Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves (Q2372831)

This paper addresses the problem of determining a complex abelian variety from its period lattice from a computational viewpoint. In the case of elliptic curves, efficient algorithms are well-known. In the case of higher-dimensional abelian varieties, there has been significant progress for hyperelliptic curves whose Jacobian varieties have a given period lattice. There exist two main directions in the proposed solutions. The first can be summarized as follows: given a normalized period matrix in the Siegel upper half space corresponding to an abelian variety, one uses Thetanullwerte to compute certain algebraic invariants of a curve whose Jacobian is isomorphic to the desired abelian variety. One then deduces from these invariants the field of moduli of the curve and finds an equation for the curve over its field of definition. This approach leads to equations with large coefficients, which must then be reduced by some additional method, and the arithmetic of the problem is obscured since the initial abelian variety and the Jacobian of the curve found are only isomorphic over an algebraic closure of the field of definition. The second proposed solution, based on [J. Algebra 253, No. 1, 112--132 (2002; Zbl 1054.14041)], uses \textit{Jacobian Nullwerte} and applies to genus 2 curves. More precisely, there exists an algorithm whose input is the period lattice associated to a basis of algebraic differential forms on an abelian surface and its output is the equation of a genus 2 hyperelliptic curve which is defined over the same field over which the given differential forms are defined. This algorithm has the advantage of preserving the arithmetic of the problem, but cannot be applied when one only knows a normalized period matrix in the Siegel upper half plane, since in general these periods do not correspond to algebraic differential forms. The paper under review originated from an attempt to overcome this difficulty. In [Commun. Pure Appl. Math. 29, 813--819 (1976; Zbl 0342.14020)], \textit{A. Weil} presented a method for determining a basis of algebraic differential forms from a given normalized period matrix. The author states that Jacobian Nullwerte are the key tool for making Weil's construction explicit. The combination of this idea with the author's algorithm for abelian surfaces mentioned above leads to a particular type of equation for hyperelliptic curves which the author has christened \textit{symmetric models} of the curve. The paper contains a geometric study of symmetric models of hyperelliptic curves \textit{of any genus}. Further, the arithmetic properties of these models are studied. The contents of the paper are as follows. After a preliminary section, symmetric equations are studied in Section 2. Section 3 summarizes known results concerning Jacobian Nullwerte for hyperelliptic curves. In Section 4, classical formulas of \textit{J. Thomae} [J. Reine Angew. Math. 71, 201--222 (1870; JFM 02.0244.01)], relating Thetanullwerte and Jacobian Nullwerte to Weierstrass points on hyperelliptic curves, are stated. Sections 5 and 6 discuss theoretical implications of these results. The construction of algebraic differential forms from a normalized period matrices is explained in Section 7. Section 8 contains a general method for finding symmetric equations of a general hyperelliptic curve having a given normalized period matrix. Sections 9 and 10 discuss genus 2 and genus 3 curves, respectively. To conclude, we quote from the paper the following interesting remarks: (a) ``An important consequence of Corollary 8.13 and Proposition 2.13 is the fact that Jacobian Nullwerte provide an effective solution to the hyperelliptic Schottky problem'', and (b) ``The simple expressions in [Proposition 9.3] [merit] attention for practical applications. In case we want to compute a (symmetric) equation for [a genus 2 curve] \(C\) from its period matrix, it happens that only six different Thetanullwerte are involved [\dots]. In particular, the computation of the Igusa invariants of \(C\) by means of these formulas requires only six numerical evaluations of the Theta function. This represents a gain of 40\% with [respect to] the methods [used] in [previous works]''.