Arithmetical or arithmetic (algebraic) geometry is a field with a less elementary definition. After the advent of scheme theory it could reasonably be defined as the study of Alexander Grothendieck's schemes of finite type over the spectrum of the ring of integersZ. This point of view has been very influential for around half a century; it has very widely been regarded as fulfilling Leopold Kronecker's ambition to have number theory operate only with rings that are quotients of polynomial rings over the integers (to use the current language of commutative algebra). In fact scheme theory uses all sorts of auxiliary constructions that do not appear at all 'finitistic', so that there is little connection with 'constructivist' ideas as such. That scheme theory may not be the last word appears from continuing interest in the 'infinite primes' (the real and complex local fields), which do not come from prime ideals as the p-adic numbers do.
The abc conjecture of Masser and Oesterlé attempts to state as much as possible about repeated prime factors in an equation a + b = c. For example 3 + 125 = 128 but the prime powers here are exceptional.
Enrico Bombieri, Serge Lang and Paul Vojta and Piotr Blass have conjectured that algebraic varieties of general type do not have Zariski dense subsets of K-rational points, for K a finitely-generated field. This circle of ideas includes the understanding of analytic hyperbolicity and the Lang conjectures on that, and the Vojta conjectures. An analytically hyperbolic algebraic varietyV over the complex numbers is one such that no holomorphic mapping from the whole complex plane to it exists, that is not constant. Examples include compact Riemann surfaces of genus g > 1. Lang conjectured that V is analytically hyperbolic if and only if all subvarieties are of general type.
Chabauty's method, based on p-adic analytic functions, is a special application but capable of proving cases of the Mordell conjecture for curves whose Jacobian's rank is less than its dimension. It developed ideas from Thoralf Skolem's method for an algebraic torus. (Other older methods for Diophantine problems include Runge's method.)
The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > dk. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.
Discriminant of a point
The discriminant of a point refers to two related concepts relative to a point P on an algebraic variety V defined over a number field K: the geometric (logarithmic) discriminantd(P) and the arithmetic discriminant, defined by Vojta. The difference between the two may be compared to the difference between the arithmetic genus of a singular curve and the geometric genus of the desingularisation. The arithmetic genus is larger than the geometric genus, and the height of a point may be bounded in terms of the arithmetic genus. Obtaining similar bounds involving the geometric genus would have significant consequences.
It was realised in the nineteenth century that the ring of integers of a number field has analogies with the affine coordinate ring of an algebraic curve or compact Riemann surface, with a point or more removed corresponding to the 'infinite places' of a number field. This idea is more precisely encoded in the theory that global fields should all be treated on the same basis. The idea goes further. Thus elliptic surfaces over the complex numbers, also, have some quite strict analogies with elliptic curves over number fields.
The Hasse principle states that solubility for a global field is the same as solubility in all relevant local fields. One of the main objectives of Diophantine geometry is to classify cases where the Hasse principle holds. Generally that is for a large number of variables, when the degree of an equation is held fixed. The Hasse principle is often associated with the success of the Hardy–Littlewood circle method. When the circle method works, it can provide extra, quantitative information such as asymptotic number of solutions. Reducing the number of variables makes the circle method harder; therefore failures of the Hasse principle, for example for cubic forms in small numbers of variables (and in particular for elliptic curves as cubic curves) are at a general level connected with the limitations of the analytic approach.
A height function in Diophantine geometry quantifies the size of solutions to Diophantine equations. Classical or naive height is defined in terms of ordinary absolute value on homogeneous coordinates: it is now usual to take a logarithmic scale, that is, height is proportional to the "algebraic complexity" or number of bits needed to store a point. Heights were initially developed by André Weil and D. G. Northcott. Innovations around 1960 were the Néron–Tate height and the realisation that heights were linked to projective representations in much the same way that ample line bundles are in pure geometry.
Infinite descent was Pierre de Fermat's classical method for Diophantine equations. It became one half of the standard proof of the Mordell–Weil theorem, with the other being an argument with height functions (q.v.). Descent is something like division by two in a group of principal homogeneous spaces (often called 'descents', when written out by equations); in more modern terms in a Galois cohomology group which is to be proved finite. See Selmer group.
The Mordell conjecture is now the Faltings theorem, and states that a curve of genus at least two has only finitely many rational points. The Uniformity conjecture states that there should be a uniform bound on the number of such points, depending only on the genus and the field of definition.
The naive or classical height of a vector of rational numbers is the maximum absolute value of the vector of coprime integers obtained by multiplying through by a lowest common denominator. This may be used to define height on a point in projective space over Q, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial.
The Néron–Tate height (also often referred to as the canonical height) on an abelian varietyA is a height function (q.v.) that is essentially intrinsic, and an exact quadratic form, rather than approximately quadratic with respect to the addition on A as provided by the general theory of heights. It can be defined from a general height by a limiting process; there are also formulae, in the sense that it is a sum of local contributions.
A replete ideal in a number field K is a formal product of a fractional ideal of K and a vector of positive real numbers with components indexed by the infinite places of K. A replete divisor is an Arakelov divisor.
The special set in an algebraic variety is the subset in which one might expect to find many rational points. The precise definition varies according to context. One definition is the Zariski closure of the union of images of algebraic groups under non-trivial rational maps; alternatively one may take images of abelian varieties; another definition is the union of all subvarieties that are not of general type. For abelian varieties the definition would be the union of all translates of proper abelian subvarieties. For a complex variety, the holomorphic special set is the Zariski closure of the images of all non-constant holomorphic maps from C. Lang conjectured that the analytic and algebraic special sets are equal.
The Tate curve is a particular elliptic curve over the p-adic numbers introduced by John Tate to study bad reduction (see good reduction).
The Tsen rank of a field, named for C. C. Tsen who introduced their study in 1936, is the smallest natural number i, if it exists, such that the field is of class Ti: that is, such that any system of polynomials with no constant term of degree dj in n variables has a non-trivial zero whenever n > ∑ dji. Algebraically closed fields are of Tsen rank zero. The Tsen rank is greater or equal to the Diophantine dimension but it is not known if they are equal except in the case of rank zero.
The uniformity conjecture states that for any number field K and g > 2, there is a uniform bound B(g,K) on the number of K-rational points on any curve of genus g. The conjecture would follow from the Bombieri–Lang conjecture.
An unlikely intersection is an algebraic subgroup intersecting a subvariety of a torus or abelian variety in a set of unusually large dimension, such as is involved in the Mordell–Lang conjecture.
The Weil conjectures were three highly influential conjectures of André Weil, made public around 1949, on local zeta-functions. The proof was completed in 1973. Those being proved, there remain extensions of the Chevalley–Warning theorem congruence, which comes from an elementary method, and improvements of Weil bounds, e.g. better estimates for curves of the number of points than come from Weil's basic theorem of 1940. The latter turn out to be of interest for Goppa codes.
Weil distributions on algebraic varieties
André Weil proposed a theory in the 1920s and 1930s on prime ideal decomposition of algebraic numbers in coordinates of points on algebraic varieties. It has remained somewhat under-developed.
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