Representations of integers as sums of an even number of squares

Modular forms and period polynomials

“…For small values of N (eg N = 2, 3, 4, 5), the polynomials ρ f are completely determined by the principal parts ρ f (I), so that the relations above are completely explicit via the computation of ρ − (R I,a )(I) in [FY09]. In the remainder of this section, we discuss in detail the case of period polynomials for Γ = Γ 0 (2), which have been studied in [IK05], [FY09], [KT11].…”

ON THE THETA SERIES ATTACHED TO $D_{m}^{+}$–LATTICES

“…T (z) (up to the factor e πiz/4 ) is the generating function for the triangular numbers. For 8|m, T m is a modular form of weight m 2 on Γ 0 (2) [3]. As is well-known…”

“…In a recent paper [3], Imamoglu and one of the authors proved that θ m for 8|m is a finite rational linear combination of products of two specific Eisenstein series of weight m 2 on the Hecke congruence subgroup Γ 0 (4) (recall that for N ∈ N one defines Γ 0 (N ) = " a b c d « ∈ Γ(1)|N |c where Γ(1) = SL 2 (Z)). In particular, r m (n) for 8|m can be expressed as a finite rational linear combination of products of two modified elementary divisor functions.…”

“…To proceed one has to make use of a result given in [2] which can be stated as saying that ϑ D + m for 8|m is the image of essentially the mth power of the generating On the Theta Series Attached to D + m -Lattices 3 function for the triangular numbers under the trace map from Γ 0 (2) to Γ(1). One then applies the main result of [3]. Note that if 8|m a similar argument can also be applied to the theta series (with levels > 1) attached to the lattices D m and its duals D * m [1, pp.…”

On theta series vanishing at ∞ and related lattices

“…Introduction. In a recent paper [8], Imamoḡlu and Kohnen have studied the mth power of the Riemann theta function ϑ in relation with the number r m (n) of representations of a positive integer n as a sum of m integral squares. Their result is interesting, since, for each m, the computation of r m (n) does not require any pre-knowledge of r m (n ′ ) for n ′ < n. One of the main tools used in this proof was that ϑ m has highest order of vanishing at one cusp or, better, that a translate of ϑ has highest order of vanishing at the cusp ∞; subsequently Kohnen and the second author extended the result to the integral representations of the lattice D + m (see [9]).…”