Quickly constructing curves of genus 4 with many points
Everett W. Howe
发表信息：
pp. 149-173 in: Frobenius Distributions: Sato-Tate and Lang-Trotter conjectures (D. Kohel and I. Shparlinski, eds.), Contemporary Mathematics 663, American Mathematical Society, Providence, RI, 2016
The "defect" of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over finite fields such that the defect of the double cover is not much more than the defect of the genus-2 curve. We give an algorithm that uses this construction to produce genus-4 curves with small defect. Heuristically, for all sufficiently large primes and for almost all prime powers q, the algorithm is expected to produce a genus-4 curve over F_q with defect at most 4 in time q^{3/4}, up to logarithmic factors.
As part of the analysis of the algorithm, we present a reinterpretation of results of Hayashida on the number of genus-2 curves whose Jacobians are isomorphic to the square of a given elliptic curve with complex multiplication by a maximal order. We show that a category of principal polarizations on the square of such an elliptic curve is equivalent to a category of right ideals in a certain quaternion order.
Given a sextic CM field $K$, we give an explicit method for finding all genus 3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng, we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus 3 hyperelliptic curves over a finite field $\mathbb{F}_p$ with a …
In this paper, we describe the construction of superelliptic curves with a rational point of prescribed order on their jacobians. The construction is based on Hensel's Lemma and produces for a given integer $N$ a superelliptic curve of genus linear in $N$ with a rational $N$-division point on the jacobian. The method is illustrated with multiple examples.
The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5, 6, and 7 with small defect. Our aim is to be able to produce, in a reasonable amount of time, curves that can be used to populate the online table of curves with many points found at http://www.manypoints.org.
We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.
The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known bounds. More precisely, we improve the lower bounds on the maximal number of points $N_q(g)$ for many values of the genus $g$ and of the cardinality $q$ of the finite field $\mathbb F_q$, by looking at coverings of all genus three smooth projective curves over $\mathbb F_q$, for $q$ is an odd prime less than 19.
I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which sp…
The purpose of this paper is to investigate the asymptotic behavior of automorphism groups of function fields when genus tends to infinity. Motivated by applications in coding and cryptography, we consider the maximum size of abelian subgroups of the automorphism group $\mbox{Aut}(F/\mathbb{F}_q)$ in terms of genus ${g_F}$ for a function field $F$ over a finite field $\mathbb{F}_q$. Although the whole group $\mbox{Aut}(F/\mathbb{F}_q)$ could have size $Ω({g_F}^4)$, the maximum size $m_F$ of abelian subgroups of the automorphism group $\mbox{Aut}(F/\mathbb{F}_q)$ is upper bounded by $4g_F+4$ fo…
In the recent paper arXiv:1807.02721, B. Lawrence and A. Venkatesh develop a method of proving finiteness theorems in arithmetic geometry by studying the geometry of families over a base variety. Their results include a new proof of both the $S$-unit theorem and Faltings' theorem, obtained by constructing and studying suitable abelian-by-finite families over $\mathbb{P}^1\setminus\{0,1,\infty\}$ and over an arbitrary curve of genus $\geq 2$ respectively. In this paper, we apply this strategy to reprove Siegel's theorem: we construct an abelian-by-finite family on a punctured elliptic curve to …
