Mathematical background#

Let \(A\) be a dimension-\(g\) abelian variety together with a symmetric ample line bundle \(L\). We say that \(L\) has level \(n>0\) if the kernel \(K\) of the polarisation \(\phi_L : A \rightarrow \hat{A}\) is isomorphic to \((\mathbb{Z}/n\mathbb{Z})^{2g}\). We have \(\operatorname{dim} \Gamma(A, L)=n^g\) and, if \(t=(t_1, \ldots, t_{n^g})\) is a basis of sections of \(\Gamma(A, L)\), we have a projective morphism \(\mu_t : A \rightarrow P^{n^g}\), \(x \mapsto (t_i(x))\). By a Theorem of Lefschetz this morphism is an embedding if \(n\geq 3\). In practise, we consider even levels in order to be able to use duplication and Riemann formulas for the arithmetic. As the dimension of the ambiant space is \(2^n-1\) if \(n\) is the level, the lower level is the best for computations. Therefore, we mainly work with:

  • level 2: we obtain (generically) an embedding of the Kummer variety associated to \(A\) that is the quotient of \(A\) by the automorphism \((-1)\)

  • level 4 : we have an embedding of \(A\) and the full arithmetic on it.

The polarisation \(L\) induces a Weil paring on \(K\). Note that the embedding \(\mu_t\) depends on the choice of the basis \(t\). But the data of:

  1. a symplectic decomposition of \(K= K_1 \oplus K_2\) for the Weil pairing;

  2. sections \(s_1 : K_i \rightarrow G(L)\) where \(G(L)\) is the theta group of \(L\)

defines a unique basis of \(\Gamma(A, L)\). The data of 1) and 2) is called a theta structure after Mumford.

If \(\Theta\) is a theta structure, we denote by \(t_\Theta=(t_1, \ldots, t_{n^g})\) the unique basis of \(\Gamma(A,L)\) defined by \(\Theta\) and \(\mu_{t_\Theta} : A \rightarrow P^{n^g}\) the morphism. The point \(\mu_{t_\Theta}(0)=(t_i(0))\) is called the theta null point of \((A,L,\Theta\)). Its importance comes from the fact that, if \(n\geq 4\),

  • the data of the theta null point is equivalent to the data of \((A, L, \Theta)\);

  • it parametrises the Riemann equations which give equations for the embedding of \(A\) into \(P^{n^g}\) as well as the arithmetic of \(A\).