We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2.
The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mo…
We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are t…
We study the variation of Selmer ranks of Jacobians of twists of hyperelliptic curves and superelliptic curves. We find sufficient conditions for such curves to have infinitely many twists whose Jacobians have Selmer ranks equal to $r$, for any given nonnegative integer $r$. This generalizes earlier results of Mazur-Rubin on elliptic curves.
Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathscr{C}$ of $C$ over the ring of integers ${\mathcal O}_K$ (i.e. the hyperelliptic involution of $C$ extends to $\mathscr{C}$ and the quotient is a smooth model of ${\mathbb P}^1_K$ over ${\mathcal O}_K$), we give necessary and sometimes sufficient conditions for $\mathscr{C}$ to be defined by a global Weierstrass equation. In particular, if $C$ has everywhere good reduction, we prove that it is defined by a global Weierstrass equation with invertible discriminant if the class number $h_K$ is prim…
We consider the family of hyperelliptic curves over $\Q$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also prove that the average size of the $ϕ$-Selmer groups of a family of isogenies associated to this family is exactly 2.
In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg representation of the group $G=\text{PSO}(2n+2)$ and a Hitchin fibration. Consistent with the result over Q of Arul Shankar and Xiaoheng Wang [Rational points on hyperelliptic curves having a marked non-Weierstrass point, in Compos. Math., 154(1):188-222, 2018], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperell…
Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $\mathop{\textrm{char}} k \neq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = \mathop{\textrm{Spec}} R$. Let $\mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $\mathop{\textrm{Art}} (\mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $\mathcal{X}$ and let $ν(Δ)$ denote the minimal discriminant of $C$. We prove that $-\mathop{\textrm{Art}} (\mathcal{X}/S) \leq ν(Δ)$. As a …
We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels--Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen--Stoll. We establish the local formula in many ins…
We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2. For curves with a rational Weierstrass point, fast explicit formulas are well known and widely available. Here we address the general case, in which we do not assume the existence of a rational Weierstrass point, using a balanced divisor approach.
For integers $N\geq 3$ and $g\geq 1$, we study bounds on the cardinality of the set of points of order dividing $N$ lying on a hyperelliptic curve of genus $g$ embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.
Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.
We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree…
We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C_1. In general, we show that a large (but in general proper) subgroup of the 2-torsion classes are given by the construction. Examples demonstrating applications to the arithmetic of hyperelliptic curves defined over number fields are given.
Let E_G be a family of hyperelliptic curves over F2^(alg cl) with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm to compute the zeta function of E_g for g in a degree n extension field of F, which has as time complexity O(n^3) and memory requirements O(n^2). With a slightly different algorithm we can get time O(n^2,667) and memory O(n^2,5), and the computation of O(n) curves of the family can be done in time and space O(n^3). All these algorithms are polynomial in the genus.
In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$ for integers $m$. Namely, we first provide a condition on $m$ that gives a bound of the size of Selmer group and then we provide a condition on $m$ that makes $L$-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch--Swinnerton-Dyer conjecture.
Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ its minimal regular model with relative dualizing sheaf $ω_{\mathcal{X}/ \mathcal{O}_K}$. We show how to directly read off a basis for $H^0(\mathcal{X},ω_{\mathcal{X}/\mathcal{O}_K})$ from the cluster picture of the roots of $f$. Furthermore we give a formula for the valuation of $λ$ such that $λ\cdot \frac{dx}{y} \land \dots \land x^{g-1}…
It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples…
Zero-cycles on surfaces dominated by products of hyperelliptic curves
CNRS Laboratoire de Mathématiques d’Orsay Université Paris-Saclay（法国国家科学研究中心奥赛数学实验室巴黎萨克雷大学）；CNRS Institut Galilée Université Sorbonne Paris Nord（法国国家科学研究中心加利亚莱研究所索邦巴黎北大学）；Institute for the Information Transmission Problems Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this factorization with the property that if it is empty then the curve C has no rational points. We shall demonstrate the effectiveness of our new method by solving the generalized Fermat equation with signature (3,4,5), which is unassailable via the previously existing methods.
Let G be the separable Galois group of a finite field F of characteristic p, and X/F an imaginary hyperelliptic curve such that G acts transitively on its set W(X) of Weierstrass points. The existence of a G-invariant 2-torsion point on the Jacobian J(X) of X depends only on the parity of |W(X)|, but for large enough |F|, there exist two such curves X and X' with |W(X)|=|W(X')|, such that J(X) has (and J(X') does not have) a G-invariant 4-torsion point. The problem is equivalent to a study of the 2-,4- and 8-rank of the class number of the maximal order in the function field of such curves, an…
Let $K$ be a number field, and $g \geq 2$ a positive integer. We define $c_K(g)$ as the smallest integer $n$ such that there exist infinitely many $\overline{K}$-isomorphism classes of genus $g$ hyperelliptic curves $C/K$ with all Weierstrass points in $K$ having potentially good reduction outside $n$ primes in $K$. We show that $c_K(g) > π_{K, \textrm{odd}}(2g) + 1$, where $π_{K, \textrm{odd}}(n)$ denotes the number of odd primes in $K$ with norm no greater than $n$, as well as present a summary of various conditional and unconditional results on upper bounds for $c_K(g)$.
We describe how one can use the "Selmer group Chabauty" method developed by the author to show that certain hyperelliptic curves of the form $$ C \colon y^2 = x^N + h(x)^2 \,, $$ where $N = 2g + 1$ is odd, $h \in \mathbb{Z}[x]$ with $\operatorname{deg} h \le g$ and $h(0)$ odd, have only the "obvious" rational points $\infty$ (the unique point at infinity on the smooth projective model of the curve) and $(0, \pm h(0))$. As an application of the method, we prove the following result. Let $c \in \mathbb{Q}$ and write $f_c(x) = x^2 + c$. We denote the iterates of $f_c$ by $f_c^{\circ n}$; i.e., we…
Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian variety over $K$. Let us consider the canonical embedding of $C$ into $J$ that sends $O$ to the zero of the group law on $J$. This embedding allows us to identify $C(K)$ with a certain subset of the commutative group $J(K)$. A special case of the famous theorem of Raynaud (Manin--Mumford conjecture) asserts that the set of torsion points in $C(K)$ is finite. It i…
Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrass points are defined over extensions of $\mathbb{Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case of elliptic curves), we prove that any line bundle of degree $0$ on $C$ which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large. This gives a positive answer, for such curves, to a question by Agboola and Pappas.
We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves have at most $B_d$ points of degree $d$. If $d < g$ is even, we conditionally bound degree $d$ points not pulled back from points of degree $d/2$ on the projective line. Furthermore, we show one may take $B_2=24$ and B_3=114$. Our proofs proceed by refining recent work of Park, which applied tropical geometry to symmetric power Chabauty, and then applying resul…
Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g tends to infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty's method that shows that certain computable conditions imply #C(Q)=1; on the o…
Let $Γ$ be the Fuchsian group of the first kind. For an even integer $m\ge 4$, we describe the space $H^{m/2}\left(\mathfrak R_Γ\right)$ of $m/2$--holomorphic differentials in terms of a subspace $S_m^H(Γ)$ of the space of (holomorphic) cuspidal modular forms $S_m(Γ)$. This generalizes classical isomorphism $S_2(Γ)\simeq H^{1}\left(\mathfrak R_Γ\right)$. We study the properties of $S_m^H(Γ)$. As an application, we describe the algorithm implemented in SAGE for testing if a cusp at $\infty$ for non-hyperelliptic $X_0(N)$ is a $\frac{m}{2}$-Weierstrass point.
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…
Let $F$ be the maximal totally real subfield of $\mathbf{Q}(ζ_{32})$, the cyclotomic field of $32$nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptio…
The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model $y^2=4x^5+c_4 x^4+ ... +c_1 x+c_0$. We show that the recurrence sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves…
Let $K$ be a field whose absolute Galois group is finitely generated. If $K$ neither finite nor of characteristic 2, then every hyperelliptic curve over $K$ with all of its Weierstrass points defined over $K$ has infinitely many $K$-points. If, in addition, $K$ is not locally finite, then every elliptic curve over $K$ with all of its 2-torsion rational has infinite rank over $K$. These and similar results are deduced from the Hales-Jewett theorem.
Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography
Institut de mathématiques de Bordeaux (Bordeaux INP, CNRS, Univ. Bordeaux)（波尔多数学研究所）；Inria Bordeaux（法国国家信息与自动化研究所波尔多中心）；Institute of Mathematics of the Romanian Academy（罗马尼亚科学院数学研究所）；CertSIGN, Bucharest（布加勒斯特CertSIGN公司）；Babeş-Bolyai University, Cluj-Napoca（博巴伊-玻艾尤大学）
Let $K$ be a number field and $S$ a finite set of places of $K$ containing all archimedean places. In this paper, we show that the second moment of the number of $S$-integral points on elliptic curves over $K$ is bounded. In particular, we prove that, for any positive real number $r\leq \log_2 5 = 2.3219 \ldots$, the $r$-th moment of the number of $S$-integral points is bounded for the family of all integral short Weierstrass curves ordered by height, or for any positive density subfamily thereof. For certain other families of elliptic curves over $\mathbb{Q}$, such as those with one or two ma…
We prove that, when genus two curves $C/\mathbb{Q}$ with a marked Weierstass point are ordered by height, the average number of rational points $\#|C(\mathbb{Q})|$ is bounded. The argument follows the same ideas as the sphere-packing proof of boundedness of the average number of integral points on (quasiminimal Weierstrass models of) elliptic curves. That is, we bound the number of small-height points by hand, the number of medium-height points by establishing an explicit Mumford gap principle and using the theorem of Kabatiansky-Levenshtein on spherical codes (this technique goes back to work…
