Number Of Rational Points Research Articles

ELEMENTARY ABELIAN P-EXTENSIONS AND CURVES WITH MANY POINTS

In the paper we give a generalization of two results of two results obtained by García and Stichtenoth ([G-S]) and use them of exhibit a method to construct curves over finite fields whose number of rational points is large compared to their genus. Such curves are induced by algebraic functions fields obtained from elementary abelain p-extensions of the rational function field Fq(x) using the trace operator TrFq/Fp.

On the number of rational points on Artin-Schreier hypersurfaces

Let Fqk denote the finite field with qk elements. In this work, we study the number of Fqk-rational points of the affine hypersurfaces given by yq−y=a1x1d1+…+asxsds+b. We prove that if b∈Fqk has a nonzero trace over Fq, then Weil's bound cannot be achieved and that a sharp bound can be explicitly provided. In the case Trqk/q(b)=0, we give necessary and sufficient conditions for Weil's bound to be attained. Furthermore, we present several new cases in which formulas for the number of Fqk-rational points can be obtained.

Rank growth of elliptic curves over 𝑁-th root extensions

Fix an elliptic curve E E over a number field F F and an integer n n which is a power of 3 3 . We study the growth of the Mordell–Weil rank of E E after base change to the fields K d = F ( d 2 n ) K_d = F(\!\sqrt [2n]{d}) . If E E admits a 3 3 -isogeny, then we show that the average “new rank” of E E over K d K_d , appropriately defined, is bounded as the height of d d goes to infinity. When n = 3 n = 3 , we moreover show that for many elliptic curves E / Q E/\mathbb {Q} , there are no new points on E E over Q ( d 6 ) \mathbb {Q}(\sqrt [6]d) , for a positive proportion of integers d d . This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert’s tenth problem has a negative solution over a positive proportion of pure sextic fields Q ( d 6 ) \mathbb {Q}(\sqrt [6]{d}) . The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the “correlation trick”, which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

Geometric quadratic Chabauty over number fields

This article generalizes the geometric quadratic Chabauty method, initiated over Q \mathbb {Q} by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell–Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.

The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields

<abstract><p>Let $ q $ be an even prime power and let $ \mathbb{F}_{q} $ be the finite field of $ q $ elements. Let $ f $ be a nonzero polynomial over $ \mathbb{F}_{q^2} $ of the form $ f = a_{1}x_{1}^{m_{1}}+\dots+a_{s}x_{s}^{m_{s}}+y_{1}y_{2}+\dots+y_{n-1}y_{n}+y_{n-2t-1}^{2}+\dots +y_{n-3}^2+y_{n-1}^{2}$ $+b_{t}y_{n-2t}^{2}+ $ $ \dots +b_{1}y_{n-2}^{2}+b_{0}y_{n}^{2} $, where $ a_{i}, b_j\in \mathbb{F}_{q^{2}}^{*}, $ $ m_i\ne 1, $ $ (m_{i}, m_{k}) = 1, $ $ i\ne k, $ $ m_{i}|(q+1), $ $ m_{i}\in \mathbb{Z}^{+}, $ $ 2|n $, $ n > 2 $, $ 0\leq t\leq \frac{n}{2}-2 $, $ {\mathrm{Tr}}_{\mathbb{F}_{q^2}/\mathbb{F}_{2}}(b_{j}) = 1 $ for $ i, k = 1, \dots, s $ and $ j = 0, 1, \dots, t $. For each $ b \in \mathbb{F}_{q^2} $, let $ N_{q^2}(f = b) $ denote the number of $ \mathbb{F}_{q^2} $-rational points on the affine hypersurface $ f = b $. In this paper, we obtain the formula of $ N_{q^2}(f = b) $ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.</p></abstract>

Projective plane curves over a finite field with conical components

We present the list of maximal projective plane curves containing conics and those which are arrangements of conics. The number of rational points and the corresponding polynomials are given. The third highest number of points of projective curves of degree d over a finite field Fq (d < [q/3]) is associated only to some linear curves. We show that for q/2 + 2 < d < q, this is no longer the case: the third highest number of points can also be obtained by some curves containing a conic. Throughout this work, we obtain some bounds concerning the number of Fq-points of curves with linear, conic and cubic factors. these bounds apply (not sharply) to irreducible curves

Counting rational points of a Grassmannian

We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt ([12]) and Thunder ([20]) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.

On the number of rational points on a class of singular hypersurfaces

On the number of rational points on a class of singular hypersurfaces

Curves on Frobenius classical surfaces in $\mathbb{P}^{3}$ over finite fields

We give an upper bound on the number of rational points on an irreducible curve $C$ of degree $\delta $ defined over a finite field $\mathbb F_q$ lying on a Frobenius classical surface $S$ embedded in $\mathbb P^3$. This leads us to investigate arithmetic

Permutation binomials over finite fields

Let Fq denote the finite field with q elements. In this paper we use the relation between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements a∈Fq for which the binomial xn(xq−13+a) is a permutation polynomial. In order to do this, we employ results on the Cartin-Manin operator and the Riemann Hypothesis for elliptic curves to present the exact number of points on a suitable elliptic curve.

Uniformity in Mordell–Lang for curves

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of $g$, $d$ and the Mordell–Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in $g$ and $d$, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel–Jacobi map. Both estimates generalize our previous work for one-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.

Number of rational points of elliptic curves

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul. 39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc. 15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].

A Proof for the Birch Swinnerton-Dyer Conjecture

The Birch Swinnerton-Dyer Conjecture states that the number of rational points on an elliptic curve are infinite. This paper develops the logical formalism for a reinforcement learning system embedded with a quantum search algorithm. The formalism proves the rational points on high dimensional elliptical curves are infinite.

On components of vectorial permutations of [formula omitted

We consider vectorial mapsF(x1,…,xn)=(f1(x1,…,xn),…,fn(x1,…,xn)):Fqn↦Fqn, which induce permutations of Fqn. We show that the degrees of the components f1,f2,…,fn∈Fq[x1,…,xn] are at least 2 when 2≤deg(F)=d<q and d|(q−1). Our proof uses an absolutely irreducible curve over Fq and the number of rational points on it that we relate to the cardinality of the value set of a polynomial.

Curves over Finite Fields and Permutations of the Form x k

We consider the polynomials of the form $P(x)=x^k-\gamma \mathrm{Tr}(x)$ over $\mathbb{F}_{q^n}$ for $n\geq 2$. We show that $P(x)$ is not a permutation of $\mathbb{F}_{q^n}$ in the case $\gcd(k, q^n-1)>1$. Our proof uses an absolutely irreducible curve over $\mathbb{F}_{q^n}$ and the number of rational points on it.

Divisibility of L-polynomials for a family of Artin–Schreier curves

In this paper we consider the curves Ck(p,a):yp−y=xpk+1+ax defined over Fp and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves Ck(p,a). Our proof involves finding an exact formula for the number of Fpn-rational points on Ck(p,a) for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.

A new method of solving certain quartic and higher degree diophantine equations

In this paper, we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be applied to some diophantine systems in five or more variables. Under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, two examples being a sextic equation in four variables and two simultaneous equations of degrees four and six in six variables. We also simultaneously obtain arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. We obtain these solutions without finding a curve of genus 0 or 1 on the variety defined by the equations concerned. It appears that there exist projective varieties on which there are an arbitrarily large number of rational points and which do not contain a curve of genus 0 or 1 with infinitely many rational points.

Remarks on the Number of Rational Points on a Class of Hypersurfaces over Finite Fields

Let 𝔽q be the finite field of q elements and f be a nonzero polynomial over 𝔽q. For each b ϵ 𝔽q, let Nq(f = b) denote the number of 𝔽q-rational points on the affine hypersurface f = b. We obtain the formula of Nq(f = b) for a class of hypersurfaces over 𝔽q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.

Counting rational points on quartic del Pezzo surfaces with a rational conic

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over {mathbb {Q}} that contains a conic defined over mathbb {Q}.

A SIMPLE PROOF OF CHEBOTAREV’S DENSITY THEOREM OVER FINITE FIELDS

We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.