# 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

*a*+

*b*=

*c*. For example 3 + 125 = 128 but the prime powers here are exceptional.

*Arakelov class group*is the analogue of the ideal class group or divisor class group for Arakelov divisors.

^{[1]}

*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]}

*Arakelov height*on a projective space over the field of algebraic numbers is a global height function with local contributions coming from Fubini–Study metrics on the Archimedean fields and the usual metric on the non-Archimedean fields.

^{[5]}

^{[6]}

*See main article arithmetic of abelian varieties*

## B

*good reduction*.

^{[7]}

*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 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.

^{[8]}

## C

**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.)

*Coates–Wiles theorem*states that an elliptic curve with complex multiplication by an imaginary quadratic field of 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.

^{[9]}

*p*coefficients in this case. It is one of a number of theories deriving in some way from Dwork's method, and has applications outside purely arithmetical questions.

## D

*Diophantine dimension*of a field is the smallest natural number

*k*, if it exists, such that the field of is class C

_{k}: that is, such that any homogeneous polynomial of degree

*d*in

*N*variables has a non-trivial zero whenever

*N*>

*d*

^{k}. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.

^{[10]}

*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]}

## E

## F

*Faltings height*of an elliptic curve or abelian variety defined over a number field is a measure of its complexity introduced by Faltings in his proof of the Mordell conjecture.

^{[13]}

^{[14]}

## G

*geometric*class field theory.

*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

*V*

_{p}over

**Z**/

*p*

**Z**. 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]}

## H

*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.

^{[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.

*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

*p*

^{n}of a fixed prime number

*p*. General rationality theorems are now known, drawing on methods of mathematical logic.

^{[18]}

*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

*p*

^{n}-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

## L

*linear torus*is a geometrically irreducible Zariski-closed subgroup of an affine torus (product of multiplicative groups).

^{[19]}

*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

*C*in its Jacobian variety

*J*can only contain a finite number of points that are of finite order in

*J*, unless

*C*=

*J*.

^{[20]}

^{[21]}

^{[22]}

^{[23]}

*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.

^{[24]}

## N

*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*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]}

*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*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

*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]}

## Q

## R

*modulo*a prime number or ideal

*good reduction*.

*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

^{[34]}suggested it around 1960. It is a prototype for Galois representations in general.

*Chabauty's method*.

*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**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

*good reduction*).

^{[39]}is the smallest natural number

*i*, if it exists, such that the field is of class T

_{i}: that is, such that any system of polynomials with no constant term of degree

*d*in

_{j}*n*variables has a non-trivial zero whenever

*n*> ∑

*d*

_{j}

^{i}. 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*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*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]}

## V

## W

^{[43]}

*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*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]}

## See also

## References

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*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. 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
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- ↑ Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".
*Inventiones Mathematicae*.**73**(3): 349–366. 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 (11968). "Good reduction of abelian varieties".
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(help) - ↑ Lang (1997) pp.43–67
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*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
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- ↑ 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

- Bombieri, Enrico; Gubler, Walter (2006).
*Heights in Diophantine Geometry*. New Mathematical Monographs.**4**. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034. - Hindry, Marc; Silverman, Joseph H. (2000).
*Diophantine Geometry: An Introduction*. Graduate Texts in Mathematics.**201**. ISBN 0-387-98981-1. Zbl 0948.11023. - Lang, Serge (1988).
*Introduction to Arakelov theory*. New York: Springer-Verlag. ISBN 0-387-96793-1. MR 0969124. Zbl 0667.14001. - Lang, Serge (1997).
*Survey of Diophantine Geometry*. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051. - Neukirch, Jürgen (1999).
*Algebraic Number Theory*. Grundlehren der Mathematischen Wissenschaften.**322**. Springer-Verlag. ISBN 978-3-540-65399-8. Zbl 0956.11021.

## Further reading

- Dino Lorenzini (1996), An invitation to arithmetic geometry, AMS Bookstore, ISBN 978-0-8218-0267-0