Scharlau's talk
Basic structure & operations
R ground ring, e.g.R=ZF F is the quotient ring ofR V a vector space overF basic structure: a quadratic module
(L,b) , whereL⊆V is a f.g.R -module s.t.FL=V b:L×L→F symmetric bilinear form
Operations
scaling :
τ(L,b):=(L,τb) multiplying :
τ·(L,b):=(τL,b) dualizing:
(L,b)#:=(L#,b) L#:={y∈V:b(x,y)∈R∀x∈L}
intersection:
L⋂M sum
L+M - sublattices
- defined by generators
- defined by congruences
in particular:
Lv,p:={x∈L:b(v,x)∈p} forp an ideal ofR andv∈L# .
radical modulo
p
Example
"partial dual" of
(L,b) m∈F , thenDm(L,b):=(L#⋂m−1L,mb) .
typically,
L⊆L# andm|level(L,b) (here
level(L,b):=exponent(L#/L) .)remark: for the theta series
ΘL , the operatorDp induces the Atkin-Lehner involution (at least in even dimension)
- A Pari function for the partial dual, taking a Gram matrix as input