Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0012231

A mass formula for unimodular lattices with no roots

Oliver King
View original source ↗Read source document ↗Export metadata ↓

Abstract

We derive a mass formula for n-dimensional unimodular lattices having any prescribed root system. We use Katsurada's formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension n < 31. In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension n < 31, verifying Bacher and Venkov's enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants. The ASCII text file table.txt (1.3 MB) at arXiv:math.NT/0012231 includes the complete table of masses of 32-dimensional even unimodular lattices with any given root system, and may be dowloaded using the source link.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Oliver King. 2001-10-17. A mass formula for unimodular lattices with no roots. https://arxiv.org/abs/math/0012231

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

Given a cusp form $f$ which is supersingular at a fixed prime $p$ away from the level, and a Coleman family $F$ through one of its $p$-stabilisations, we construct a $2$-variable meromorphic $p$-adic $L$-function for the symmetric square of $F$. We prove that this new $p$-adic $L$-function interpolates values of complex imprimitive symmetric square $L$-functions, for the various specialisations of the family $F$. We use this $p$-adic $L$-function to prove a $p$-adic factorisation formula, expressing the geometric $p$-adic $L$-function attached to the Rankin--Selberg convolution of $f$ with itself as a the product of the $p$-adic symmetric square $L$-function of $f$ and a Kubota-Leopoldt $L$-function. This extends a result of Dasgupta in the ordinary case.

math.NT↗
arXiv2026

Exceptional poles of archimedean Rankin-Selberg L-functions for irreducible generic representations of GL(n,R)

For irreducible generic representations $π_1$ and $π_2$ of $\operatorname{GL}_n(\mathbb R)$, we prove that the notions of exceptional pole of type $1$ and type $2$ coincide at every level. When both representations are in general position, we use this identification to express the Rankin--Selberg $L$-function $L(s,π_1\timesπ_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of their derivatives.

math.NT↗