A Remark on Mordell's Conjecture

Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field $$ \mathbb{F}_q $$ of q elements, we obtain the asymptotic formula q g+o(g) for the size of set of the $$ \mathbb{F}_q $$ -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.

Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k-rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K. Further let (τ, B) be the K/k:-trace of J (see §2 below and).

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

We develop algorithms for three problems. Starting with a complex torus of dimension g ≥ 2, isomorphic to a principally polarized, simple abelian variety A/C, the first problem is to find an algorithmic solution of the hyperelliptic Schottky problem: Is there a hvperelliptic curve C of genus g whose jacobian variety Ic is isomorphic to A over C? Our solution is based on [Poor 1994]. If such a hyperelliptic curve C exists, the next problem is the construction of the Rosenhain model C : y2 = X(X−1)(X−λ1)(X−λ2)…(X−λ2g−1) for pairwise distinct numbers λj E ∈ C \\ {0, 1}. Applying the theory of hyperelliptic theta functions we show that these numbers λj can easily be computed by using theta constants with even characteristics. If the abelian variety A is defined over a field k (this field could be the field of rational numbers, an algebraic number field of low degree, or a finite field), we show only in the case k = Q for simplicity, how the method in [Mestre 1991] can be generalized to get a minimal equation over Z[½] for the hyperelliptic curve C with jacobian variety Ic ≅c A. This is our third problem. For some hyperelliptic, principally polarized and simple factors with dimension g = 3, 4, 5 of the jacobian variety J0(N) = IX0(N) of the modular curve X0(N) we compute the corresponding curve equations by applying our algorithms to this special situation.

Recent workers [1, 3] have proved density theorems about the rational points on K3 surfaces of the form $$ V:\;X_0^4+cX_1^4=X_2^4+cX_3^4 $$ for certain non-zero values of c. Their arguments depend on the presence of at least two pencils of curves of genus 1 on V. Unfortunately the values of c for which the argument works are constrained by the need to exhibit explicitly a rational point on V which satisfies certain extra conditions; these in particular require it to lie outside the four obvious rational lines on V. It is therefore natural to ask whether there are other curves of genus 0 or 1 defined over Q on V. In the case c = 1 there are known to be infinitely many such curves (see [2]), and for general rational c the quadratic form Q on the Néron–Severi group whose value is the self-intersection number takes the values 0 and -2 infinitely often. Naively one might expect the case c = 1 to be typical; but this is not so. The main object of this paper is to prove the following result.

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

When Newton met Diophantus: A Study of Rational-Derived Polynomials and Their Extension to Quadratic Fields

Introduction.We can state the problem of the moduli of algebraic curves in the following way.Consider all irreducible algebraic curves, defined over a ground field k, of a fixed genus g.Can we, in some natural fashion, associate to each birational class of these curves a point of an algebraic variety N, defined over k, in such a way that for distinct birational classes we obtain distinct points?We impose the further condition that those points of N which do not represent any of the above birational classes form a proper algebraic subvariety of N, defined over k.We shall consider this problem only for the special case of hyperelliptic curves, van der Waerden [3] has attacked the general problem.There is, however, an important gap in the proof of Theorem 7 [p.698, loc.cit.j.We shall use the notation of van der Waerden in pointing this out.I, is an irreducible algebraic system of regular plane curves of genus g [1J4, p. 698, loc.cit.].The algebraic correspondence 93, transforms 7, onto itself.The difficulty arises from the fact that, as far as we know, an irreducible regular plane curve V of genus g may be fundamental for the correspondence 93,.This situation has not been ruled out since V may correspond, under 93,., to curves in I, which do not belong to ^Jr.These curves could not be irreducible regular curves of genus g.Such "limiting curves" can be ignored when considering I, as the domain of 93" but not when considering it as the range of this correspondence.If T is fundamental for 93, then the birational class of r would be represented by infinitely many points on the moduli-variety My.In the special case of hyperelliptic curves the objects involved are much simpler, hence the corresponding difficulty can be handled easily.We shall assume the basic concepts and definitions of the theory of algebraic varieties.Our terminology conforms to that of Zariski [6].We suppose that a universal domain ft has been fixed, once and for all.All quantities which occur will lie in ft, and any ground field k that we choose will be such that ft has infinite transcendence degree over it.

A note on Artin's conjecture in algebraic number fields

Square-free values of cubic polynomials in algebraic number fields

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

I present this paper for publication to an American journal and in English for the following reason :The theory of linear algebras has been greatly extended through the work of American mathematicians.Of late, German mathematicians have become active in this theory.In particular, they have succeeded in obtaining some apparently remarkable results by using the theory of algebraic numbers, ideals, and abstract algebra, highly developed in Germany in recent decades.These results do not seem to be as well known in America as they should be on account of their importance.This fact is due, perhaps, to the language difference or to the unavailability of the widely scattered sources.This paper develops a new application of the above mentioned theories to the theory of linear algebras.Of particular importance is the fact that purely algebraic results are obtained from deep-lying arithmetical theorems.In the middle part, an account is given of the fundamental algebraic basis for these arithmetical methods.This account is more extended than is necessary for this paper, and should obviate an extended study of several German papers.I am very grateful to Professor F. T. Engstrom (New Haven) for going through the manuscript and proof with me and anglicising my many literal translations from the German.I. Statement of theorems to be proved!1. Cyclic algebras.In I, the reference field is assumed to be an algebraic number field of finite degree.}A cyclic algebra of degree over Q, is defined as an algebra A of the following type: * Presented to the Society, September 9, 1931; received by the editors May 29, 1931.f This section has also appeared recently in German in the Nachrichten von der Gesellschaft der Wissenschaften zu Gttingen, 1931. In the following, such notations as are only made complete by naming a definite reference field, may be implicitly understood to be with respect to unless another reference field is explicitly named. || See also Dickson (1 App. II).* It is unnecessary to distinguish between isomorphism (equivalence) and identity unless subalgebras of the same algebra are considered.t In the following, the general norm residue theory, recently developed by the author, is of the greatest importance.See therefore the papers Hasse (11,12, 13 II). This alternative form has been introduced by the editors to simplify typography. Hasse (1-4).I' a,Z\ Ia,Z\ I ,Z\(3-2) hr)-(-r)(ir> By (3.1) and (3.2), the symbol ((a, Z)/p) is indeed fixed except an arbitrary exponent prime to n and independent of a which may be attached.fIt is a pity that one is not able to-day to fix that exponent in a quite natural manner, i.e., without having to consider also the congruence behavior of a for prime spots different from p and with it, in principle, the law of reciprocity.Therefore, in order to obtain the complete definition of the symbol, one has to take the following round-about way : Let f be the conductor (Fhrer) of Z,% fp the exact power of p contained in f, and a0 a number in Q with the following properties : * Hasse (12 p. 150).t This follows immediately from the fact that Z is cyclic.X I.e., the conductor of the ideal class group to which Z belongs as class corps (Klassenkrper) See Hasse (7, 9). Hasse (7 Satz 13).Incidentally (3.5) may also be omitted; see Hasse (11, 13 6).I adjoined it here only in order to get a formal simplification in the later proof of Theorem 1, (i).* This alternative form has been introduced by the editors to simplify typography.t See Hasse (7.Erl. 30, 9 8).J See also Dickson (1 51,3 78), Artin (2). See Wedderburn (1), Artin (2).