Manjul Bhargava, Jonathan Hanke, Arul Shankar
We prove that, on average, the monogenicity or $n$-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.
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Bhargava, Hanke, and Shankar have recently shown that the asymptotic average $2$-torsion subgroup size of the family of class groups of monogenized cubic fields with positive and negative discriminants is $3/2$ and $2$, respectively. In this paper, we provide strong computational evidence for these asymptotes. We then develop a pair of novel conjectures that predicts, for $p$ prime, the asymptotic average $p$-torsion subgroup size in class groups of monogenized cubic fields.
The second moment of the size of the $2$-class group of monogenized cubic fields
本文证明按高度排序的单基因化三次数域中，全实与复情形的2-类群大小二阶矩分别至多为3和6，且全实窄2-类群二阶矩至多为9，为单基因性改变类群分布提供了新证据。
In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce all possible torsion groups.
We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\Z /21 \Z$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.
This is the third part of a series of papers discussing the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=39$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) $G$ can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves $E/\Q$ together with cubic fields $K$ so that $G= E(K)_{tors}$.
Merel's result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell-Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=49,40,25$ or $22$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
Let $E$ be an elliptic defined over a number field $K$. Then its Mordell-Weil group $E(K)$ is finitely generated: $E(K)\cong E(K)_{tor}\times\mathbb{Z}^r$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=169,143,91,65,77$ or $55$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
On the average $2$-torsion in class groups and narrow class groups of cubic orders with prescribed shape
本文研究了具有指定形状的三次域和三次阶的类群和狭义类群中2- torsion的平均值，通过改进Bhargava和Varma的方法得出了实二次域和复二次域的类群2- torsion平均大小以及实三次域狭义类群2- torsion平均大小的结论。
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ be the compositum of all cubic extensions of $\mathbb{Q}$. In this article we show that the torsion subgroup of $E(\mathbb{Q}(3^\infty))$ is finite and determine 20 possibilities for its structure, along with a complete description of the $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-inv…
We prove a generalization of a result of Bhargava regarding the average size $\mathrm{Cl}(K)[2]$ as $K$ varies among cubic fields. For a fixed set of rational primes $S$, we obtain a formula for the average size of $\mathrm{Cl}(K)/\langle S \rangle[2]$ as $K$ varies among cubic fields with a fixed signature, where $\langle S \rangle$ is the subgroup of $\mathrm{Cl}(K)$ generated by the classes of primes of $K$ above primes in $S$. As a consequence, we are able to calculate the average sizes of $K_{2n}(\mathcal{O}_K)[2]$ for $n > 0$ and for the relaxed Selmer group $\mathrm{Sel}_2^S(K)$ as $K$ …
We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.
Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $|\rm Cl_2({\cal…
The mean number of $2$-torsion elements in the class groups of cubic orders
本文确定了三次序类群中2-挠元的平均个数，并将其推广到任意奇数维情形，给出了可约SL_N(Z)-轨道的渐近计数公式。
Torsion of elliptic curves over cyclic cubic fields
本研究明确了循环三次域、非循环全实三次域与复三次域上椭圆曲线的全部可能挠群，填补了相关数论研究的空白。
We study Tamagawa numbers of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over cubic fields and of elliptic curves with an $n-$isogeny over $\mathbb{Q}$, for $n\in\{6,8,10,12,14,16,17,18,19,37,43,67,163\}$. Bruin and Najman proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}$. We find that Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cu…
Let E be an elliptic curve defined over Q. We study the relationship between the torsion subgroup E(Q)_tors and the torsion subgroup E(K)_tors, where K is a cubic number field. In particular, We study the number of cubic number fields K such that E(Q)_tors\neq E(K)_tors.
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as $S$. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves that appear over the field with smallest absolute value of its discriminant and having Galois group $S_3$ and over the field with smallest absolute value of its discriminant and having Galois group $\Z…
Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics
本文提出算法，通过三次曲线与方程组描述并计算平面四次曲线雅可比簇的3-挠子群，并利用其伽罗瓦结构计算局部野导子指数。
Over 200 years ago, Gauss discovered a composition law on the $SL_2({\mathbb Z})$-equivalence classes of primitive binary quadratic forms. Since then, bijections of classes of binary forms have been found with ideal class groups of quadratic rings. This paper uses one such bijection given by Bhargava, relating classes of projective binary cubic forms to the $3$-torsion of an ideal class group, to find an explicit form for a cubic analogue of Gaussian composition.
The minimal integral Mahler measure of a number field $K$, $M(\mathcal{O}_K)$, is the minimal Mahler measure of a non-torsion primitive element of $\mathcal{O}_K$. Upper and lower bounds, which depend on the discriminant, are known. We show that for cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute $M(\mathcal{O}_K)$ for all cubics with absolute value of the discriminant bounded by $N$.
A short remark on the $\ell$-torsion part of class groups
本文改进了纯三次数域类群ℓ- torsion部分的界，并推广到任意奇数次纯数域，同时回答了Ellenberg关于小高度原始元数量的问题。
In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the …
In this paper we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over $\mathbb{Q}$ satisfying $\max\{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava-Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves
In this note, we present a complete and corrected version of Kenku's method in the proof of the nonexistence of 27-torsion on elliptic curves over quadratic fields and show why this method can not be applied to the cases of 32- and 24-torsion on elliptic curves over cubic fields.
Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles
通过椭圆曲线算术研究三次族 $f_{b,a}(x)=x^3+bx+a$ 的有理 2-循环，给出其与椭圆曲线 $E_a:Y^2=X^3+4X^2+16a^2$ 有理点之间的显式双有理关系，并证明对每个非零 $a$ 存在无穷多有理 $b$ 使 $f_{b,a}$ 具有有理 2-循环。
We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a conjecture of Datskovsky-Wright and Roberts. We also prove a variety of generalizations, including to arithmetic progressions, where we discover a curious bias in the secondary term. Roberts' conjecture has also been proved independently by Bhargava, Shankar, and Tsimerman. In contrast to their work, our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms, dev…
$\ell$-Torsion in Class Groups via Dirichlet $L$-functions
本文利用Dirichlet L-函数的性质，研究了判别式光滑的二次和三次数域类群中$\ell$-挠部分的上界。
We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.
We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ by Brauer--Siegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves; 2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves; 3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves; and 4) bounds of Baily and Wong on the number of $A_4$-quarti…
A parametrization of $3$-class groups of quadratic rings over Dedekind domains
本文将 Bhargava 的二元三次形式参数化推广到特征不为 3 的 Dedekind 整环，通过构造 GL_2(K) 的算术子群刻画二次环类群中的 3-挠理想类。
Sporadic Cubic Torsion
本文完成了三次数域上椭圆曲线扭转子群的分类，通过确定特定模曲线的三次点及相关雅可比簇的秩，并为广义扭转猜想提供了证明证据。
Monogenicity and 2-torsion in the class group of number fields of odd degree
研究奇数次单生成域类群中的平均2-挠，证明其非平凡2-挠元素平均数量至多为Cohen--Lenstra--Martinet--Malle启发式预测值两倍，在特定条件下确定Bhargava--Hanke--Shankar发现的加倍现象在各奇数次和符号中均存在。
Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.
We give simple proofs of the Davenport--Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.
Heavenly Elliptic Curves over Cubic Number Fields
本文研究三次数域上的天椭圆曲线，引入完全平衡概念，证明其挠表示迹非满射，给出平衡性素数界，并与CM曲线比较。
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.
Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{11}$.
Cocycle Relations and Elliptic Gamma Values
本文利用平滑椭圆 Gamma 函数的上循环关系，证明了复三次域椭圆单位猜想解析表达式不依赖于挠点 $h$ 的选择，从而支持了显式类域论框架下的猜想。
The Klein-Vélu septic, 2-Division, and modular function fields of level 14
该研究明确了Klein-Vélu七次式的2-除法与14级模函数域的关联，通过$X(7)$上的三次方程及theta函数，揭示了其分裂域对应$X(14)$的函数域。
$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields
本文给出了非阿贝尔三次和四次域上椭圆曲线族的两种构造，在驯顺符号核中分别获得三个和四个线性无关的“积分”元素，证明了不忽略挠元的新积分性判据，并对90余条曲线数值验证了 Beilinson 猜想。
We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of …
For every odd integer $n \geq 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of $\mathbb{P}^1$ cut out by integral binary $n$-ic forms. By obtaining upper bounds on the mean number of $2$-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$) of such fields have trivial $…
Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics ("old a…
We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generatin…
A proof of Sylvester's conjecture
本文证明了Sylvester猜想：每个素数$p\equiv4,7,8\pmod9$是两个有理数的立方和，通过证明相关椭圆曲线解析秩一及Heegner点非挠完成剩余类$p\equiv8\pmod9$的证明。
We study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\Z/m\Z \oplus \Z/n\Z$, where $m|n$, be a torsion group such that the modular curve $X_1(m,n)$ is an elliptic curve. Let $K$ be a number field such that there is a positive and finite number of elliptic curves $E_T$ over $K$ having $T$ as a subgroup. We call such pairs $(E_T, K)$ \emph{exceptional}. It is known that there are only finitely many exceptional pairs when $K$ varies through all quadratic or cubic fields. We prove that when $K$ varies t…
In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain torsion in the abelianized fundamental group is a derived invariant. We prove that the collection of Hodge-Witt cohomology groups is a derived invariant. In particular, Hodge-Witt reduction and ordinary reduction are preserved by derived equivalence when the characteristic is sufficiently large. Finally, using the techniques of non-commutative algebraic geometr…
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We analyze the minimal modular form $Θ_{F_4}$ on the double cover of $F_4$, following Loke--Savin and Ginzburg. Using $Θ_{F_4}$, we define a modular form of weight $\frac{1}{2}$ on (the double cover of) $G_2$. We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$-torsion in the narrow class groups of totally real cubic fields.
Counting integral points on symmetric varieties with applications to arithmetic statistics
本文结合 Bhargava 几何数论与动力点计数方法，提出对称簇整点计数新技术，并应用于三次数域类群 2-挠子群和椭圆曲线 2-Selmer 群的薄族分布，揭示了小范数生成元对平均大小的增大效应。
Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant C\_p>0 such that log(\#T\_K) $\le$ C\_p log(\sqrt(D\_K)) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer--Siegel Theorem, wearing here on the valuation of the residue at s=1 (essentially equal to \#T\_K) of the p-adic …
Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification
本文利用 p-adic 伪测度的亏格理论，比较实阿贝尔域中 tame 核与阿贝尔 p-分歧挠群，证明 Deng--Li 猜想并给出 p-秩下界。
General inverse theory for the $\mathsf{U}^4$ norm
本文提出了一种一般有限阿贝尔群中Gowers四次范数的逆理论，通过近立方多项式和结构结果，揭示了均匀性障碍与多项式扩展的联系，并解决了有限torsion群的逆理论问题。
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