This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions.
As explained by Huisman and Mahe, a given monic squarefree positive polynomial in two variables x and y of degree in y divisible by 4 is a sum of three squares of rational fractions if and only if the jabobian variety of some hyperelliptic curve (associated to P) has an "antineutral" point.
Using this criterium, we follow a method developped by Cassels, Ellison and Pfister to solve our problem : at first we show the Mordell-Weil rank of the jacobian variety J associated to some polynomial is zero (this step is done by doing a 2-descent), and then we check that the jacobian variety J has no antineutral torsion point.
The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non-reduced components in the special fibre of their Néron model. In this present paper we explain how the real period can be computed, and how the verification has been extended to many more hyperelliptic curves, some of genus 3, 4 and 5, without using modular methods.
We present algorithms for computing the squared Weil and Tate pairings on elliptic curves and the squared Tate pairing for hyperelliptic curves. The squared pairings introduced in this paper have the advantage that our algorithms for evaluating them are deterministic and do not depend on a random choice of points. Our pairings save about 20-30% over the usual pairings.
For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.
We study the character sums \[ϕ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } ψ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left((x^{m}+a)(x^{n}+b)\right)\] where $ϕ$ is the quadratic character defined over $\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties.
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
We apply a variant of the square-sieve to produce a uniform upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over the projective line, whose general fibre is a hyperelliptic curve.
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …
We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.
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…
Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional complex abelian variety $P$ be a Prym variety of $C_f$ that corresponds to a unramified double cover of $C_f$. Suppose that there exists a subfield $K$ of $\C$ such that $f(x)$ lies in $K[x]$, is irreducible over $K$ and its Galois group is the full symmetric group. Assuming that $g>2$, we prove that $End(P)$ is either the ring of integers $Z$ or the direct sum…
The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ is distributed as the trace of a random $2g\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with p…
Erdős, Graham, and Selfridge considered, for each positive integer $n$, the least value of $t_n$ so that the integers $n+1, n+2, \dots, n+t_n $ contain a subset the product of whose members with $n$ is a square. An open problem posed by Granville concerns the size of $t_n$, under the assumption of the ABC Conjecture. We establish some results on the distribution of $t_n$, and in the process solve Granville's problem unconditionally.
Let g >= 1 and let Q be a monic, squarefree polynomial of degree 2g + 1 in Z[x]. For an odd prime p not dividing the discriminant of Q, let Z_p(T) denote the zeta function of the hyperelliptic curve of genus g over the finite field F_p obtained by reducing the coefficients of the equation y^2 = Q(x) modulo p. We present an explicit deterministic algorithm that given as input Q and a positive integer N, computes Z_p(T) simultaneously for all such primes p < N, whose average complexity per prime is polynomial in g, log N, and the number of bits required to represent Q.
Let $f(x)$ be a degree $(2g+1)$ monic polynomial with coefficients in an algebraically closed field $K$ with $char(K)\ne 2$ and without repeated roots. Let $\mathfrak{R}\subset K$ be the $(2g+1)$-element set of roots of $f(x)$. Let $\mathcal{C}: y^2=f(x)$ be an odd degree genus $g$ hyperelliptic curve over $K$. Let $J$ be the jacobian of $\mathcal{C}$ and $J[2]\subset J(K)$ the (sub)group of its points of order dividing $2$. We identify $\mathcal{C}$ with the image of its canonical embedding into $J$ (the infinite point of $\mathcal{C}$ goes to the identity element of $J$). Let $P=(a,b)\in \ma…
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 $c_K(g)$.
It is known that for a smooth hyperelliptic curve to have a large $a$-number, the genus must be small relative to the characteristic of the field, $p>0$, over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve $C$ to have $a_C=g-1$, the genus is bounded by $g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g <p$. The method of proof is to force the Cartier-Manin matrix to have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. We then use this bound to summarize what is kn…
A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.
Let $\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\mathbb{Q}(T)$ of genus $g\geq 1$. Assume that the jacobian of $\mathcal{X}$ over $\mathbb{Q}(T)$ has no subvariety defined over $\mathbb{Q}$. Denote by $\mathcal{X}_t$ the specialization of $\mathcal{X}$ to an integer $T=t$, let $a_{\mathcal{X}_t}(p)$ be its trace of Frobenius, and $A_{\mathcal{X},r}(p) = \frac{1}{p}\sum_{t=1}^p a_{\mathcal{X}_t}(p)^r$ its $r$-th moment. The first moment is related to the rank of the jacobian $J_\mathcal{X}\left(\mathbb{Q}(T)\right)$ by a generalization of a conjecture of Nagao: $$\lim_{X \to \inft…
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…
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…
For a complex polynomial $D(t)$ of even degree, one may define the continued fraction of $\sqrt{D(t)}$. This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients. In this paper we shall prove that, however, a correct analogue of Lagrange's theorem still exists in full generality: pre-periodicity of the {\it degrees} of the partial quotients al…
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $\sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in \cite{zannier} that the sequence of the…
Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of hyperelliptic curves whose genera tend to infinity, these measures tend to the natural measure on the space of rank two bundles. This is a function field analogue of Duke's theorem on the equidistribution of Heegner points, and can be proven similarly: it follows from a manipulation of zeta functions, plus the Riemann Hypothesis for curves. Likewise, a sequence o…
In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two pri…
Problems of (i) precise (exact) bound for families of hyperelliptic curves over prime finite fields and (ii) equidistribution of angles of Kloosterman sums are discussed.
An asymptotic formula for the sum $\sum L(1,χ)$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ as $g\rightarrow\infty$ making use of the analogue of the approximate functional equation for such $L$--functions. As a corollary, we obtain a formula for the average of the class number of the associated rings $\mathbb{F}_{q}[T,sqrt{D}]$.
Poonen and Stoll have shown that the reduced Shafarevich-Tate group of a principally polarized abelian variety over a global field can have order twice a square (the odd case) as well as a square (the even case). For a curve over a global field, they give a local diophantine criterion for its jacobian to be even or odd. In this note we use the Cherednik-Drinfeld p-adic uniformization to study the local points and divisors on certain Atkin-Lehner quotients of Shimura curves. We exhibit an explicit family of such quotient curves over the rationals with odd jacobians, with genus going to infinity…
Let $f\in\Q[x]$ be a square-free polynomial of degree $\geq 3$ and $m\geq 3$ be an odd positive integer. Based on our earlier investigations we prove that there exists a function $D_{1}\in\Q(u,v,w)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{1}y^2=f(x),\quad C_{2}:\;y^2=D_{1}x^m+b,\quad C_{3}:\;y^2=D_{1}x^m+c, \end{equation*} have all positive ranks over $\Q(u,v,w)$. Similarly, we prove that there exists a function $D_{2}\in\Q(u,v,w)$ such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{2}y^2=h(x),\quad C_{2}:\;y^2=D_{2}x^m+b,\quad C_{3}:\;y^2=x^m+cD_{2}, …
We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square with side length $M$. We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curves $Y^2=X^{2g+1} + a_{2g-1}X^{2g-1} +...+ a_1X+a_0$ over the finite field $\F_p$ of $p$ elements, with coefficients in a $2g$-dimensional cube $ (a_0,..., a_{2g-1})\in [R_0+1,R_0+M]\times...\times [R_{2g-1}+1,R_{2g-1}+M]$ that are is…
We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can be written as the sum of 3 squares.
We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.
In this paper, we evaluate a smoothed sum of the form $\displaystyle \sideset{}{^*}\sum_{d \leq X}\left(\sum_{n \leq Y} \left(\frac{8d}{n} \right ) \right)^2$, where $\left ( \frac{8d}{\cdot} \right )$ is the Kronecker symbol and $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers.
The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Châtelet surfaces defined over the rationals.
We show that there are short intervals $[x,x+y]$ containing $\gg y^{1/10}$ numbers expressible as the sum of two squares, which is many more than the average when $y=o( (\log{x})^{5/9})$. We obtain similar results for sums of two squares in short arithmetic progressions.
Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares for all $n \in \mathbb{N}$ if and only if $a$ is a perfect square. We also prove that for $a\equiv 0,1,2\pmod{4},$ if $a^{n} + 1$ is the sum of two squares, then $a^δ + 1$ is the sum of two squares for all $δ| n, \ δ>1$. Using Aurifeuillian factorization, we show that if $a$ is a prime and $a\equiv 1 \pmod{4}$, then there are either zero or infinitely many o…
We examine the representation of numbers as the sum of two squares in $\mathbb{Z}_n$ for a general positive integer $n$. Using this information we make some comments about the density of positive integers which can be represented as the sum of two squares and powers of $2$ in $\mathbb{N}$.
Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$ which are a sum of two squares. Our result confirms a special case of a conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero (e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two squares), then $s^{-1}(\mathcal{A})$ also has asymptotic densi…
We improve a previous unconditional result about the asymptotic behavior of $\sum_{n\le x} r(n)r(n+m)$ with $r(n)$ the number of representations of $n$ as a sum of two squares when $m$ may vary with $x$.
We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.
Inspired by the fact that the sum of the cubes of the first $n$ naturals is equal to the square of their sum, we explore, for each $n$, the Diophantine equation representing all non-trivial sets of $n$ integers with this property. We find definite answers to the standard question of infinitude of the solutions as well as several other surprising results.
Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.
We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two non-negative integer powers of 2.
We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.
