Glossary of arithmetic and diophantine geometry

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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.

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.

Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers.[1] Arithmetic geometry has also been defined as the application of the techniques of algebraic geometry to problems in number theory.[2]

See also the glossary of number theory terms at Glossary of number theory.


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 class group
The Arakelov class group is the analogue of the
Arakelov divisors.[3]
Arakelov divisor
An Arakelov divisor (or replete divisor
infinite places having real coefficients.[3][5][6]
Arakelov height
The
non-Archimedean fields.[7][8]
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
l-adic cohomology
groups.

B

Bad reduction
See good reduction.
Birch and Swinnerton-Dyer conjecture
The
Kolyvagin's theorem.[9]

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
Runge's method
.)
Coates–Wiles theorem
The Coates–Wiles theorem states that an
class number 1 and positive rank has L-function with a zero at s = 1. This is a special case of the Birch and Swinnerton-Dyer conjecture.[10]
Crystalline cohomology
Dwork's method
, and has applications outside purely arithmetical questions.

D

Diagonal forms
local zeta-functions are computed in terms of Jacobi sums. Waring's problem
is the most classical case.
Diophantine dimension
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.[11]
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
desingularisation.[13] 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.[13]
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. He first proved the rationality of local zeta-functions, the initial advance in the direction of the Weil conjectures.

E

Étale cohomology
The search for a Weil cohomology (q.v.) was at least partially fulfilled in the
local zeta-functions
, and was basic in the formulation of the Tate conjecture (q.v.) and numerous other theories.

F

Faltings height
The
Mordell conjecture.[14][15]
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
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
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
abelian coverings
to varieties of dimension at least two is often called geometric class field theory.
Good reduction
Fundamental to
semistable elliptic curve, Serre–Tate theorem.[16]
Grothendieck–Katz conjecture
The Grothendieck–Katz p-curvature conjecture applies reduction modulo primes to algebraic differential equations, to derive information on algebraic function solutions. The initial result of this type was Eisenstein's theorem.

H

Hasse principle
The
cubic curves
) are at a general level connected with the limitations of the analytic approach.
Hasse–Weil L-function
A
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.[17]
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
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
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
Lichtenbaum conjecture
.

L

Lang conjecture
compact Riemann surfaces of genus g > 1. Lang conjectured that V is analytically hyperbolic if and only if all subvarieties are of general type.[19]
Linear torus
A linear torus is a geometrically irreducible Zariski-closed subgroup of an affine torus (product of multiplicative groups).[20]
Local zeta-function
A
Riemann zeta-function, including the Riemann hypothesis
.

M

Manin–Mumford conjecture
The
Manin–Mumford conjecture, now proved by Michel Raynaud, states that a curve C in its Jacobian variety J can only contain a finite number of points that are of finite order in J, unless C = J.[21][22]
Mordell conjecture
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.
Mordell–Lang conjecture
The Mordell–Lang conjecture, now proved by
semiabelian variety.[23][24]
Mordell–Weil theorem
The
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.[25]

N

Naive height
The naive height 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.[26]
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.[27][28][29] The global Néron symbol, which is the sum of the local symbols, is just the negative of the height pairing.[30]
Néron–Tate height
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.[30]
Nevanlinna invariant
The
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.[31] 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.[32]

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.[33]

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.
Replete ideal
A replete ideal in a number field K is a formal product of a
Arakelov divisor.[4]

S

Sato–Tate conjecture
The
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[35] 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;[36] another definition is the union of all subvarieties that are not of general type.[19] For abelian varieties the definition would be the union of all translates of proper abelian subvarieties.[37] 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.[38]
Subspace theorem
Schmidt's
S-unit equation.[39]

T

Tamagawa numbers
The direct
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
Diophantine dimension but it is not known if they are equal except in the case of rank zero.[41]

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.[42]
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.[43]

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
l-adic cohomology.[44]
Weil cohomology
The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a
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 Algebraic geometry 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.[45] They are used in the construction of the local components of the Néron–Tate height.[46]
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).[47]

See also

References

  1. ^ Arithmetic geometry at the nLab
  2. ^ Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 22 March 2019.
  3. ^
    Zbl 1188.11076
    .
  4. ^ a b Neukirch (1999) p.189
  5. ^ Lang (1988) pp.74–75
  6. Zbl 1030.11063
    .
  7. ^ Bombieri & Gubler (2006) pp.66–67
  8. ^ Lang (1988) pp.156–157
  9. ^ Lang (1997) pp.91–96
  10. Zbl 0359.14009
    .
  11. ISBN 978-3-540-37888-4
    .
  12. ^ Lang (1997) p.146
  13. ^ a b c Lang (1997) p.171
  14. S2CID 121049418
    .
  15. ISBN 0-387-96311-1
    . → Contains an English translation of Faltings (1983)
  16. Zbl 0172.46101
    .
  17. ^ Lang (1997)
  18. Zbl 0287.43007
    .
  19. ^ a b Hindry & Silverman (2000) p.479
  20. ^ Bombieri & Gubler (2006) pp.82–93
  21. Zbl 0581.14031
    .
  22. Zbl 1098.14030
    .
  23. S2CID 120053132
    .
  24. ^ 2 page exposition of the Mordell–Lang conjecture by B. Mazur, 3 Nov. 2005
  25. ^ Lang (1997) p.15
  26. Zbl 1145.11004
    .
  27. ^ Bombieri & Gubler (2006) pp.301–314
  28. ^ Lang (1988) pp.66–69
  29. ^ Lang (1997) p.212
  30. ^ a b Lang (1988) p.77
  31. ^ Hindry & Silverman (2000) p.488
  32. Zbl 0679.14008
    .
  33. ^ Lang (1997) pp.161–162
  34. ^ Neukirch (1999) p.185
  35. ^ 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).
  36. ^ Lang (1997) pp.17–23
  37. ^ Hindry & Silverman (2000) p.480
  38. ^ Lang (1997) p.179
  39. ^ Bombieri & Gubler (2006) pp.176–230
  40. Zbl 0015.38803
    .
  41. ISBN 978-0-387-72487-4
    .
  42. Zbl 0872.14017
    .
  43. ISBN 978-0-691-15371-1
    .
  44. ^ Pierre Deligne, Poids dans la cohomologie des variétés algébriques, Actes ICM, Vancouver, 1974, 79–85.
  45. ^ Lang (1988) pp.1–9
  46. ^ Lang (1997) pp.164,212
  47. ^ Hindry & Silverman (2000) 184–185

Further reading
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