<<TableOfContents>>

= 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]]