我们确定了在固定有限域F_q（奇特征）上的超椭圆曲线的皮卡群4秩的大特征极限分布。这是Fouvry和Klüners结果的一个函数域类比。我们的计算与（皮卡群类比的）Cohen-Lenstra-Gerth启发式在q≡3 mod 4的情况下一致，即在基础域中没有根的幂时。当存在根的幂时，结果形式与包含根的幂的二次扩张数域类群的猜想分布相同。极限分布在对曲线的分歧行为施加有限多个条件时不会改变。在此过程中，我们确定了由对称条件确定的特定随机矩阵族在有限域上的秩分布。
英文摘要：
We determine the large-genus limiting distribution of the 4-rank of the Picard group of hyperelliptic curves over a fixed finite field $\mathbb F_q$ of odd characteristic. This is a function field analogue of a result of Fouvry and Klüners. Our computation agrees with (the Picard group analogue of) the Cohen--Lenstra--Gerth heuristics in the case $q \equiv 3\pmod{4}$, i.e., in the absence of roots of unity in the base field. When roots of unity are present, the result is of the same form as conjectured distribution for class groups of quadratic extensions of number fields containing roots of unity. The limiting distribution does not change when imposing finitely many conditions on the ramification behavior of the curves. In the process, we determine the rank distribution of a certain class of random matrix ensembles over finite fields determined by symmetry conditions.
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.
Let $ε>0$. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve $C$ of genus $g$ over a finite field $k$ of cardinality $q$ given by $y^2+h(x)y=f(x)$ such that the $x$-coordinate map is ramified at $\infty$. In time $O(g^{2+ε} q^{1/2+ε})$ the algorithm outputs a set of generators of the Picard group $\mathrm{Pic}^0_k(C)$. This extends results which others have obtained when $g=1$. In this article we introduce a combinatorial tool, the `shape parameter', which we use together with character sum estimates from class field theory t…
We present an algorithm that computes the Hasse-Witt matrix of given hyperelliptic curve over Q at all primes of good reduction up to a given bound N. It is simpler and faster than the previous algorithm developed by the authors.
We present an efficient algorithm to compute the Hasse-Witt matrix of a hyperelliptic curve C/Q modulo all primes of good reduction up to a given bound N, based on the average polynomial-time algorithm recently introduced by Harvey. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for N = 2^26.
We study the inverse Jacobian problem for the case of Picard curves over $\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small period matrix $Ω\in \mathbb{C}^{3\times 3}$ corresponding to a principally polarized abelian threefold equipped with an automorphism of order $3$, returns a Legendre-Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike-Weng in [Math. Comp., 74(249):499-518, 2005] which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain (numericall…
Let C/Q be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of Q, but may not have a hyperelliptic model of the usual form over Q. We describe an algorithm that computes the local zeta functions of C at all odd primes of good reduction up to a prescribed bound N. The algorithm relies on an adaptation of the "accumulating remainder tree" to matrices with entries in a quadratic field. We report on an implementation, and compare its performance to previous algorithms for the ordinary hyperelliptic case.
We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the Katz-Sarnak random matrix model) between the distributions of Lp(T) and of characteristic polynomials of random matrices in the compact Lie group USp(2g). We then formulate an analogue of the Sato-Tate conjecture for curves of genus 2, in which the generic distribution is augmented by 22 exceptional distributions, each corresponding to a compact subgroup o…
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $ψ_f\in \mathbb{Q}[x]$ such that the splitting field of $ψ_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $ψ_f$; we also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffice…
In this paper, by the tools of circulant matrices and hyperelliptic curves over finite fields, we study some arithmetic properties of certain determinants involving the Legendre symbols and $k$-th residues.
We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.
We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to obtain class groups for number fields $\mathbb{Q}(P)$, $P\in V(\Qbar)$, for certain families of points P. In particular, we show how the problem of constructing quadratic number fields with a large-rank ideal class group can be reduced to the problem of finding a hyperelliptic curve with a rational Weierstrass point and a large rational torsion subgroup in its…
The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives onl…
Let $q$ be an odd prime power, and $H_{d,q}$ denote the set of square-free monic polynomials $D(x) \in F_q[x]$ of degree $d$. Katz and Sarnak showed that the moments, over $H_{d,q}$, of the zeta functions associated to the curves $y^2=D(x)$, evaluated at the central point, tend, as $q \to \infty$, to the moments of characteristic polynomials, evaluated at the central point, of matrices in $USp(2\lfloor (d-1)/2 \rfloor)$. Using techniques that were originally developed for studying moments of $L$-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments…
In recent algorithms that use deformation in order to compute the number of points on varieties over a finite field, certain differential equations of matrices over p-adic fields emerge. We present a novel strategy to solve this kind of equations in a memory efficient way. The main application is an algorithm requiring quasi-cubic time and only quadratic memory in the parameter n, that solves the following problem: for E a hyperelliptic curve of genus g over a finite field of extension degree n and small characteristic, compute its zeta function. This improves substantially upon Kedlaya's resu…
We study the growth rate of the summatory function of the Möbius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, which was proved by Ng in 2004. We find an expression for a bound of the summatory function, which becomes sharp when the zeta zeros of the curve satisfy a certain linear independence property. Extending a result of Kowalski in 2008, we prove that most curves in the family of universal hyperelliptic curves satisfy this property. Then, we consider a certain geometric average of such bound in this …
If $π: Y \to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_π$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$-ranks can occur for $P_π$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_π$. As an application, when $p \equiv 5 \bmod 6$, we prove that there exists a curve $X$ of genus $3…
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work…
