.

Corollary 4. Let the notation and the assumption be as in Theorem 1.

(a) We have

[??]/dt = [[infinity].summation over (m, n=0] (2m + 3n)!/(m + 2n)!m!n! [A.sup.m] [B.sup.n] [t.sup.4m+6n]

(b) We have the equalities

x = - [[infinity].summation over (m, n=0) (2m + 3n - 2)!/(m + 2n - 1)!m!n! [A.sup.m] [B.sup.n] [t.sup.4m+6n-2]

and

y = 2 [[infinity].summation over (m, n=0) (2m + 3n - 2)!/(m + 2n - 1)!m!n! [A.sup.m] [B.sup.n] [t.sup.4m+6n-3]

in R((i)). Here we understand ((2m+3n-2)!/(m+2n-1)!m!n! = - 1 when m = n = 0. (Observe that , (2m+3n-2)!/(m+2n-1)!m!n! is an integer for any integers m, n [greater than or equal to] 0.)

(c) For any integer p, q [member of] Z satisfying k := p + q [not equal to] 0, the monomial [x.sup.p][y.sup.q] is equal to -[(-2).sup.q]/[t.sup.2p+3q] times

[summation over (m, n [greater than or equal to] 0] k[(m + 2n - k + 1).sub.m+n-1]/m!n! [A.sup.m] [B.sup.m] [t.sup.4m+6n]

in R((t)).(Observe that k[(m+2n-nk+1).sub.m+n-1]/m!n! is an integer for any integers m, n [greater than or equal to] 0.)[]

Remark 5. To be precise, the formulae for x, y, and [x.sup.p][y.sup.q] in Corollary 4 are not consequences of Theorem 1 but immediate consequences of the proof of Theorem 1 given in Section 2.

Corollary 6. Suppose that R is a Q-algebra.

(a) Let [log.sub.[??]] [member of] R[[t]] denote the formal logarithm associated to [[??].E] with respect to the formal parameter t. By definition [log.sub.[[??].E]] is the unique formal power series satisfying [dlog.sub.[??]] = [??] and [log.sub.[??]] (0)=0. We then have

[log.sub.[??]] = [[infinity].summation over (m, n=0] (2m + 3n)!/m + 2n)!m!n! [A.sup.m] [B.sup.n] [t.sup.4m+6n+1]/4m + 6n + 1.

(b) Let [??] [member of] R((t)) be a formal Laurent power series satisfying d[??] = -x[??]. Then [??] is equal to

c - [[infinity].summation over (m, n=0) (2m + 3n - 1)!/(m + 2n - 1)!m!n! [A.sup.m] [B.sup.n] [t.sup.4m=6n-1]/4m = 6n -1

for some constant c [member of] R. Here we understand (2m+3n-1)!/(m=2n-1)!m!n! = 1 when m = n = 0.

Remark 7. Corollary 6 (a) was announced (with the author's name) without proof in p. 289 of [4].

Corollary 8. Let the notation and assumption be as in Theorem 1.

(a) Suppose that B = 0. We then have

[??]/dt = 1/[square root of 1 - 4 A[t.sup.4]]

and

x[??]/dt = 1/2[t.sup.2][square root of 1 - 4A[t.sup.4] = 1/2[t.sup.2].

(b) Suppose that A = 0. (Observe that 6 is invertible in R in this case.) We then have

[??]/dt = [sub.2][F.sub.1] (1/3, 2/3; 1/2; 27/4 B[t.sup.6])

and

x[??]/dt = 2/3[t.sup.2][sub.2][F.sub.1] (1/3, 2/3; 1/2; 27/4 B[t.sup.6]) + 1/3[t.sup.2],

where [sub.2][F.sub.1]([alpha], [beta]; y; z) is Gau[beta] hypergeometric series

[sub.2][F.sub.1]([alpha], [beta]; y; z) = [[infinity].summation over (n=0 [([alpha]).sub.n] [([beta]).sub.n]/[(y).sub.n]n! [z.sub.n]. []

Remark 9. The claim (a) in Corollary 8 can be proved directly without using Theorem 1. We include this for completeness.

Suppose that R is a field which is complete with respect to an absolute value [absolute value]. When the absolute value [absolute value] is archimedean, let [alpha] be the unique real root of

4[[absolute value of A].sub.3]([T.sup.2] - 4) - 27[[absolute value of B].sup.2][T.sup.3]

satisfying 0 [less than or equal to] [alpha] [less than or equal to] 1 and set

r = 1/[square root of 6] [[(4 - [alpha]).sup.(4-[alpha])][(3[alpha]/[absolute value of A]).sup.3[alpha]][(2 - 2[alpha]/[absolute value of B])2-2[alpha].sup.)].sup.1/12]

When [absolute value] is non-archimedean, we set r = 1/ max([[absolute value of A].sup.1/4], [[absolute value of B].sup.1/6]).We use the terminology "analytic " to stand for real analytic, complex analytic, and rigid analytic in the case when [absolute value] is real archimedean, complex archimedean, and non-archimedian, respectively. When [absolute value] is archimedean (resp. non-archimedean), we let [E.sup.an] denote E(R) regarded as an analytic manifold (resp. an analytic space over R associated to E). It then can be checked easily that there exists a unique open neighborhood (resp. a unique admissible open neighborhood with respect to the strong G-topology) of the origin O in [E.sup.an] such that the rational function t on E gives an isomorphism from U to the open disk {t [absolute value] of t] < e}.

Corollary 10. Let the notation and assumption be as above.

(a) The formulae in Theorem 1 and Corollary 4, with [??] replaced with w, are valid on U.

(b) Suppose that R is of characteristic zero. Then there exists a unique analytic function [log.sub.E] on U and an analytic function [zeta] on U \ {O} such that d [log.sub.E] = Q, d[zeta] = -xw and that the value of [log.sub.E] at the origin is equal to zero. The formulae (a) and (b) in Corollary 6, with [log.sub.[??]] and [??] replaced with [log.sub.E] and [zeta], are valid on U and U \ {O}, respectively.

Almost all the material of this manuscript is a translation into English of my handwritten notes and my emails to Shinichi Kobayashi, all of which were written in Japanese on January 2004.

2. Proof of Theorem 1. In this section we give a proof of Theorem 1. Let the notation and assumption be as in Theorem 1. If suffices to prove the claim for the universal case when R is the localization R = Z[1/2, A, B, 1/(4[A.sup.3] + 27[B.sup.2])] of the polynomial ring over Z[1/2] of the two variables A and B. By choosing an injective ring homomorphism [??] : Z[1/2, A, B, 1/(4[A.sup.3] + 27[B.sup.2])] [??] C such that [??](A) and [??](B) are real numbers, we can reduce the proof to that in the case when R = C and both A and B are real numbers.

Let us assume that R = C and both A and B are real numbers. For k [member of] Z, we let [F.sub.k](t) denote the formal power series (1) with coefficients in R. Observe that the formal power series [t.sup.2k][F.sub.k](t) is absolutely convergent on [absolute value of t] < [c.sub.k] for a sufficiently small [c.sub.k] > 0. Hence it suffices to prove that for each integer k, the value of [x.sup.k]w/dt at t = a is equal to [F.sub.k](a) for infinitely many complex numbers a with 0 < [absolute value of a] < [c.sub.k].

Observe that, if (x, y) [member of] [C.sup.x] x [C.sup.x] satisfies [y.sub.2] = 4([x.sub.3] + Ax + B),then (t, u) = (-2x/y, 1/x) satisfies the equality

1 = [t.sup.2]/u + A[ut.sup.2] + B[u.sup.2][t.sup.2].

Let us fix t [member of] [C.sup.x] and set

f(u) = u(1 - [t.sup.2] - A[ut.sup.2] - B[u.sup.2][t.sup.2]),

which we regard as a holomorphic function of u. For [absolute value of t] sufficiently small, the function f(u) have a unique zero on [absolute value of u] < 1 , which we denote by [u.sub.0] .Then (x, y)= 1/u0, -2/([tu.sub.0])) is a point of E(C) with -2x/y = t.

We prove the claim for k = 0. By Jensen's formula (cf. [1, p. 208])

log[absolute value of f(0)] = -log [absolute value of 1/[u.sub.0]] = 1/2[pi] [[integral].sup.2[pi].sub.0] log[absolute value of f([e.sup.i[theta]])]d[theta]

we have

log [absolute value of [-t.sup.2] = log [absolute value of [u.sub.0] + 1/2[pi] Re [[integral].sup.2[pi].sub.0] log(1 - g([e.sup.i[theta]]))d[theta]

where

g(u) = [t.sup.2]/u + A[t.sup.2]u + B[t.sup.2][u.sup.2].

Since we have assumed that A and B are real numbers, [u.sup.0] is a positive real number if t is a sufficiently small real number. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

log [u.sub.0]

= log[t.sup.2] + [summation over (m, n [greater than or equal to] 0 (m, n)[not equal to](0, 0) (2m + 3n)!/m + 2n)!m!n! [A.sup.m] [B.sup.n] [([t.sup.2]).sup.2m+3n]/2m = 3n

if t is a sufficiently small real number. By differentiating with respect to t and by using w = td[u.sub.0] /(2[u.sub.0]), we obtain the desired equality Q/dt = [F.sub.0](t) for any sufficiently small real number t, which proves the claim for k = 0.

Next we consider the case when k [not equal to] 0. Let [r.sub.k] denote the residue of [u.sup.-k](uf(u))'/(uf(u)) at u = 0. By the residue theorem we have

(2) [x.sup.k] + [r.sub.k] = 1/2[pi]i [[integral].sub.[absolute value of u]=1] [u.sup.-k](uf(u))'/uf(u)du

for [absolute value of t] sufficiently small. We set

h(u) = u/[t.sup.2] - A[u.sup.2] - B[u.sup.3].

Since

(uf(u))'/uf(u) = (-1/[t.sup.2] - 2Au - 3B[u.sup.2])/1 - h(u)

= (- 1/[t.sup.2] - 2Au - 3B[u.sup.2]) [summation over (n [greater than or equal to] 0 h[(u).sup.n]

where the last infinite sum is absolutely convergent if [absolute value of u] is much smaller than [[absolute value of t].sup.2], we have [r.sub.k] = 0 for k < 0 and

(3) [r.sub.k] = - 1/[t.sup.2][C.sub.k,1] + 2A[C.sub.k,2] + 3B[C.sub.k,3]

for k > 0. Here [C.sub.kj] is the finite sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for j = 1, 2, 3, where [l.sub.k,j] (m, n) = (k - j) - (2m + 3n). On the other hands, since

(uf(u))'/uf(u) = [u.sup.-1] - 2A[t.sup.2] - 3B[t.sup.2]u/1 - g(u)

= ([u.sup.-1] - 2A[t.sup.2] - 3B[t.sup.2]u) [summation over (n [greater than or equal to] 0 g[(u).sup.n]

where the last infinite sum is absolutely convergent if [absolute value of u] = 1 and [absolute value of t] is sufficiently small, the right hand side of (2) is equal to

(4) [D.sub.k,0] - 2A[t.sup.2][D.sub.k,1] - 3B[t.sup.2] [D.sub.k,2].

Here [D.sub.k,j] is the infinite sum

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for j = 0, 1, 2, where [l'.sub.k,j](m, n)= m + 2n -(k - j). By (2), (3), and (4), the value of -[x.sup.k]/k is equal to the sum

[summation over (m, n [greater than or equal to] 0 (m + 2n - k + 1)/m!n! [A.sup.m] [B.sup.n] [t.sup.2(2m+3n-k)]

for [absolute value of t] sufficiently small. Since [x.sup.k]w = t/2 x d(-[x.sup.k]/k), we have the equality [x.sup.k]w/dt = [F.sub.k](t) for [absolute value of t] sufficiently small, which proves the claim for k [not equal to] 0.

3. Some other formulae. The method of the proof, given in Section 2, of Theorem 1 can be applied to a more general situation. Especially we can obtain in many cases explicit expansions of the pullbacks of functions or 1-forms on a plane curve over a field with respect to a local parameter at some closed point.

In this section we give several examples of such formulae. We omit the proofs of these formulae, since the main idea of the proofs is essentially the same as that of Theorem 1.

Theorem 11. Let m = 2g + 1 be a positive odd integer. Let C be a hyperelliptic curve over Q whose affine form is given by [y.sup.2] = [x.sup.m] - 1. We set t = -[x.sup.g]/y, which is a local parameter of C at the infinity. Let [[??].C] denote the completion of C at the infinity, which is canonically isomorphic to the formal spectrum of Q[[t]]. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in Q((t)), and the pullback [??] of w = [x.sup.g]-1 dx/(2y) to [??] has the following explicit description:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us go back to the situation in Theorem 1 and suppose that R is a subring of the field C of complex numbers. Let r and U be as in the paragraph just before Corollary 10. Let logE be the complex analytic function on U introduced in (b) of Corollary 10. We regard [log.sub.E] as a complex analytic function of t on the open disk {t | [absolute value of t] < r}. Let [LAMBDA] [subset] C denote the lattice generated by the periods of E(C) with respect to w. Let a be the Weierstrafi a-function on C with respect to the lattice A. We end this paper with two formulae on the t-expansions of some functions related to [sigma]. The author expect that they are useful for explicit computation related to the formal group law or the canonical height.

Theorem 12. Let the notation and assumption be as above. Let S be the set of quadruples (a, b, c, d) of integers a, b, c, d [less than or equal to] 0 satisfying (a, b, c, d) [not equal to] (0, 0, 0, 0).For (a, b, c, d) [member of] S, we set

[V.sub.a,b,c,d] = (2a + 3b - 1)! (2c + 3d)!/a + 2b - 1)! (c + 2d)! a!b!c!d!.

Here we understand (2a + 3b - 1)!/(a + 2b - 1)! = 1 when (a, b) = (0, 0).(Observe that [V.sub.a,b,c,d] is an integer for any a, b, c, d [less than or equal to] 0.) Then -log([sigma]([log.sub.E](t))/t) is equal to the sum

[summation over ((a, b, c, d)[member of]S [V.sub.a, b, c, d][A.sup.a + c] [B.sup.b + d] [t.sup.4a + 6b + 4c + 6d]/(4a + 6b - 1)(4a + 6b + 4c +6d)

for any complex number t with [absolute value of t] < r.

In order to state the last formula in this paper, we need to introduce some more notation. For non-negative integers m, n, a, b [less than or equal to] 0 satisfying the condition

(*) 2m + 3n = a + b,

let us introduce two integers [E.sub.m, n, a, b], [F.sub.m, n, a, b] [member of] Z.

Let m, n, a, b [less than or equal to] 0 be integers satisfying the condition (*). Let [XI](m, n, a, b) denote the set of pairs ([m.sub.1], [n.sub.1]) [member of] Z x Z satisfying the following conditions:

0 [greater than or equal to] [m.sub.1] [greater than or equal to] m, 0 [greater than or equal to] [n.sub.1] [greater than or equal to] n,

[m.sub.1] + [n.sub.1] [greater than or equal to] a [greater than or equal to] 2[m.sub.1] + 3[n.sub.1] - 1.

For ([m.sub.1], [n.sub.1]) [member of] [XI](m, n, a, b), we let [e.sub.m, n, a, b] ([m.sub.1], [n.sub.1]) denote the integer

(2[m.sub.1] + 3[n.sub.1] - a)a!b! (a -([m.sub.1] + [n.sub.1]))! (b -([m.sub.2] + [n.sub.2]))![m.sub.1]![n.sub.1]![m.sub.2]![n.sub.2]!,

where [m.sub.2] = m - [m.sub.1] and [n.sub.2] = n - [n.sub.1] .We set

[E.sub.m, n, a, b] = [summation over ([m.sub.1], [n.sub.1])[member of][XI](m, n, a, b) [e.sub.m, n, a, b]([m.sub.1], [n.sub.1]).

If either a = 0 or b = 0,then we have [E.sub.m, n, a, b] = 0 since the set [XI](m, n, a, b) is an empty set. Let [theta] (m, n, a, b) denote the set of integers m1 satisfying the conditions

max{0, a - 3n} [greater than or equal to] [2.sub.m1] [greater than or equal to] min{2m, a, 2m + b - 1}, 2[m.sub.1] = a mod 3.

For [m.sub.1] [member] [theta](m, n, a, b), we let [f.sub.m, n, a, b] ([m.sub.1]) denote the integer

a!(b - 1)!/(a - ([m.sub.1] + [n.sub.1]))!((b - 1) - ([m.sub.2] + [n.sub.2]))![m.sub.1]![n.sub.1]![m.sub.2]![n.sub.2]!,

where [m.sub.2] = m - [m.sub.1], [n.sub.1] = a-2[m.sub.1]/3, and [n.sub.2] = n - [n.sub.1]. We set

[F.sub.m, n, a, b] = [summation over ([m.sub.1][member of][theta](m, n, a, b) [f.sub.m, n, a, b](mi)

If b = 0, then we have [F.sub.m, n, a, b] = 0 since there exists no integer [m.sub.1] satisfying the condition above. When b [less than or equal to] 1, we also set [F'.sub.m, n, a, b] = (2 - 1/b) [E.sub.m, n, a, b] + [F.sub.m, n, a, b].

Theorem 13. Let the notation be as above. We then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for (s, t) [member of] C x C satisfying [absolute value of s], [absolute value of t] < r. []

Acknowledgments. The author is grateful to my former classmate Takehiro Kaneko. A problem on probability theory which he brought up in 1993 have lead the author to the formulae presented in this paper. The author had not noticed any importance or applicability of the result for several years. He would like to thank Shinichi Kobayashi and Takuya Yamauchi for having suggested him of possible importance of the results in this paper. He would like to thank Shinichi Kobayashi also for careful reading of the manuscript of the paper, for a lot of helpful comments. Finally, the author would like to give his heartfelt thank to Noriko Hirata-Kohno. Without her enthusiastic persuading for publication, the author would never make up his mind to write up this manuscript.

doi: 10.3792/pjaa.89.123

References

[1] L. V. Ahlfors, Complex analysis, third edition, McGraw-Hill, New York, 1979.

[2] K. Aomoto and M. Kita, Theory of hypergeometric functions, translated from the Japanese by Kenji Iohara, Springer Monographs in Mathematics, Springer, Tokyo, 2011.

[3] F. Beukers, Une formule explicite dans la theorie des courbes elliptiques. (Preprint).

[4] K. Bannai and S. Kobayashi, Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers, Duke Math. J. 153 (2010), no. 2, 229-295.

[5] M. Sadek, Formal groups and combinatrial objects, arXiv:1303.6706.

[6] J.H.Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, New York, 1986.

[7] J. Stienstra and F. Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. Ann. 271 (1985), no. 2, 269-304.

Seidai YASUDA

Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-8502, Japan

(Communicated by Heisuke HIRONAKA, M.J.A., Oct. 15, 2013)

2000 Mathematics Subject Classification. Primary 14H52; Secondary 33E05, 33C75.