Über eine zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlicher

Some elementary aspects of modular functions in several variables

“…Thus it can be shown by a generalization of the Siegel-Götzky method [9], that it suffices to prove an identity by showing it holds on the zero manifolds (14.1) up to a finite number of derivatives with respect to the distance from the zero manifold. The details are quite laborious, but the following "zeroth approximation" will suffice to demonstrate the formalism : Let \l/(c, d\T) be a quadruple satisfying the same functional equations as 0(c, d\Ty (like (10.11)) for fe = 2 or 3.…”

The fifteen theorem for universal Hermitian lattices over imaginary quadratic fields

“…In 1928, Götzky [7] proved that x 2 1 + x 2 2 + x 2 3 + x 2 4 is universal over Q( √ 5). In 1941, Maass [20] proved the three square theorem, which states: the quadratic form x 2 1 + x 2 2 + x 2 3 is universal over Q( √ 5).…”

“…Recently, Bhargava and Hanke enunciated that they proved the 290-conjecture which characterizes the universality of (nonclassical) quadratic forms. That is, if a (nonclassical) quadratic form represents the 29 numbers, 1, 2, 3, 5,6,7,10,13,14,15,17,19,21,22,23,26,29,30,31,34,35,37,42,58,93,110,145,203,290, then it is universal (see [9]). One can use this theorem to prove the universality of a Hermitian lattice.…”

Universal octonary diagonal forms over some real quadratic fields

“…More concretely, she showed there are 178 quaternary quadratic forms f (x, y, z, w) up to equivalence such that f is positive definite integral quadratic form, the coefficients of cross terms of f are always even and the equation f = n is solvable for all n ∈ Z + . On the other hand, the study of positive universal quadratic integral forms over totally real number fields was initiated by F. Götzky [3]. In 1928, he proved that the sum of four squares is universal over Q( √ 5).…”