On the Functional Independence of Ratios of Theta Functions

Voi. 13, 1927

MA THEMA TICS: S. LEFSCHETZ

657

Moore, R. L., Trans. Amer. Math. Soc., 21, 1920 (345). Whyburn, G. T., "Cyclicly Connected Continuous Curves," these PROCEEDINGS, 13, 1927 (31-38). 4Moore, R. L., Fund. Math., 3, 1921 (233-237). 5 Moore, R. L., Math. Zeit., 15, 1922 (254-260). 6 Schoenflies, A., Die Entwickelung der Lehre von den Punktmannigfaltigkeiten, Leipzig, 1908, S. 237. 7Whyburn, G. T., "Concerning Continua in the Plane," Trans. Amer. Math. Soc., 29, 1927 (369-400). 8 Whyburn, G. T., these PROCEIDINGS, 12, 1926 (761-767). 9 Wilder, R. L., "Concerning Continuous Curves," Fund. Math., 7, 1925 (340-377), theorems 11 and 6. 10 For proofs of these and other related theorems for the case of the boundary of a complementary domain of a plane continuous curve, see Wilder, R. L., loc. cit. 1I Cf., for example, the proof of theorem 4 of my paper cited in ref. 8. Use will also be made here of Sierpinski's theorem that any continuum having property, S, is a continuous curve. See Sierpinski, W., Fund. Math., 1, 1920 (44-66). 12 That this set of points is a continuous curve has recently been proved by H. M. Gehman in his paper "Some Relations between a Continuous Curve and Its Subsets," Annals of Math. Ser. 2, 28, 1927 (103-111), theorem 8. 2 I

ON THE FUNCTIONAL INDEPENDENCE OF RATIOS OF THETA FUNCTIONS By S. LsvSCHPTZ DEPARTMSNT OF MATHEMATICS, PRINCXTON UNIVERSITY Communicated July 26, 1927

1. In the theory of multiply periodic functions there is a theorem which for many reasons it would be desirable to prove directly. Existing proofs either are not completely general or are very roundabout.' It may be stated thus: Consider a family of linearly independent e's of the same order and characteristic, in maximum number, attached to a definite period matrix, D. When the number of functions >p, their genus, then among their ratios to one of them there are p functionally independent. Let e, e2, . . . eGr be the functions and ul, . . ., up, the variables. The above theorem is equivalent to proving that the matrix eh

ash___

(1)

is of rank p + 1 when the (u)'s are arbitrary. This we shall do under the assumption that the order n is sufficiently great. Apparently restrictive, it is really immaterial in most cases. The proof, merely outlined here, is, as we shall see, not only direct but also very elementary.

MA THEMA TICS: S. LEFSCHETZ

658

PROC. N. A. S.

2. We take for Q the well-known normal form:

|lo, . .*, 0,1,0, .. .0; ajl, ... . and for e's those of characteristic zero: eh

ch)

=E exp [n7ri E aik(Mr + cj)(Mk + j,k

h(n

=

aj k

+2

ak,

E

(mj + cj)uJ;

= qh positive integer less than n).

The first summation is with respect to the m's which run through all integral values from -co to + co, while the q's are arbitrary within their limits. It will be observed that Q is not the most general normal matrix. In the general case the proof is the same but the formulas a little more involved. To establish our theorem it suffices to show that for fixed q's when the u's are all zero and n - co, (1) - a matrix of rank p + 1. We need, therefore, the limits of eh (0), 8eh(0), as n -

au,

co.

I say that the first expression-+ 1. Let a1k denote the imaginary part of akkX Since E a'jxk is a definite, positive, quadratic form, there exists a positive w such that 7rE ajkXjXk > heh( -)-1 | 2, since owing to c < 1/2, the bracket is always negative. + X

| (3h() *Ev

where

e

-

1 |