On Theta Series Vanishing at ∞ and Related Lattices
نویسندگان
چکیده
In this paper we consider theta series with the highest order of vanishing at the cusp ∞. When the level is a power of 2, these theta series are the m-th powers of a certain theta function with characteristic, related to the lattice √ 2kZ. Instead if the level is a power of 3, these theta series are the m/2-th powers of a theta series associated to the 2−dimensional root lattice A2 with characteristic. These modular forms have also many representations as theta series related to different lattices; we prove that the lattices of level a power of 2, respectively a power of 3, that afford these theta series, are lattices constructed from binary, respectively ternary codes.
منابع مشابه
New Look at Hecke’s Indefinite Theta Series
where Q is an indefinite quadratic form on Z, f(m,n) is a doubly periodic function on Z such that the sums of f(m,n)q over all vertical and all horizontal lines in Z vanish. Some of these series appeared as coefficients in univalued triple Massey products on elliptic curves computed via homological mirror symmetry in [3]. In particular, in this context the condition of vanishing of sums over ve...
متن کاملTheta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials
We find two combinatorial identities on the theta series of the root lattices of the finitedimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of the well-known Essenstein series E4(z) and Ramanujan series ∆24(z). Using these two identities, we determine the theta series of certain infinite families of postive definite even unimodular lattices obtained b...
متن کاملDecoding in Compute-and-Forward Relaying: Real Lattices and the Flatness of Lattice Sums
Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However, the theta series is in general not known in closed form, excluding a small set of very special lattices. In this article, motivated by the practical applicatio...
متن کاملA New Look at Hecke’s Indefinite Theta Series
where Q is an indefinite quadratic form on Z, f(m,n) is a doubly periodic function on Z such that the sums of f(m,n)q over all vertical and all horizontal lines in Z vanish. Some of these series appeared as coefficients in univalued triple Massey products on elliptic curves computed via homological mirror symmetry in [3]. In particular, in this context the condition of vanishing of sums over ve...
متن کاملModular forms for the even unimodular lattice of signature (2,10)
More than forty years ago, Igusa proved in [Ig] a “Fundamental Lemma”, according to which, theta constants give an explicit generically injective map from some modular varieties related to the symplectic group Sp(n,R) into a projective space. The embedded varieties satisfy some quartic relations, the so called Riemann’s relations. We expect that similar results hold for orthogonal instead of sy...
متن کامل