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

partial_dual.gp