We show that the minimal $g$ for which the degree $g$ theta series of positive-definite even unimodular lattices of rank $m\geq24$ are linearly independent is bounded between $\frac{m}{2}$ and $\frac{3m}{4}$.
We look at genera of even unimodular lattices of rank $12$ over the ring of integers of $\mathbb{Q}(\sqrt{5})$ and of rank $8$ over the ring of integers of $\mathbb{Q}(\sqrt{3})$, using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank $12$ over the Eisenstein integers, even and unimodular over $\mathbb…
On the degree-$2$ Siegel theta series of extremal even unimodular lattices of ranks $48$, $72$, $96$, and $120$
Harvard Business School（哈佛商学院）；Department of Economics and Center of Mathematical Sciences and Applications, Harvard University（哈佛大学经济学系与数学科学及应用中心）；a16z crypto
In this paper, we study a family of rank two false theta series associated to the root lattice of type $A_2$. We show that these functions appear as Fourier coefficients of a meromorphic Jacobi form of negative definite matrix index. Hypergeometric $q$-series identities are also obtained.
We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension $n$, but it is practical and more efficient than canonical forms based on Minkowski reduction.
Theta functions for definite signature lattices constitute a rich source of modular forms. A natural question is then their generalization to indefinite signature lattices. One way to ensure a convergent theta series while keeping the holomorphicity property of definite signature theta series is to restrict the sum over lattice points to a proper subset. Although such series do not have the modular properties that a definite signature theta function has, as shown by Zwegers for signature $(1,n-1)$ lattices, they can be completed to a function that has these modular properties by compromising o…
Let $Q$ be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer $n$ that is locally represented by $Q$ but not represented. Assuming that $n$ is relatively prime to $D(Q)$, the determinant of the Gram matrix of $Q$, we show that $n$ is represented provided that \[ n \gg \max \{ N(Q)^{3/2 + ε} D(Q)^{5/4 + ε}, N(Q)^{2 + ε} D(Q)^{1 + ε} \}. \] Here $N(Q)$ is the level of $Q$. We give three other bounds that hold under successively weaker local conditions on $n$. These results are proven by bounding the Peter…
A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$
Harvard Business School（哈佛商学院）；Department of Economics and Center of Mathematical Sciences and Applications, Harvard University（哈佛大学经济与数学科学应用中心）；a16z crypto（a16z 加密）
In this paper we study new invariants $\widehat{Z}_{\boldsymbol{a}}(q)$ attached to plumbed $3$-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable $q$-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed $3$-manifold. Here we investigate the series $\widehat{Z}_{0}(q)$ for unimodular plumbing ${\tt H}$-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, $\widehat{Z}_{0}(q)$ is a depth two quantum modular form on $\mathbb{Q}$.
We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to representation theory of W-algebras. We derive modular transformation properties of regularized Kostant's partial and certain higher rank false theta functions. Modulo conjectures in representation theory, as an application, we compute regularized quantum dimensions of atypical …
Let L and L' be two integral Euclidean lattices in the same genus. We give an asymptotic formula for the number of Kneser p-neighbors of L which are isometric to L', when the prime p goes to infinity. In the case L is unimodular, and if we fix furthermore a subgroup A of L, we also give an asymptotic formula for the number of p-neighbors of L containing A and which are isomorphic to L'. These statements explain numerical observations in the recent classifications of unimodular lattices of rank 26, 27 and 28, by B. Allombert and the author. In an Appendix, O. Taïbi shows how to deduce from Arth…
We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdotθ(y))$, where $θ$ is an involution of $K$. We classify all pairs $K$, $θ$ such that $\mathscr O$ is similar to either an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also classify all pairs $K$, $θ$ such that $\mathscr O$ is a root lattice. In addition to this, we show that $\mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $\leqslant 48$, in particular, $…
