Glossary of arithmetic and Diophantine geometry
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.
Diophantine geometry in general is the study of algebraic varieties V over fields K that are finitely generated over their prime fields—including as of special interest number fields and finite fields—and over local fields. Of those, only the complex numbers are algebraically closed; over any other K the existence of points of V with co-ordinates in K is something to be proved and studied as an extra topic, even knowing the geometry of V.
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 integers Z. 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.
A
- abc conjecture
- 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.
- Arakelov theory
- Arakelov theory is an approach to arithmetic geometry that explicitly includes the 'infinite primes'.
- Arithmetic of abelian varieties
- See main article arithmetic of abelian varieties
- Artin L-functions
- Artin L-functions are defined for quite general Galois representations. The introduction of étale cohomology in the 1960s meant that Hasse-Weil L-functions (q.v.) could be regarded as Artin L-functions for the Galois representations on l-adic cohomology groups.
B
- Bad reduction
- See good reduction.
- Birch and Swinnerton-Dyer conjecture
- The Birch and Swinnerton-Dyer conjecture on elliptic curves postulates a connection between the rank of an elliptic curve and the order of pole of its Hasse-Weil L-function. It has been an important landmark in Diophantine geometry since the mid-1960s, with important results such as the Coates–Wiles theorem, Gross–Zagier theorem and Kolyvagin's theorem
- Bombieri–Lang conjecture
- Enrico Bombieri, Serge Lang and Paul Vojta 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 holomorphic algebraic variety V 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.
C
- Canonical height
- The canonical height on an abelian variety is a height function that is a distinguished quadratic form. See Néron-Tate height.
- Chabauty's method
- 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.)
- Crystalline cohomology
- Crystalline cohomology is a p-adic cohomology theory in characteristic p, introduced by Alexander Grothendieck to fill the gap left by étale cohomology which is deficient in using mod p coefficients in this case. It is one of a number of theories deriving in some way from Dwork's method (q.v.), and has applications outside purely arithmetical questions.
D
- Diagonal forms
- Diagonal forms are some of the simplest projective varieties to study from an arithmetic point of view (including the Fermat varieties). Their local zeta-functions are computed in terms of Jacobi sums. Waring's problem is the most classical case.
- Dwork's method
- Bernard Dwork used distinctive methods of p-adic analysis, p-adic algebraic differential equations, Koszul complexes and other techniques that have not all been absorbed into general theories such as crystalline cohomology (q.v.). He first proved the rationality of local zeta-functions, the initial advance in the direction of the Weil conjectures (q.v.)
E
- Étale cohomology
- The search for a Weil cohomology (q.v.) was at least partially fulfilled in the étale cohomology theory of Alexander Grothendieck and Michael Artin. It provided a proof of the functional equation for the local zeta-functions, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories.
F
- Fermat's last theorem
- Fermat's last theorem, the most celebrated conjecture of Diophantine geometry, was proved by Andrew Wiles and Richard Taylor.
- Flat cohomology
- 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 reduce modulo all prime numbers p. In the typical situation this presents little difficulty for almost all p; 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 therefore excludes 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. 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, Ogg-Néron-Shafarevich criterion, Serre-Tate theorem.
- Grothendieck-Katz conjecture
- The Grothendieck-Katz p-curvature conjecture applies reduction modulo primes to algebraic differential equations, to derive information on algebraic function solutions. It is an open problem as of 2005[update]. The initial result of this type was Eisenstein's theorem.
H
- Hasse principle
- 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.
- Hasse-Weil L-function
- A Hasse-Weil L-function, sometimes called a global L-function, is an Euler product formed from local zeta-functions. The properties of such L-functions remain largely in the realm of conjecture, with the proof of the Taniyama-Shimura conjecture being a breakthrough. The Langlands philosophy is largely complementary to the theory of global L-functions.
- Height function
- A height function in Diophantine geometry quantifies the size of solutions to Diophantine equations. It is standard to take a logarithmic scale: that is, the height is proportional to the number of bits a computer needs to store a point in homogeneous coordinates. Heights were initially developed by André Weil and D. G. Northcott. Innovations around 1960 were the Néron-Tate height (q.v.) and the realisation that heights were linked to projective representations in much the same way that ample line bundles are in pure geometry.
- Hilbertian fields
- A Hilbertian field K is one for which the projective spaces over K are not thin sets in the sense of Jean-Pierre Serre. This is a geometric take on Hilbert's irreducibility theorem which shows the rational numbers are Hilbertian. Results are applied to the inverse Galois problem. Thin sets (the French word is mince) are in some sense analogous to the meagre sets (French maigre) of the Baire category theorem.
I
- Igusa zeta-function
- An Igusa zeta-function, named for Jun-ichi Igusa, is a generating function counting numbers of points on an algebraic variety modulo high powers pn of a fixed prime number p. General rationality theorems are now known, drawing on methods of mathematical logic.
- Infinite descent
- 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 variety J 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.
K
- K-theory
- Algebraic K-theory is on one hand a quite general theory with an abstract algebra flavour, and, on the other hand, implicated in some formulations of arithmetic conjectures. See for example Birch-Tate conjecture, Lichtenbaum conjecture.
L
- Local zeta-function
- A local zeta-function is a generating function for the number of points on an algebraic variety V over a finite field F, over the finite field extensions of F. According to the Weil conjectures (q.v.) these functions, for non-singular varieties, exhibit properties closely analogous to the Riemann zeta-function, including the Riemann hypothesis.
M
- Mordell conjecture
- The Mordell conjecture is now the Faltings theorem, and states that a curve of genus at least two has only finitely many rational points.
- Mordell-Lang conjecture
- The Mordell-Lang conjecture is a complex of a number of conjectures of Serge Lang unifying the Mordell conjecture and Manin-Mumford conjecture in an abelian variety or semi-abelian variety.
- Mordell-Weil theorem
- 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.
N
- Néron-Tate height
The Néron-Tate height (also often referred to as the canonical height) on an abelian variety A 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 field contributions.
Q
- Quasi-algebraic closure
- The topic of quasi-algebraic closure, i.e. solubility guaranteed by a number of variables polynomial in the degree of an equation, grew out of studies of the Brauer group and the Chevalley–Warning theorem. It stalled in the face of counterexamples; but see Ax-Kochen theorem from mathematical logic.
R
- Reduction modulo a prime number or ideal
- See good reduction.
S
- Sato-Tate conjecture
- The Sato-Tate conjecture on elliptic curves is a conjectural result on the distribution of Frobenius elements in the Tate module. It is a prototype for Galois representations in general.
- Skolem's method
- See Chabauty's method.
T
- Tamagawa numbers
- The direct Tamagawa number definition works well only for linear algebraic groups. There the Weil conjecture on Tamagawa numbers was eventually proved. For abelian varieties, and in particular the Birch-Swinnerton-Dyer conjecture (q.v.), the Tamagawa number approach to a local-global principle fails on a direct attempt, though it has had heuristic value over many years. Now a sophisticated equivariant Tamagawa number conjecture is a major research problem.
- Tate conjecture
- 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).
V
- Vojta conjecture
- The Vojta conjecture is a complex of conjectures by Paul Vojta, making analogies between Diophantine approximation and Nevanlinna theory.
W
- Weights
- The yoga of weights is a formulation by Alexander Grothendieck of analogies between Hodge theory and l-adic cohomology.[1]
- Weil cohomology
- The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a cohomology theory applying to algebraic varieties over finite fields that would both be as good as singular homology at detecting topological structure, and have Frobenius mappings acting in such a way that the Lefschetz fixed-point theorem could be applied to the counting in local zeta-functions. For later history see motive (algebraic geometry), motivic cohomology.
- Weil conjectures
- 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.
Notes
- ^ Pierre Deligne, Poids dans la cohomologie des variétés algébriques, Actes ICM, Vancouver, 1974, 79-85.
See also
Further reading