The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field

Abstract

An embedding of the Jacobian variety of a curve of genus 2 is given, together with an explicit set of defining equations. A pair of local parameters is chosen, for which the induced formal group is defined over the same ring as the coefficients of . It is not assumed that has a rational Weierstrass point, and the theory presented applies over an arbitrary ground field (of characteristic ╪ 2, 3, or 5).

Explicit computations on the desingularized Kummer surface

“…One can regard a point of the Jacobian J(C) as the class of a divisor X = r + u, where r = (x, y), u = (u, v) is a pair of points on C. Starting from this, Flynn constructs a projective embedding of the Jacobian as follows ( [4]). For a point X = {r, u} on C (2) (symmetric product) with r = (x, y), u = (u, v), define:…”

“…The Jacobian J (C) can be embedded in P 15 and is described by 72 quadratic equations ( [4]). More computable objects, S and its twists, appeared in recent attempts by M. Stoll and N. Bruin, to compute the Mordell-Weil group of J (C).…”

An Explicit Theory of Heights

“…Suppose that, for some finite extension L of K, there exist matrices Wx,W2,W3e W{L) such that N = W]xW2xW3\. Then we can take the height constant C2 = H{W~l)H{W-{)2H{W-1) 4 for HK on G.…”

“…A /^-rational point on the Jacobian of ^, denoted ^ = Jr(^7) = Pic°(^), can be represented by a divisor of the form (xi, y\ ) + {x2, y2) -oo+ -oo~ , where (xi, y\ ), {x2, y2) are any points on 'W including oo+ and oo~ . We shall follow [4] and use as a shorthand notation the unordered pair {{x\, y\), {x2, y2)} . This representation sets up a one-to-one correspondence with members of f{K), except that the canonical equivalence class of pairs of the form {(x, y), (x, -y)} represents a single menber of ß{K), denoted cf.…”

Computing the Mordell-Weil rank of Jacobians of curves of genus two

“…One missing ingredient there was the defining equations for /, which have now been worked out by Flynn in [10] and the second author in [11]. For computational reasons, more work has to be done beyond that stated in [6]: indeed, it is beneficial to find equations that define projective models of the homogeneous spaces of J.…”