Michael Stoll
Based on computational evidence, we formulate a number of conjectures on the distribution of rational points on curves of genus 2 over the rational numbers, in terms of the size of the coefficients of an equation of the form y^2 = f(x) >.
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In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.
Extended genus fields of abelian extensions of rational function fields
本文研究了全局有理函数域有限阿贝尔扩张的广义亏格域，通过分析素数幂次循环扩张及分圆扩张复合域的性质，显式给出了基于“分圆投影”的一般有限阿贝尔扩张广义亏格域的求解结果。
Counting odd genus $2$ curves with a marked rational $3$-torsion point
研究有理数域上特定条件的亏格2曲线，通过朴素高度排序计数，核心方法是基于曲线的首一魏尔斯特拉斯模型及雅可比行列式的有理3 - 挠点条件。
In this paper we show that the maximum number of rational points possible for a smooth, projective, absolutely irreducible genus 4 curve over a finite field F_7 is 24. It is known that a genus 4 curve over F_7 can have at most 25 points. In this paper we prove that such a curve can have at most 24. On the other hand we provide an explicit example of a genus 4 curve over F_7 having 24 points.
We determine the maximum number of rational points on a curve over $\mathbb{F}_2$ with fixed gonality and small genus.
In this paper, we remove certain hypothesis in the theorem of Bertolini-Darmon on the rationality of Stark-Heegner points over narrow genus class fields of real quadratic fields. Along the way, we establish that certain normalized special values of $L$-functions are squares of rational numbers, a result that is of independent interest, and can be regarded as instances of the rank zero case of the Birch and Swinnerton-dyer conjecture modulo squares.
A note on extended genus fields of Kummer extensions of rational function fields
本文将 Clement 于 1992 年关于有理函数域素数次循环 Kummer 扩张扩展亏格域的构造推广到一般 Kummer 扩张，并证明其原方法依然适用。
Rational and isolated quadratic points on hyperelliptic curves of rank 0 and small genus
本文提出一种利用Kummer簇上Mordell-Weil群像来计算秩0超椭圆曲线有理点和孤立二次点的方法，并用Magma实现，在大量曲线上验证。
Building on work of Balakrishnan, Dogra, and of the first author, we provide some improvements to the explicit quadratic Chabauty method to compute rational points on genus $2$ bielliptic curves over $\mathbb{Q}$, whose Jacobians have Mordell-Weil rank equal to $2$. We complement this with a precision analysis to guarantee correct outputs. Together with the Mordell-Weil sieve, this bielliptic quadratic Chabauty method is then the main tool that we use to compute the rational points on the $411$ locally solvable curves from the LMFDB which satisfy the aforementioned conditions.
We extend the computations from our previous paper arXiv:2005.07054 to determine the maximum number of rational points on a curve over $\mathbb{F}_3$ and $\mathbb{F}_4$ with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus $g$ has gonality at most $\lfloor \frac{g+3}{2} \rfloor$. Second, a curve of gonality $γ$ and large genus over $\mathbb{F}_q$ has $γ(q+1)$ rational points.
We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).
It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve that classifies periodic points of period~4 is birational to $X_1(16)$, whose rational points had been previously computed. We prove there are none with $N=5$. Here the relevant curve has genus~$14$, but it has a genus~$2$ quotient, whose rational points we compute by performing a~$2$-descent on its Jacobian and applying a refinement of the method of Chabau…
In this paper we first obtain the genus field of a finite abelian non-Kummer $l$--extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields and our previous results, we deduce the general expression of the genus field of a finite abelian extension of a global rational function field.
In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.
We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P^1 of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X^4 + Y^4 + Z^4) - 96914(X^2 Y^2 + X^2 Z^2 + Y^2 Z^2) = 0, whose Jacobian has 864 rational torsion points. This paper has appeared in Forum Math. 12 (2000) 315-364.
For a cyclic Kummer extension $K$ of a rational function field $k$ is considered, via class field theory, the extended Hilbert class field $K_H^+$ of $K$ and the corresponding extended genus field $K_g^+$ of $K$ over $k$, along the lines of the definitions of R. Clement for such extensions of prime degree. We obtain $K_g^+$ explicitly. Also, we use cohomology to determine the number of ambiguous classes and obtain a reciprocity law for $K/k$. Finally, we present a necessary and sufficient condition for a prime of $K$ to decompose fully in $K_g^+$.
Rational Points of some genus $3$ curves from the rank $0$ quotient strategy
本文提出一种算法，用于计算亏格3曲线中作为秩0亏格1曲线二次覆盖情形的全部有理点，并在Magma中实现，在约4万条曲线上验证，找到了达到Stoll上界的实例。
We obtain the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses class field theory. First we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields and compare this approach with the one given by …
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
In the published version of this paper [Finite Fields and Their Applications {\bf 20} (2013) 40--54], there is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statments for Theorems 4.2, 4.5 and 5.2 We give a construction of genus fields for congruence function fields. First we consider the cyclotomic function field case following the ideas of Leopoldt and then the general case. As applications we give explicitly the genus fields of Kummer, Artin--Schreier and cyclic $p$--extensions. Kummer extensions were obtained previously by G. Peng and Artin…
We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.
In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a cyclotomic funtion field, the construction is given using Leopoldt's ideas by means of Dirichlet characters. Following the definition of Anglès and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian $l$-extension of a field of rational functions.
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
In this article, we enumerate all Shimura curves X^D_0(N) of genus at most two.
In this paper, we give a universal family of curves of genus 2 whose jacobians have $\sqrt2$ multiplication fixed by the Rosati involution, and several results based on it, including isogenies between jacobians of curves, and jacobians of curves with quaternionic multiplication.
Using class field theory one associates to each curve C over a finite field, and each subgroup G of its divisor class group, unramified abelian covers of C whose genus is determined by the index of G. By listing class groups of curves of small genus one may get examples of curves with many points; we do this for all curves of genus 2 over the fields of cardinality 5,7,9,11,13 and 16, giving new entries for the tables of curves with many points (www.manYPoints.org).
We give a generalization of the method of "Elliptic Curve Chabauty" to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell-Weil sieve to provide a complete solution to the problem of determining the set of rational points of an algebraic curve $Y$.
Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated since the full torsion subgroup has rank 2g. In this paper we prove that distortion maps always exist for supersingular curves of genus g>1 and we construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).
In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of $L$-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these famili…
Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show…
Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two
本文证明了对于任何数域K和固定亏格g≥2，存在无穷多个非同构的亏格为g的双曲曲线，其雅可比曲线的秩为0、1或2。
A projective, smooth, absolutely irreducible algebraic curve X of genus g defined over a finite field F_q is called optimal if for every other such genus g curve Y over F_q one has $\#Y(F_q)\le \#X(F_q)$. In this paper we show that for $g\le 5$ there is a unique optimal genus g curve over F_2. For g=6 there are precisely two and for g=7 there are at least two.
The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.
The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genus $2$ defined over the reflex field. We solve the problem by showing that the list given in Bouyer and Streng [LMS J. Comput. Math., 18(1):507{538, 2015] is complete.
Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We also provide an intrinsic characterization of so-called Legendre elliptic curves.
Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $\text{PGL}_2(\mathbb{F}_q)$ or $\text{PSL}_2(\mathbb{F}_q)$. We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.
We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4.
We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.
In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its jacobian,…
We give a complete list of rational functions $A$ such that the genus $g$ of the Galois closure of $\mathbb C(z)/\mathbb C(A)$ equals zero. We also provide a geometric description of $A$ for which $g=1.$
For affine plane curves defined over the rationals of genus at least two, we show that a decision algorithm for the existence of solutions also yields a search algorithm for all solutions.
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.
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and…
Moduli for rational genus 2 curves with real multiplication for discriminant 5
本文针对判别式5的实乘法情形，给出了有理亏格2曲线对应的希尔伯特模曲面有理点的简单通用刻画，并提供了Weierstrass模型及二次型化简技术。
Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner poi…
Determining the rational points on a curve of genus 2 and Mordell-Weil rank 1
本文给出在 Mordell–Weil 秩为 1 时，利用 Jacobi 簇上有限指数生成点高效计算亏格 2 曲线全部有理点的方法。
The number A(q) shows the asymptotic behaviour of the quotient of the number of rational points over the genus of non-singular absolutely irreducible curves over a finite field Fq. Research on bounds for A(q) is closely connected with the so-called asymptotic main problem in Coding Theory. In this paper, we study some generalizations of this number for non-irreducible curves, their connection with A(q) and its application in Coding Theory.
Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfac…
On the Prym variety of genus 3 covers of genus 1 curves
本文针对亏格1曲线的非超椭圆亏格3覆盖，构造显式亏格2曲线，使雅可比簇同源于原曲线与覆盖曲线的雅可比簇乘积，推广了Nils Bruin的结果。
The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in $\mathbb P^4$. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat's diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namel…
We find explicit equations for two-coverings of Jacobians of genus two curves over an arbitrary ground field of characteristic different from two.
In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fields
For any number field, we prove that there exists a stacky curve of genus one half defined over the ring of its integers violating the local-global principle for integral points.
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.
We describe the ring of modular forms of degree 2 in characteristic 2 using its relation with curves of genus 2.
Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whos…
We prove a genus formula for modular curves of $D$-elliptic sheaves. We use this formula to show that the reductions of modular curves of $D$-elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of $D$ tends to infinity.
We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks on Frobenius non-classical quartics over finite fields are given.
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