Quadratic Minima and Modular Forms Ii

Carl Ludwig Siegel showed in [Siegel 1969] (English translation, [Siegel 1980]) that the constant terms of certain level one negative-weight modular forms Th are non-vanishing (“ Satz 2 ”), and that this implies an upper bound on the least positive exponent of a non-zero Fourier coefficient for any level one entire modular form of weight h with a non-zero constant term. Level one theta functions fall into this category. Their Fourier coefficients code up representation numbers of quadratic forms. For positive even h, Siegel’s result gives an upper bound on the least positive integer represented by a positive-definite even unimodular quadratic form in n = 2h variables. This bound is sharper than Minkowski’s for large n. (Mallows, Odlyzko and Sloane have improved Siegel’s bound in [Mallows, Odlyzko, and Sloane 1975].) John Hsia [private communication to Glenn Stevens] suggested that Siegel’s approach might be extended to higher levels. Following this hint, we constructed an analogue of Th for Γ0(2), which we denote as T2,h. To prove Satz 2, Siegel controlled the sign of the Fourier coefficients in the principal part of Th. In [Brent 1998] (henceforth, “part I”), following Siegel, we found upper bounds for the first positive exponent of a non-zero Fourier coefficient occuring in the expansion at infinity of an entire modular form with a non-zero constant term for Γ0(2) in the case h ≡ 0 (mod 4). Siegel’s method carried over intact. In part I, we also stated that it was not clear that Siegel’s method forces the non-vanishing of the T2,h constant terms when h ≡ 2 (mod 4). But it turns out that we can tweak our definition of the T2,h and carry out Siegel’s strategy. Let us denote the vector space of entire modular forms of weight h for Γ0(N) as M(N, h). In part I, we proved that the second non-zero Fourier coefficient of an an element of M(2, h) with non-zero constant term must have exponent at most dimM(2, h) = 1 + ⌊ h 4 ⌋