Talk on Gross-Keating invariants

Siegel-Weil formula

thm

For a positive definite even unimodular lattice $L$, $$\left( \sum_{M\in {\rm gen}(L)}\frac{\Theta_M^{(g)}(Z)}{|{\rm Aut}(M)|}\right)\,\cdot\, \left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1}= E^{(g)}_{k}(Z),$$

Moreover, the Fourier coefficients $a_{E}(N)$ of $E$ can be expressed as an infinite product of local densities $$ a_{E}(N)=\prod_{p:\text{primes}}\beta_{L,p}(N) \label{lp} $$

mass formula

• for a half-integral $N$,

$$ a_{E}(N)=\left( \sum_{M\in {\rm gen}(L)}\frac{r_M(N)}{|{\rm Aut}(M)|}\right)\,\cdot\, \left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} $$ where $\Theta_M^{(g)}(Z)=\sum_{N}r_M(N)\exp\left(2\pi i \operatorname{Tr}(N\tau)\right)$

• if $2N$ is a Gram matrix of $L$, then we obtain

$$ a_{E}(N)=\left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} $$ as $$ r_M(N) = \begin{cases} |\operatorname{Aut}(L)|, & \text{if }L\sim M \\ 0, & \text{if }L\nsim M \\ \end{cases} $$

• then we can express

$$ a_{E}(N)=\left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} $$ in terms of local densities \ref{lp}, which gives the Smith-Minkowski-Siegel mass formula

related items

• Gross-Keating invariants of a quadratic form
• Siegel-Weil formula
• Local density of quadratic form
• Local Siegel series and Katsurada Fp polynomial
• 틀:수학노트
• 틀:수학노트