Linear independence of theta series of positive-definite, unimodular even lattices

TL;DR

The paper addresses the problem of when the degree $g$ theta series $\\theta_L^{(g)}$ of positive-definite, unimodular even lattices of rank $m\\ge 24$ become linearly independent. It proves the bound $\\frac{m}{2}\\le g_m\\le\\frac{3m}{4}$ for the minimal $g_m$ that yields injectivity of $\\vartheta_m^{(g)}$, by translating lattice data into automorphic representations of orthogonal groups, applying Arthur's multiplicity formula, and using Böcherer-type theta correspondence to relate $g_m$ to zeros/poles of associated $L$-functions. The method hinges on the standard parameter $\\psi(\\pi,St)$, the theta lift $\\vartheta_m^{(g)}$, and the analytic behavior of $L(s,\\pi)$ at integers to control possible theta-lift occurrences. The result is conditional on stabilizing the twisted trace formula, and it highlights how conjectural nonvanishing of $L(\tfrac12,f)$ could sharpen the bound. Overall, the work builds a bridge between lattice theta series and the automorphic spectrum of orthogonal groups to bound linear independence of theta series.

Abstract

We show that the minimal $g$ for which the degree $g$ theta series of positive-definite, unimodular even lattices of rank $m\geq24$ are linearly independent is bounded between $\frac{m}{2}$ and $\frac{3m}{4}$.

Theorems & Definitions (10)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Theorem 2
• Theorem 3: Theorem 8.5.8 and Conjecture 8.1.2 in CL, cf. Taibi
• Theorem 4: Theorem 3.4 in Boecherer
• Proposition 5
• proof
• proof : Proof of Theorem \ref{['thm:mainthm']}