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Register a new userNumerical reconstruction of curves from their Jacobians
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We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We verify this methods via numerical experiments with curves up to genus 7.
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Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $mathfrak{b} in J(K)$ there are exactly $2^{2g}$ elements $mathfrak{a} in J(K)$ such that $2mathfrak{a}=mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $mathfrak{a}$, in terms of the Mumford representation of $mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $mathfrak{a}$, when $mathfrak{b}in J(K)$ is given by $P=(a,b) in C(K)subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.
Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)in C(K)$ there are $2^{2g}$ points $frac{1}{2}P in J(K)$. We describe explicitly the Mumford represesentations of all $frac{1}{2}P$. The rationality questions for $frac{1}{2}P$ are also discussed.
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
Let $K$ be a field of prime characteristic $p$, $n>4 $ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Let $l$ be an odd prime different from $p$, $Z[zeta_l]$ the ring of integers in the $l$th cyclotomic field, $C_{f,l}:y^l=f(x)$ the corresponding superelliptic curve and $J(C_{f,l})$ its jacobian. We prove that the ring of all endomorphisms of $J(C_{f,l})$ coincides with $Z[zeta_l]$ if $J(C_{f,l})$ is an ordinary abelian variety and $(l,n) e (5,5)$.
Let ${cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected etale cover ${cal M}^l$ of ${cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is almost anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let $pi_1({cal M}^l_{ol{k}})$ be the geometric algebraic fundamental group of ${cal M}^l$ and let ${Out}^*(pi_1({cal M}^l_{ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${cal M}^l$ (this is the $ast$-condition motivating the almost above). Let us denote by ${Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${cal M}^l$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({cal M}^l)cong{Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.
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