Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Question

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.

I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of dimension 2g) and elements of $(\mathbb{Z}/2\mathbb{Z})^{2g}$ (that I also call half-integer characteristics).

This is very well-know by the experts so good references would be definetly enough. Let us fix the 4x4 matrix, divided in 2x2 blocks, $M:=\left(\begin{array}{cc} 0 & Id_g \\ -Id_g & 0 \\ \end{array}\right)$. With coefficients in $\mathbb{Z}/2\mathbb{Z}$, it is also symmetric. Now I think it is true to say that all the quadratic forms on $V:=(\mathbb{Z}/2\mathbb{Z})^{2g}$ can be written as follows

$$q(x)=x_1x_{g+1} + \dots + x_gx_{2g} + \sum_{i=1}^{2g} \epsilon_i x_i^2$$

where $\epsilon_i \in \mathbb{Z}/2\mathbb{Z}$. Is this true? Remark that the first part of the formula is the quadratic form ssociated to $M$.

Hence there is a clear action of $(\mathbb{Z}/2\mathbb{Z})^{2g}$ on the set of quadratic forms and we can identify them non-canonically.

Is this correct? where can I find this properly explained?

Answer 1

Let me call $m$ the symmetric biIinear form associated to $M$. In your first question you mean "all quadratic forms whose associated bilinear form is $m$"; then the answer is yes. There is indeed a canonical, simply transitive action of $V$ on the set of such quadratic forms; for $v\in V$, the form $v\cdot q$ is given by $x\mapsto q(x)+m(v,x)$. That does not mean that "we can identify them non-canonically" : these quadratic forms fall into two different types, distinguished by the Arf invariant.

I suppose you can find this in any book on quadratic forms. Since you mention theta-characteristics let me suggest this paper by Saavedra, which gives a smooth introduction to the subject.