This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
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.
An Arakelov divisor (or replete divisor[2]) on a global field is an extension of the concept of divisor or fractional ideal. It is a formal linear combination of places of the field with finite places having integer coefficients and the infinite places having real coefficients.[1][3][4]
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 holomorphic if and only if all subvarieties are of general type.[8]
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.[10]
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) discriminant[11]d(P) and the arithmetic discriminant, defined by Vojta.[12] 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.[12] 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.[12]
Flat cohomology is, for the school of Grothendieck, one terminal point of development. It has the disadvantage of being quite hard to compute with. The reason that the flat topology has been considered the 'right' foundational topos for scheme theory goes back to the fact of faithfully-flat descent, the discovery of Grothendieck that the representable functors are sheaves for it (i.e. a very general gluing axiom holds).
Function field analogy
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.
G
Geometric class field theory
The extension of class field theory-style results on abelian coverings to varieties of dimension at least two is often called geometric class field theory.
Good reduction
Fundamental to local analysis in arithmetic problems is to reducemodulo all prime numbers p or, more generally, prime ideals. In the typical situation this presents little difficulty for almost allp; for example denominators of fractions are tricky, in that reduction modulo a prime in the denominator looks like division by zero, but that rules out only finitely many p per fraction. With a little extra sophistication, homogeneous coordinates allow clearing of denominators by multiplying by a common scalar. For a given, single point one can do this and not leave a common factor p. However singularity theory enters: a non-singular point may become a singular point on reduction modulo p, because the Zariski tangent space can become larger when linear terms reduce to 0 (the geometric formulation shows it is not the fault of a single set of coordinates). Good reduction refers to the reduced variety having the same properties as the original, for example, an algebraic curve having the same genus, or a smooth variety remaining smooth. In general there will be a finite set S of primes for a given variety V, assumed smooth, such that there is otherwise a smooth reduced Vp over Z/pZ. For abelian varieties, good reduction is connected with ramification in the field of division points by the Néron–Ogg–Shafarevich criterion. The theory is subtle, in the sense that the freedom to change variables to try to improve matters is rather unobvious: see Néron model, potential good reduction, Tate curve, semistable abelian variety, semistable elliptic curve, Serre–Tate theorem.[15]
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.[16] 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.[17] 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.
Iwasawa theory
Iwasawa theory builds up from the analytic number theory and Stickelberger's theorem as a theory of ideal class groups as Galois modules and p-adic L-functions (with roots in Kummer congruence on Bernoulli numbers). In its early days in the late 1960s it was called Iwasawa's analogue of the Jacobian. The analogy was with the Jacobian varietyJ of a curve C over a finite field F (qua Picard variety), where the finite field has roots of unity added to make finite field extensions F′ The local zeta-function (q.v.) of C can be recovered from the points J(F′) as Galois module. In the same way, Iwasawa added pn-power roots of unity for fixed p and with n → ∞, for his analogue, to a number field K, and considered the inverse limit of class groups, finding a p-adic L-function earlier introduced by Kubota and Leopoldt.
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 Mordell–Weil theorem is a foundational result stating that for an abelian variety A over a number field K the group A(K) is a finitely-generated abelian group. This was proved initially for number fields K, but extends to all finitely-generated fields.
Mordellic variety
A Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field.[24]
N
Naive height
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.[25]
Néron symbol
The Néron symbol is a bimultiplicative pairing between divisors and algebraic cycles on an Abelian variety used in Néron's formulation of the Néron–Tate height as a sum of local contributions.[26][27][28] The global Néron symbol, which is the sum of the local symbols, is just the negative of the height pairing.[29]
Néron–Tate height
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.[29]
Nevanlinna invariant
The Nevanlinna invariant of an ample divisorD on a normalprojective varietyX is a real number which describes the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor.[30] It has similar formal properties to the abscissa of convergence of the height zeta function and it is conjectured that they are essentially the same.[31]
O
Ordinary reduction
An Abelian variety A of dimension d has ordinary reduction at a prime p if it has good reduction at p and in addition the p-torsion has rank d.[32]
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.[33] A replete divisor is an Arakelov divisor.[2]
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 varieities;[35] another definition is the union of all subvarieties that are not of general type.[8] For abelian varieties the definition would be the union of all translates of proper abelian subvarieties.[36] 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.[37]
Subspace theorem
Schmidt's subspace theorem shows that points of small height in projective space lie in a finite number of hyperplanes. A quantitative form of the theorem, in which the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by Schlickewei (1977) to allow more general absolute values on number fields. The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation.[38]
The Tate conjecture (John Tate, 1963) provided an analogue to the Hodge conjecture, also on algebraic cycles, but well within arithmetic geometry. It also gave, for elliptic surfaces, an analogue of the Birch–Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of its importance.
Tate curve
The Tate curve is a particular elliptic curve over the p-adic numbers introduced by John Tate to study bad reduction (see good reduction).
Tsen rank
The Tsen rank of a field, named for C. C. Tsen who introduced their study in 1936,[39] 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.[40]
U
Uniformity conjecture
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.[41]
Unlikely intersection
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.[42]
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 co-ordinates of points on algebraic varieties. It has remained somewhat under-developed.
The Weil height machine is an effective procedure for assigning a height function to any divisor on smooth projective variety over a number field (or to Cartier divisors on non-smooth varieties).[46]
^van der Geer, G.; Schoof, R. (2000). "Effectivity of Arakelov divisors and the theta divisor of a number field". Selecta Mathematica, New Series. 6 (4): 377–398. arXiv:math/9802121. doi:10.1007/PL00001393. Zbl1030.11063.
^Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN3-540-37888-X.
^Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer. ISBN0-387-96311-1. → Contains an English translation of Faltings (1983)
^Raynaud, Michel (1983). "Sous-variétés d'une variété abélienne et points de torsion". In Artin, Michael; Tate, John (eds.). Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic. Progress in Mathematics (in French). Vol. 35. Birkhauser-Boston. pp. 327–352. Zbl0581.14031.
^Roessler, Damian (2005). "A note on the Manin-Mumford conjecture". In van der Geer, Gerard; Moonen, Ben; Schoof, René (eds.). Number fields and function fields — two parallel worlds. Progress in Mathematics. Vol. 239. Birkhäuser. pp. 311–318. ISBN0-8176-4397-4. Zbl1098.14030.
^Marcja, Annalisa; Toffalori, Carlo (2003). A Guide to Classical and Modern Model Theory. Trends in Logic. Vol. 19. Springer-Verlag. pp. 305–306. ISBN1402013302.
^It is mentioned in J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry, pages 93–110 (1965).
