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We first prove some equivalent statements on J-stability of families of critically finite entire functions. Then, with these in hand, a conjecture concerning stability of the family of exponential functions is affirmatively answered in some cases.

We consider families of abelian varieties parametrized by classical theta-functions, and show that specifying the family and a CM point in Siegel space determines the grössencharacter of the corresponding CM abelian variety. We associate an adelic group to the family, and describe the kernel of the grössencharacter as the pull-back of the group under the map in Shimura’s Reciprocity Law.

is uniform, the solution is a rational function, or a rational function of eIIz (u constant), or an elliptic function. It is part of a famous theorem of Weierstrass that every uniform function with an algebraic addition theorem is of one of the three types just enumerated. It is a result of Picard that if two meromorphic functions are algebraically related to one another, either the relation is of genus zero, in which case the two functions are rational functions of a single meromorphic function, or the relation is of genus unity, in which case the two functions are elliptic functions of a single integral function.t The present paper treats a problem which might easily be suggested by any of the above theorems. We seek all meromorphic functions f(z) such that, for some linear X(z), an algebraic relation

Then the function f(z) has both a rational multiplication and a rational addition theorem. We propose to determine all such functions. Our problem is a generalization of one treated in a paper by Ritti1), in which all periodic meromorphic functions having a multiplication theorem were determined. This is equivalent to taking S(z) =z in (2). For this case, Ritt proved that \m\ ^ 1 ; and if \m\ > 1, then f(z) is a linear function of one of the functions eaz, cos(az+s), p(z+s); when gs = 0, g»2(z+j3); when g2 = 0, S>'(z+s) and j?3(z+j3). Here a is arbitrary while s is restricted to certain values. If \m\ =1, then it was shown that m is either —1 or a third, fourth or sixth root of unity. The forms for the function f(z) were also explicitly given in this case. Use will be made of the results quoted and also of the methods of Ritt's paper. We distinguish the following three cases : (A) m >1; (B) m ?1, with m not a rational root of unity; (C) m a primitive wth root of unity. For Case (A), which is the one usually considered in multiplication theorems, we can restate the problem in a way which shows its connection with the theory of a class of functions first systematically studied by Poincare(2). He provided an existence theorem for meromorphic functions satisfying (1) assuming that 7?(z) has a fixed point(3) a for which 7?'(a) has a modulus greater than unity. Thus our problem is equivalent to that of finding all Poincare functions having a rational addition theorem.

A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let Ω \Omega be a bounded Euclidean region. With the underlying space taken as the class M \mathcal {M} of all nonnegative functions u on Ω \Omega admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over λ ≥ 0 \lambda \geq 0 of the regularized reduced functions for ( u − λ ) + {(u - \lambda )^ + } . Quasi-bounded and singular functions are then defined as those u for which S u = 0 Su = 0 and S u = u Su = u , respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita(1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on Ω \Omega is quasi-bounded if and only if there exists a nonnegative, increasing, convex function φ \varphi on [ 0 , ∞ ] [0,\infty ] such that φ ( x ) / x → + ∞ \varphi (x)/x \to + \infty as x → ∞ x \to \infty and φ ∘ u \varphi \circ u admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in M \mathcal {M} satisfying S u ≤ u Su \leq u .

The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function a e y z a{e^{yz}} on the disk | z | ≦ ρ |z| \leqq \rho . It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for a e y z a{e^{yz}} on | z | ≦ ρ |z| \leqq \rho converge uniformly to a e y z a{e^{yz}} on each bounded subset of the plane.

of degree n in w and of genus p, and seek to determine those cases in which w can be expressed in terms of z by means of radicals. Why the results obtained here should not have been found before this time is a question which has puzzled us as much as it will puzzle the reader. Our results for the case in which n is prime are fairly complete. We show first that if n is prime, and if w can be expressed in terms of radicals, then for every genus except zero there exists an upper bound for n. For any given genus the mechanisms of the possible Riemann surfaces for w can be determined in a finite number of steps. For the case of genus zero our problem becomes that of finding all rational functions whose inverses can be expressed in terms of radicals. Given one such function, others can be obtained by performing linear transformations upon the function and upon the variable. We show that all rational functions of prime degree with inverses expressible in terms of radicals can be thus obtained from functions of the following types: (a) The powers of w. (b) The polynomials which occur in the formulas for the multiplication of the argument in the function cos u. (c) The fractional rational functions which occur in the formulas for the transformations of the periods of the function pu. (d) For the case of n = 1 (mod 4), the fractional rational functions met, in the lemniscatic case (g3 = 0), in the formulas for the multiplication of the argument of &o u by a i 3i, where a2 + 32 = n2. (e) For the case of n 1 (mod 6), the fractional rational functions met, in the equianharmonic case (g2 = 0), in the formulas for the multiplication of the arguments of p'u and p"2 u by a i fi -63, where a2 + 332 = n2. The functions just described are already known to have inverses expressible

The paper contains some new extensions of the well-known theorem of F. Carlson for entire functions of exponential type.

A strengthening of Siegel's proof of the Hermite-Lindemann Theorem is given. The results are used to obtain lower bounds for the values of exponential polynomials at algebraic points. The question of how well the