The Conway-sloane Calculus for 2-adic Lattices

We develop the notational system developed by Conway and Sloane for working with quadratic forms over the 2-adic integers, and prove its validity. Their system is far better for actual calculations than earlier methods, and has been used for many years, but it seems that no proof has been published before now. Throughout, an integer means an element of the ring Z2 of 2-adic integers, and we write Q2 for Z2’s fraction field. A lattice means a finitedimensional free module over Z2 equipped with a Q2-valued symmetric bilinear form. Our goal is to develop the Conway-Sloane notational system for such lattices [4, ch. 15]. This calculus is widely used and much simpler than previous systems, but the literature contains no proof of its validity. Conway and Sloane merely mention that they established the correctness of their system of moves by “showing that they suffice to put every form into B. W. Jones’ canonical form [5] yet are consistent with G. Pall’s complete system of invariants [6]”. The only written proof is Bartels’ dissertation [2], which remains unpublished and lacks the elementary character of [4, ch. 15]. And although the foundations are correct, Conway and Sloane made an error defining their canonical form, resolved in [1]. We hope that the present self-contained treatment will make their calculus more accessible. Briefly, the Conway-Sloane notation attaches a “2-adic symbol” to each Jordan decomposition of a lattice; a complicated example is 1II [2 4]316 1 1 32 2 II 64 −2 II [128 1 256]0512 −4 II A term 1±n II or 1 ±n t represents a unimodular lattice of dimension n, with the decorations specifying which such lattice. (II is a formal symbol and t is an integer mod 8.) If q is a power of 2 then we replace 1±n II and 1 ±n t by q±n II and q ±n t for the lattice got by scaling inner products by q. The chain of symbols represents a direct sum. Terms gathered in brackets “share” their values of the invariant (oddity) appearing in subscripts, Date: November 14, 2015. 2010 Mathematics Subject Classification. 11E08. First author supported by NSF grant DMS-1101566.