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.
An Arakelov divisor (or replete divisor[4]) 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.[3][5][6]
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.[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[12]d(P) and the arithmetic discriminant, defined by Vojta.[13] 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.[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]
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.
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.
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
HardyâLittlewood 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.
Infinite descent was
Pierre de Fermat's classical method for Diophantine equations. It became one half of the standard proof of the MordellâWeil 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 FâČ The local zeta-function (q.v.) of C can be recovered from the points J(FâČ) as Galois module. In the same way, Iwasawa added pn-power roots of unity for fixed p and with n â â, 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.
Enrico Bombieri (dimension 2),
Serge Lang and
Paul Vojta (integral points case) 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.[19]
Linear torus
A linear torus is a geometrically irreducible Zariski-closed subgroup of an affine torus (product of multiplicative groups).[20]
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
MordellâWeil 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.[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]
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.[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]
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.[34] A replete divisor is an
Arakelov divisor.[4]
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 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.[39]
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 BirchâSwinnerton-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,[40] 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 > ÎŁ 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.[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
BombieriâLang 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
MordellâLang conjecture.[43]
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]
^Neukirch, JĂŒrgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.).
Springer-Verlag. p. 361.
ISBN978-3-540-37888-4.
^Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer.
ISBN0-387-96311-1. â Contains an English translation of Faltings (1983)
^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 93â110 (1965).
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.
An Arakelov divisor (or replete divisor[4]) 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.[3][5][6]
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.[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[12]d(P) and the arithmetic discriminant, defined by Vojta.[13] 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.[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]
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.
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.
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
HardyâLittlewood 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.
Infinite descent was
Pierre de Fermat's classical method for Diophantine equations. It became one half of the standard proof of the MordellâWeil 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 FâČ The local zeta-function (q.v.) of C can be recovered from the points J(FâČ) as Galois module. In the same way, Iwasawa added pn-power roots of unity for fixed p and with n â â, 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.
Enrico Bombieri (dimension 2),
Serge Lang and
Paul Vojta (integral points case) 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.[19]
Linear torus
A linear torus is a geometrically irreducible Zariski-closed subgroup of an affine torus (product of multiplicative groups).[20]
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
MordellâWeil 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.[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]
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.[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]
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.[34] A replete divisor is an
Arakelov divisor.[4]
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 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.[39]
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 BirchâSwinnerton-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,[40] 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 > ÎŁ 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.[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
BombieriâLang 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
MordellâLang conjecture.[43]
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]
^Neukirch, JĂŒrgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.).
Springer-Verlag. p. 361.
ISBN978-3-540-37888-4.
^Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer.
ISBN0-387-96311-1. â Contains an English translation of Faltings (1983)
^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 93â110 (1965).