arXiv subjects
Publications and source records attributed to Juan Marcos Cerviño.
We study the theta map which assigns to a real quadratic form its theta series. We introduce two invariants reflecting whether the differential of the theta map vanishes or is degenerate. We provide examples of lattices where this differential is zero. These invariants turn out to be modular forms for integral lattices. We illustrate this in the rank two case.
Given an integral lattice $Λ$ of rank $n$ and a finite sequence $m_1 \leq m_2 \leq ... \leq m_k$ of natural numbers we construct a modular form $Θ_{m_1,m_2,...,m_k,Λ}$ of level $N=N(Λ)$. The weight of this modular form is $nk/2+\sum_{i=1}^k m_k$. This construction generalizes the theta series $Θ_Λ$ of integral lattices, because $Θ_Λ= Θ_{0,Λ}$. We give the $q$-expansions of the modular forms $Θ_{m,m,Λ}$, and $Θ_{1,1,1,Λ}$ and show that (up to some scaling) they are given by power series with integer coefficients.
We derive lattice invariants from the heat flux of a lattice. Using systems of harmonic polynomials, we obtain sums of products of spherical theta functions which give new invariants of integer lattices which are modular forms. In particular, we show that the modular forms $Θ_{nn,Λ}$ depend only from lengths and angles in the lattice.
We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic p=29,...,97.