Scharlau's talk
Basic structure & operations
R ground ring, e.g. R = ZZ_F
F is the quotient ring of R
V a vector space over F
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
L_{v,p} := \{x\in L \;:\; b(v,x) \in pR \}
for p an ideal of R, 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