John Armitage
It was conjectured by Gauss that any negative discriminant with at least 32 genera has at least two classes of binary quadratic forms in each genus. We prove that such a discriminant cannot have a particularly large prime factor, by an argument similar to that of Baker's solution to Gauss' class number one problem.
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We assume a condition on the zeros of Dirichlet L-functions related to the GUE distribution to show there are no discriminants greater than d_{66}\approx 1.9 x 10^{130} with one class per genus.
We study genus one curves of degree 5 defined by Pfaffians. We give new formulae for the invariants, and prove the equivalence of two different definitions of minimality. As an application we show that transformations between models with square-free discriminant are necessarily integral. This result is used by Bhargava and Shankar in their work on the average ranks of elliptic curves.
In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant -19. We prove that any such curve can be given by an explicit equation of certain form. Using these equations we obtain a table of maximal and minimal curves over finite fields with discriminant -19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.
Capitulation discriminants of genus one curves
本文将$n$次亏格1模型概念推广到任意奇数$n$，证明$\mathbb{Q}$上处处局部可解的亏格1曲线存在极小积分模型，并将结果应用于椭圆曲线Tate-Shafarevich群的capitulation问题，得到相关分裂数域的判别式上界。
This paper studies two topics concerning on the orthogonal complement of one dimensional subspace with respect to a given quadratic form on a vector space over a number field. One is to determine the invariants for the isomorphism class of such a complement in the sense of Shimura. The other is to investigate an ideal of the base field, which may be viewed as a difference between the genus of maximal lattices and an integral lattice in the complement. We shall discuss about the class number of the genus of maximal lattices as an application.
We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the modul…
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。
We prove that approximately $96.23\%$ of cubic fields, ordered by discriminant, have genus number one, and we compute the exact proportion of cubic fields with a given genus number. We also compute the average genus number. Finally, we show that a positive proportion of totally real cubic fields with genus number one fail to be norm-Euclidean.
Genus 2 curves with bad reduction at one odd prime
本文研究了由 $y^2=xh(x)$ 描述的亏格2曲线，证明了当 $h(x)$ 可约时绝对判别式为奇素数的曲线仅有两族，并在 $h(x)$ 不可约时给出了显式曲线族示例。
Let d1 and d2 be discriminants of distinct quadratic imaginary orders O_d1 and O_d2 and let J(d1,d2) denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1,d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of O_d1 and O_d2 into endomorphism rings of supersingular curves, and then solve this counting problem. Interestingly, this counti…
Explicit height estimates for CM curves of genus 2
本文显式化了亏格2曲线对应循环Galois CM域判别式界定的常数，并简化了部分论证。
We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results give effective exponential versions of the generalized conjectures for some curves, including all curves $X$ of genus one or two. We obtain in particular exponential versions of Szpiro's classical discriminant conjecture for elliptic curves over $K$. In course of our proofs we establish explicit results for certain Arakelov invariants of hyperelliptic curves (e…
On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle
本研究针对一族j不变量为零的椭圆曲线及其3-同源曲线，结合整二元三次型等价类，在Tate-Shafarevich群有限假设下推导秩的奇偶性与上下界，并构造出显式违反哈塞原理的亏格1曲线类。
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).
Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant
本文研究有限域上具(n,n)分裂雅可比矩阵的2类曲线轨迹算术性质，计算多有限域有理点并推导n=2、3时的泽塔函数，据此提出分裂雅可比矩阵高效检测法，推动后量子2类系统的曲线选择、安全分析与协议设计。
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
An invariant of a model of genus one curve is a polynomial in the coefficients of the model that is stable under certain linear transformations. The classical example of an invariant is the discriminant, which characterizes the singularity of models. The ring of invariants of genus one models over a field is generated by two elements. Fisher normalized these invariants for models of degree n=2,3,4 in such a way that these invariants are moreover defined over the integers. We provide an alternative way to express these normalized invariants using modular forms. This method relies on a direct co…
The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.
The automorphism groups of a maximal function field with the second largest genus and its algebraic geometry codes
本文研究了具有第二大数据特征有限域上最大亏格的函数域的自同构群，并探讨了其在代数几何码中的应用。
Some PARI programs have bringed out a property for the non-genus part of the class number of the imaginary quadratic fields, with respect to $(\sqrt D\,)^{\varepsilon}$, where $D$ is the absolute value of the discriminant and $\varepsilon \in ]0, 1[$, in relation with the $\varepsilon$-conjecture. The general Conjecture 3.1, restricted to quadratic fields, states that, for $\varepsilon \in ]0, 1[$, the successive maxima, as $D$ increases, of $\frac{H}{2^{N-1} \cdot (\sqrt D\,)^{\varepsilon}}$, where $H$ is the class number and $N$ the number of ramified primes, occur only for prime discriminan…
Generic models for genus 2 curves with real multiplication
本文针对 Q 上带实乘法的亏格2曲线，借助 P² 上二次曲线束极小化算法，新增12种通用模型并覆盖 D<40，同时简化 D<100 的相关 Mestre 二次曲线方程。
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property 1) that they are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d. A slightly stronger condition is given in Section 3. Such primes are primes of bad reduction for the genus 2 curve and primes of …
Prime isogenous discriminant ideal twins
本文研究了素数等价判别理想双生的分类，证明在数域中，当p∈{2,3,5,7,13}时，存在有限个这样的双生。
We present explicit models for non-elliptic genus one Shimura curves X_0(D, N) with Gamma_0(N)-level structure arising from an indefinite quaternion algebra of reduced discriminant D, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan's work in his Ph. D Thesis on points with complex multiplication on Shimura curves.
We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for minimising bidegree (2,2)-forms, 3 x 3 x 3 cubes and 2 x 2 x 2 x 2 hypercubes. We also prove a theorem relating the minimal discriminant to that of the Jacobian elliptic curve.
Maximal curves of genus 5 over finite fields
研究有限域上亏格5且判别式为-19的极大曲线的存在性，利用自同构群证明当q≡2,3,4 mod 5时不存在，并给出q≡1 mod 5时的模型及q≡0 mod 5时仅可能q=5^7。
The average genus number for pure fields of prime degree
本文研究素数次数纯数域的平均亏格数，证明了亏格数为1的比例渐近于$(A_\ell\log X)^{-1}$，平均亏格数渐近于$B_\ell(\log X)^{\ell-1}$，其中常数可显式表示为素数乘积。
Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus
本文确定具有第三大亏格的极大曲线$\mathcal{X}_3$在所有点处的Weierstrass半群与完整自同构群，揭示其Weierstrass点异常丰富，并证明除小$q$外其自同构群完全继承自Hermitian曲线，为唯一性公开问题提供关键线索。
We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2: x,y,zin Z}|, just to mention one among many similar positive results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author stating that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 7y^2 + 7z^2: x,y,z in Z}|. We prove a variety of identities for certain ternary forms with discriminants 1…
Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$
针对 q≡1 mod 3 时具有第三大亏格的极大函数域 Y_3，本文显式确定其所有位的 Weierstrass 半群与完整自同构群，发现其 Weierstrass 位结构异常丰富，但自同构群除小 q 外均继承自 Hermitian 函数域。
Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zero quotients of the upper half plane. It is therefore natural to ask if there is an analogue of this property in the Math…
Let $N$ be a positive square-free integer such that the discrete group $Γ_{0}(N)^{+}$ has genus one. In a previous article, we constructed canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field associated to $Γ_{0}(N)^{+}$ as well as an algebraic equation $P_{N}(x_{N},y_{N}) = 0$ with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to $x_{N}$ and $y_{N}$, by which we mean the arithmetic nature of the numbers $x_{N}(τ)$ and $y_{N}(τ)$ for any CM point $τ$ in the upper half plane $\mathbb{H}$. If $τ$…
Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$
本文针对 $q\equiv0\pmod3$，显式确定第三大亏格的 $\mathbb{F}_{q^2}$-极大函数域 $Z_3$ 的各位 Weierstrass 半群与完整自同构群。
An arithmetic intersection for squares of elliptic curves with complex multiplication
本研究针对雅可比簇同构于复乘椭圆曲线平方的亏格2曲线，证明其稳定模型坏约化素数满足$p \leq d/4$，给出基于精细化Humbert不变量的素数计算方法，结合已有公式算出素数在稳定模型判别式中的指数，并完成d<100的显式计算与验证。
Genus-One Fibrations and the Jacobian of Linear Slices in the Quintic Equal-Sum Problem
本文研究五次等和方程在特定约束下的解结构，证明了雅可比纤维的秩界及非奇异特性的条件。
Let X be a smooth projective algebraic curve of genus g minus $r\geq 1$ points defined over an algebraically closed field k of characteristic $p\geq 0$. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.
Arithmetic aspects of the Burkhardt quartic threefold
本文研究 Burkhardt 四次三维簇的算术性质，证明其在特征非 3 域上有理、计算有限域 zeta 函数，并显式给出 genus 2 泛曲线模型及 j-平面与 3 阶子群、Hesse 铅笔的关系。
Detecting split surfaces, RM surfaces, and minimum walks on isogeny graphs
我们利用精细Humbert不变量将同源计算转化为本原表示问题，提出检测分裂曲面与RM嵌入的算法，并在小素数及千比特素数上验证，分裂度满足对数关系。
If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number field of degree 120 has class number one. This field is the unique $A_5 \times C_2$ extension of the rationals that is ramified only at 653 with ramification index 2. It is the largest degree number field unconditionally proven to have class number 1. The proof uses the algorithm of Guàrdia, Montes and Nart to calculate an integral basis and then finds integral…
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