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 coordinates 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.
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 or, more generally, prime ideals. 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 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]
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 2016. The initial result of this type was Eisenstein's 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.[18]
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.
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 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 contributions.[29]
Nevanlinna invariant
The Nevanlinna invariant of an ample divisor D on a normal projective variety X 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]
R
Reduction modulo a prime number or ideal
See good reduction.
Replete ideal
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]
S
Sato–Tate conjecture
The Sato–Tate conjecture describes the distribution of Frobenius elements in the Tate modules of the elliptic curves over finite fields obtained from reducing a given elliptic curve over the rationals. Mikio Sato and, independently, John Tate[34] suggested it around 1960. It is a prototype for Galois representations in general.
Skolem's method
See Chabauty's method.
Special set
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;[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]
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).
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]
W
Weights
The yoga of weights is a formulation by Alexander Grothendieck of analogies between Hodge theory and l-adic cohomology.[43]
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 coordinates of points on algebraic varieties. It has remained somewhat under-developed.
Weil function
A Weil function on an algebraic variety is a real-valued function defined off some Cartier divisor which generalises the concept of Green's function in Arakelov theory.[44] They are used in the construction of the local components of the Néron–Tate height.[45]
Weil height machine
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]
References
- 1 2 Schoof, René (2008). "Computing Arakelov class groups". In Buhler, J.P.; P., Stevenhagen. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications. 44. Cambridge University Press. pp. 447–495. ISBN 978-0-521-20833-8. MR 2467554. Zbl 1188.11076.
- 1 2 Neukirch (1999) p.189
- ↑ Lang (1988) pp.74–75
- ↑ 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. Zbl 1030.11063.
- ↑ Bombieri & Gubler (2006) pp.66–67
- ↑ Lang (1988) pp.156–157
- ↑ Lang (1997) pp.91–96
- 1 2 Hindry & Silverman (2000) p.479
- ↑ Coates, J.; Wiles, A. (1977). "On the conjecture of Birch and Swinnerton-Dyer". Inventiones Mathematicae. 39 (3): 223–251. Bibcode:1977InMat..39..223C. doi:10.1007/BF01402975. Zbl 0359.14009.
- ↑ Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.
- ↑ Lang (1997) p.146
- 1 2 3 Lang (1997) p.171
- ↑ Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae. 73 (3): 349–366. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432.
- ↑ Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer. ISBN 0-387-96311-1.
→ Contains an English translation of Faltings (1983)
- ↑ Serre, Jean-Pierre; Tate, John (November 1968). "Good reduction of abelian varieties". The Annals of Mathematics. Second. 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722. Zbl 0172.46101.
- ↑ Lang (1997) pp.43–67
- ↑ Bombieri & Gubler (2006) pp.15–21
- ↑ Igusa, Jun-Ichi (1974). "Complex powers and asymptotic expansions. I. Functions of certain types". Journal für die reine und angewandte Mathematik. 1974 (268–269): 110–130. doi:10.1515/crll.1974.268-269.110. Zbl 0287.43007.
- ↑ Bombieri & Gubler (2006) pp.82–93
- ↑ Raynaud, Michel (1983). "Sous-variétés d'une variété abélienne et points de torsion". In Artin, Michael; Tate, John. Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic. Progress in Mathematics (in French). 35. Birkhauser-Boston. pp. 327–352. Zbl 0581.14031.
- ↑ Roessler, Damian (2005). "A note on the Manin–Mumford conjecture". In van der Geer, Gerard; Moonen, Ben; Schoof, René. Number fields and function fields — two parallel worlds. Progress in Mathematics. 239. Birkhäuser. pp. 311–318. ISBN 0-8176-4397-4. Zbl 1098.14030.
- ↑ Marcja, Annalisa; Toffalori, Carlo (2003). A Guide to Classical and Modern Model Theory. Trends in Logic. 19. Springer-Verlag. pp. 305–306. ISBN 1402013302.
- ↑ 2 page exposition of the Mordell–Lang conjecture by B. Mazur, 3 Nov. 2005
- ↑ Lang (1997) p.15
- ↑ Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. 9. Cambridge University Press. p. 3. ISBN 978-0-521-88268-2. Zbl 1145.11004.
- ↑ Bombieri & Gubler (2006) pp.301–314
- ↑ Lang (1988) pp.66–69
- ↑ Lang (1997) p.212
- 1 2 Lang (1988) p.77
- ↑ Hindry & Silverman (2000) p.488
- ↑ Batyrev, V.V.; Manin, Yu.I. (1990). "On the number of rational points of bounded height on algebraic varieties". Math. Ann. 286: 27–43. doi:10.1007/bf01453564. Zbl 0679.14008.
- ↑ Lang (1997) pp.161–162
- ↑ Neukirch (1999) p.185
- ↑ 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).
- ↑ Lang (1997) pp.17–23
- ↑ Hindry & Silverman (2000) p.480
- ↑ Lang (1997) p.179
- ↑ Bombieri & Gubler (2006) pp.176–230
- ↑ Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper". J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803.
- ↑ Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126. ISBN 978-0-387-72487-4.
- ↑ Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). "Uniformity of rational points". Journal of the American Mathematical Society. 10 (1): 1–35. doi:10.2307/2152901. Zbl 0872.14017.
- ↑ Zannier, Umberto (2012). Some Problems of Unlikely Intersections in Arithmetic and Geometry. Annals of Mathematics Studies. 181. Princeton University Press. ISBN 978-0-691-15371-1.
- ↑ Pierre Deligne, Poids dans la cohomologie des variétés algébriques, Actes ICM, Vancouver, 1974, 79–85.
- ↑ Lang (1988) pp.1–9
- ↑ Lang (1997) pp.164,212
- ↑ Hindry & Silverman (2000) 184–185
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