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Theory of Generalized Bernoulli-Hurwitz Numbers for the Algebraic Functions of Cyclotomic Type

arXiv:math/0304377v2 [math.NT] 30 Apr 2003

? Yoshihiro Onishi

1. Introduction. Let C be the projective curve of genus g de?ned by (1.1) y 2 = x2g+1 ? 1 (or y 2 = x2g+1 ? x).

The unique point of C at in?nity is denoted by ∞. Let consider the integral

x

(1.2)

u=

∞

xg?1 dx . 2y

This is an integral of a di?erential of ?rst kind on C which does not vanish at ∞. The integral converges everywhere. If g = 0 the inverse funcion of (1.2) is ?1/ sin2 (u). As is well-known, if g = 1 the inverse function of (1.2) is just the Weierstrass function ?(u) with ?′ (u)2 = 4?(u)3 ? 4 (or ?′ (u)2 = 4?(u)3 ? 4). The Bernoulli numbers {B2n } are the coe?cients of the Laurent expansion of ?1/ sin2 (u) at u = 0: (1.3) 22n B2n u2n?2 ?1 1 (?1)n . = 2? 2 u 2n (2n ? 2)! sin (u) n=1

∞

The Hurwitz numbers {H4n } are the coe?cients of the expansion (1.4) ?(u) = 1 24n H4n u4n?2 + . 2 u 4n (4n ? 2)! n=1

∞

These kinds of numbers are quite important, especially, in number theory. Because the inverse function on the neighborhood of u = 0 (1.2) can not extend globally with respect to u if g > 1, most of mathematician never considered (1.2) for g > 1, usually based on the classical idea of Jacobi, and worked in several variable functions. Surprisingly, for the case g > 1, the Laurent coe?cients of the inverse function of (1.2) have properties which properly resemble von Staudt-Clausen’s theorem and Kummer’s congruence for Bernoulli numbers ([C], [vS], [K]) and such theorems for Hurwitz numbers ([H2], [L]). To explain these facts is the aim of this paper. ? Detailed and extended exposition should be refered to [O]. 1

2

? Yoshihiro Onishi

2. Main results. Here we describe only for the curve C de?ned by y 2 = x5 ? 1. We consider the integrals

(x,y)

(2.1)

u1 =

∞

dx , 2y

(x,y)

u2 =

∞

xdx 2y

of the elements of a natural base of the di?erentials of ?rst kind. Since these integrals are converges, there exsists a inverse function (x(u), y(u)) of (x, y) → u = (u1 , u2 ) on the range of all values u of (2.1). On a neighborhood of u = (0, 0), the functions x(u) and y(u) are functions of the second variable u2 only. Hence we have the following di?erential equation: (2.2) If we denote (2.3)

dx du2

du2 x = . dx 2y = x′ (u), then x′ (u) = 2y . x

After squaring the two sides and substituting y(u)2 = x(u)5 ? 1, by removing the denominator, we have (2.4) x(u)2 x′ (u)2 = 4x(u)5 ? 4

This (2.4) is just a good analogy of ?′′ (u) = 6?(u)2 (or ?′′ (u) = 6?(u)2 ? 2),

obtained by ?′ (u)2 = 4?(u)3 ? 1 (or ?′ (u)2 = 4?(u)3 ? 4?(u)). Indeed, if we de?ne the numbers C10n and D10n by x(u) = (2.5) y(u) = C10n u2 10n?2 1 + , 2 u2 10n (10n)! n=1 ?1 D10n u2 10n?5 + , u2 5 n=1 10n (10n)!

∞ ∞

then we have two Theorems 2.7 and 2.8 below. Here, using the property that (2.6) x(?ζu1 , ?ζ 2 u2 ) = ζx(u1 , u2 ), y(?ζu1 , ?ζ 2 u2 ) = ?y(u1 , u2 ),

we know that only the terms in (2.5) appear. The ?rst Theorem is

Generalized Bernoulli-Hurwitz Numbers

3

Theorem 2.7 For each of C10n and D10n , there exist integers G10n and H10n such that C10n =

p≡1 mod 5 p?1|10n

Ap 10n/(p?1) p

+ G10n ,

D10n =

p≡1 mod 5 p?1|10n

(4!?1 mod p) Ap 10n/(p?1) p

+ H10n ;

where Ap = (?1)(p?1)/10

(p ? 1)/2 . (p ? 1)/10

This is a generalization of von Staudt-Clausen’s theorem. The second theorem is Theorem 2.8 For any prime p ≡ 1 modulo 5, and positive integers a and n such that 10n ? 2 ≧ a, if (p ? 1) | 10n, then

a

(?1)r

r=0 a

C10n+r(p?1) a ≡ 0 mod pa , Ap a?r 10n + r(p ? 1) r D10n+r(p?1) a Ap a?r ≡ 0 mod pa ; r (10n + r(p ? 1) (p ? 1)/2 . (p ? 1)/10

(?1)r

r=0

where Ap = (?1)(p?1)/10

This is a natural generalization of Kummer’s original congruence in [K] and of such a congruence for Hurwitz numbers ([L], p.190, (20)). To prove Theorem 2.8, we need Theorem 2.9 For any prime number p ≡ 1 mod 5 and any positive integer n, if p ? 1 | 10n then C10n /10n and D10n /10n belong to Z(p) .

3. About Proof of Theorems. We could not use any addition or multiplication formulae in the classical theory of Abelian or Jacobin varieties. To prove Therem 2.7 we use a technique based on a method of L. Carlitz [Car1], and to prove Theorem 2.9 we use a new technique that is an improved method from that of Carlitz. Theorem 2.8 is rather easily shown by Theorem 2.8. By using his method, Carlitz succeeded to prove Hurwitz’s theorem out side the prime 2 in his paper [Car1]. Furthermore, although he was trying to ?nd a generalization of von Staudt-Clausen’s type theorem, Kummer’s

4

? Yoshihiro Onishi

type congruence for hyperelliptic functions, he could not succeed. Our results is not only for hyperelliptic curve but also for any algebraic curves of type y a = xb ? 1, or y 2 = xb ? x

with gcd(a, b) = 1. ? The detailed proof and a lot of numerical examples are given in [O]. References

[Car1] L. Carlitz, The coe?cients of the reciprocal of a series, Duke Math. J., 8 (1941), 689-700. [Car2] L. Carlitz, Some properties of Hurwitz series, Duke Math. J., 16 (1949), 285-295. [Car3] L. Carlitz, Congruences for the coe?cients of the Jacobi elliptic functions, Duke Math. J., 16 (1949), 297-302. [Car4] L. Carlitz, Congruences for the coe?cients of hyperelliptic and related functions, Duke Math. J., 19 (1952), 329-337. [Cl] T. Clausen, Lehrsatz aus einer Abhandlung uber die Bernoullishen Zahlen, Astron. Nar¨ chr. 17 (1840), 325-330. ¨ [H1] A. Hurwitz, Uber die Entwicklungskoe?zienten der leminiskatishen Funktionen, Nachr. Acad. Wiss. G¨ttingen, (1897), 273-276 (Werke, Bd.II, pp.338-341). o ¨ [H2] A. Hurwitz, Uber die Entwicklungskoe?zienten der leminiskatishen Funktionen, Math. Ann., 51 (1899), 196-226 (Werke, Bd.II, pp.342-373). ¨ [K] E.E. Kummer, Uber eine allgemeine Eigenschaft der rationalen Entwickelungsco¨?e cienten einer bestimmten Gattung analytischer Functionen, J. f¨r die reine und angew. u Math. 41 (1851), 368-372. [L] H. Lang, Kummersche Kongruenzen fur die normierten Entwicklungskoe?zienten der Weierstrasschen ?-Funktionen, Abh. Math. Sem. Hamburg 33 (1969), 183-196. ? ? [O] Y. Onishi, Theory of Generalized Bernoulli-Hurwitz Numbers in the Algebraic Functions of Cyclotomic Type (in Japanese, 87 pages), downloadable from http://jinsha2.hss.iwateu.ac.jp/ ~onishi/ (2003). [vS] K.G.C. von Staudt, Bewies einer Lehrsatzes, die Bernoullishen Zahlen btre?rend, J. Reine Angew. Math. 21 (1840), 372-374. 3-18-34, Ueda, Morioka, 020-8550, Japan E-mail address: onishi@iwate-u.ac.jp

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