An explicit theorem of the square for hyperelliptic Jacobians

The action of the Frobenius map on rank 2 vector bundles over a supersingular genus 2 curve in characteristic 2

“…one obtains a basis (unique up to scalar if we ask for these conditions) of W (see [5], Following [6] (Lemma 3.3), one can express, using the Abel-Jacobi map and an explicit theorem of the square (see [1]), the rational functions…”

“…Namely, we consider a family X of genus 2 curves parameterized by a discrete valuation ring R with ordinary generic fiber and special fiber isomorphic to X . In order to find an R-basis of the free R-module W := H 0 (J X , 2 ) and to express the canonical theta functions of order 2 in that basis, we pull back 2 on the product X × X via the Abel-Jacobi map X × X → J X and make use of an explicit theorem of the square for hyperelliptic Jacobians (see [1]). As for the two other cases, we can easily deduce a full description of the Verschiebung V : M X 1 (2) M X (2) (Proposition 5.1).…”

“…the set {( p +p), p ∈ C}). From[1], we know that b * ( ) = M + n N , where n is a nonnegative integer, and that σ * (M) = C × {∞} + {∞} × C. Furthermore, as σ −1 (N ) =¯ , σ * (N ) = k¯ , where k is an nonnegative integer satisfying the equation deg(σ )(N )2 = k 2 (¯ ) 2 .On the one hand, since N is the exceptional curve of the blowing up b : Sym 2 C → J C, we have (N ) 2 = −1. On the other hand, the self-intersection number (¯ ) 2 is coincides with deg¯ (O(¯ ) ⊗ O¯ ).…”