Jean-Marc Couveignes, Tony Ezome
We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute $(l,l)$ isogenies between jacobians of genus two curves.
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We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
Quotients of Jacobians
本文证明亏格≥5的曲线雅可比簇在有限群作用下的商不是单直纹的，并分类了亏格3和4中商为单直纹的所有曲线。
Quotients of Jacobians
本文研究了曲线的Jacobi簇在有限自同构群作用下的商，证明了亏格大于20时商具有典范奇点且Kodaira维数为0，并给出了亏格小于5时商为单有理的例子。
Non-torsion algebraic cycles on the Jacobians of Fermat quotients
本文研究Fermat商曲线Jacobi簇中Ceresa循环的Abel–Jacobi像，证明在任意基点和$k\leq g-2$条件下其为非挠，并推出模有理等价的非挠性。
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Pic…
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
We construct and study two series of curves whose Jacobians admit complex multiplication. The curves arise as quotients of Galois coverings of the projective line with Galois group metacyclic groups $G_{q,3}$ of order $3q$ with $q \equiv 1 \mod 3$ an odd prime, and $G_m$ of order $2^{m+1}$. The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the CM-types and show that the Jacobians are simple abelian varieties.
We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multi-variable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over $\Q$ for such curves.
We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves $C$ currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points ov…
We give some necessary and sufficient conditions for the existence of Néron models of jacobians of semistable morphisms of arbitrary relative dimension over base schemes of arbitrary dimension. To do this, we introduce a notion of alignment for semistable morphisms over any regular base scheme, and show that the jacobian of an aligned projective semistable morphism admits a separated model with the Néron mapping property. When the Picard scheme is smooth over the base scheme along its unit section we show that the converse holds.
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
Smooth quotients of principally polarized abelian varieties
该研究刻画了保持主极化阿贝尔簇$\u0398$数值类的有限自同构群$G$，使$A/G$光滑的条件，并完成曲线雅可比簇光滑商的分类。
Klein's simple group $H$ of order $168$ is the automorphism group of the plane quartic curve $C$, called Klein quartic. By Torelli Theorem, the full automorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$, obtained by adding minus identity to $H$. The quotient variety $J/G$ can be alternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex $3$-space by the complex crystallographic group $\tilde G$, the extension of $G$ by the period lattice of the Klein quartic. Moreover, it turns out that $\tilde G$ is generated by affine complex reflections. According …
We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m of them of genus 2. We show that several equations conjectured by Kurihara are correct and compute for them the kernel of Ribet's isogeny J_0(D)^{new} --> J_D between the new part of the Jacobian of the modular curve X_0(D) and the Jacobian of X_D.
Lefschetz properties and the Jacobian algebra of 3-dimensional hyperplane arrangements
本文研究高维商空间和1维几乎完全交的弱与强莱夫谢茨性质，并将结果应用于超平面构型的雅可比代数分析。
Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.
We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are several applications. We discuss one in particular: the calculation of the projection in the tautological ring of the moduli space of abelian varieties of the class of the locus of Jacobians.
Tjurina spectrum and graded symmetry of missing spectral numbers
研究超曲面孤立奇点的Tjurina子谱与缺失谱数的分级对称性，基于Jacobian环的自对偶性，改进了Briançon-Skoda指数估计。
Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions
本文证明了由反射生成的不可约复晶体反射群在三维复仿射空间上的商是权重为1,2,4,7的加权射影空间，其中相伴共线群是168阶克莱因单群，主要基于不变theta函数代数的计算。
We compute modular Galois representations associated with a newform $f$, and study the related problem of computing the coefficients of $f$ modulo a small prime $\ell$. To this end, we design a practical variant of the complex approximations method presented in the book edited by B. Edixhoven and J.-M. Couveignes. Its efficiency stems from several new ingredients. For instance, we use fast exponentiation in the modular jacobian instead of analytic continuation, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, we introduce an eff…
We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.
Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry of the singularity $\mathbb{C}^n/G$, without further knowledge of its resolutions. We explain the use of our implementation of the algorithms in Singular. As an application, we determine the Cox rings of resolutions $X\rightarrow \mathbb{C}^3/G$ for all $G\subseteq GL(3)$ with the aforementioned property and of order $|G|\leq 12$. We also provide examples in dimension 4.
For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed using arithmetic intersection theory on a regular model of the curve in practice. In the case of hyperelliptic curves we present a complete algorithm that has been implemented in Magma. Several examples are computed and the behavior of the running time is discussed.
Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of JX up to isogeny. The most prominent example of such a situation is the group G of two elements. Let X --> Y denote the corresponding quotient map. Then JX is isogenous to the product of JY with the Prym variety of X/Y. In this paper some general results on group actions on abelian varieties are given and applied to deduce a decomposition of the jacobian JX for arbitrary group actions. Several examples are given.
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the resulting weighted metric graph. In general, it is not known how to find the abstract tropicalization of a curve defined by polynomial equations, since an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction in practice. We solve this problem in the case of hyperelliptic curves by studying admissible co…
The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Wit…
An algorithm for computing the limit of a quotient of bivariate real analytic functions has been developed by one of the authors in (Limits of quotients of bivariate real analytic functions, Journal of Symbolic Computation, 50, 2013, 197 207). In this paper we provide a theoretical method based on the work developed in that article to determine the existence of the limit of a quotient of two polynomial functions of three variables. An algorithm to compute such limits, in the case where the polynomials have rational coefficients, or more generally, coefficients in a real finite extension of the…
A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.
Unipotent quotients in compactified Coxeter varieties
本文推广了Lusztig关于Coxeter簇幂么商的描述，证明有限域上连通约化群的相关结果，并应用于证明Wang的上同调不变量猜想及导出Alvis-Curtis对偶的性质。
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