<> = Scharlau's talk = == Basic structure & operations == * $R$ ground ring, e.g. $R = Z_F$ * $F$ is the quotient ring of $R$ * $V$ a vector space over $F$ * basic structure: a quadratic module $(L,b)$, where * $L\subseteq V$ is a f.g. $R$-module s.t. $FL=V$ * $b : L\times L\rightarrow F$ symmetric bilinear form === Operations === * scaling : $^\tau(L,b) := (L, \tau b)$ * multiplying : $\tau\cdot(L,b) := (\tau L, b)$ * dualizing: $(L,b)^\# := (L^\#, b)$ $L^\# := \{ y\in V \;:\; b(x,y)\in R\;\forall x\in L \}$ * intersection: $L\cap M$ * sum $L+M$ * sublattices * defined by generators * defined by congruences in particular: $L_{v,p} := \{x\in L \;:\; b(v,x) \in p \}$ for $p$ an ideal of $R$ and $v\in L^\#$. * radical modulo $p$ === Example === * "partial dual" of $(L,b)$ $m\in F$, then $D_m(L,b) := (L^\#\cap m^{-1} L, mb)$. typically, $L\subseteq L^\#$ and $m\mid level(L,b)$ (here $level(L,b):=exponent(L^\#/L)$.) * remark: for the theta series $\Theta_L$, the operator $D_p$ induces the Atkin-Lehner involution (at least in even dimension) * A Pari function for the partial dual, taking a Gram matrix as input [[attachment:partial_dual.gp]]