arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, alg ...

, the uniform boundedness conjecture for rational points asserts that for a given

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

K

and a positive integer

g \geq 2

that there exists a number

N(K,g)

depending only on

K

and

g

such that for any

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

C

defined over

K

having

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

equal to

g

has at most

N(K,g)

K

rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...

s. This is a refinement of

Faltings's theorem In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of Genus (mathematics), genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by ...

, which asserts that the set of

K

-rational points

C(K)

is necessarily finite.

Progress

The first significant progress towards the conjecture was due to Caporaso,

Harris Harris may refer to: Places Canada * Harris, Ontario * Northland Pyrite Mine (also known as Harris Mine) * Harris, Saskatchewan * Rural Municipality of Harris No. 316, Saskatchewan Scotland * Harris, Outer Hebrides (sometimes called the Isle o ...

, and Mazur. They proved that the conjecture holds if one assumes the

Bombieri–Lang conjecture In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. Statement The weak Bombie ...

Mazur's Conjecture B

A variant of the conjecture, due to Mazur, asserts that there should be a number

N(K,g,r)

such that for any algebraic curve

C

defined over

K

having genus

g

and whose

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...

J_C

has Mordell–Weil rank over

K

equal to

r

, the number of

K

-rational points of

C

is at most

N(K,g,r)

. This variant of the conjecture is known as Mazur's Conjecture B. Michael Stoll proved that Mazur's Conjecture B holds for

hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...

s with the additional hypothesis that

r \leq g - 3

. Stoll's result was further refined by

Katz Katz or KATZ may refer to: Fiction * Katz Kobayashi, a character in Japanese anime * "Katz", a 1947 Nelson Algren story in '' The Neon Wilderness'' * Katz, a character in ''Courage the Cowardly Dog'' Other uses * Katz (surname) * Katz, British C ...

, Rabinoff, and Zureick-Brown in 2015. Both of these works rely on

Chabauty's method 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 ...

. Mazur's Conjecture B was resolved by Dimitrov,

Gao Gao , or Gawgaw/Kawkaw, is a city in Mali and the capital of the Gao Region. The city is located on the River Niger, east-southeast of Timbuktu on the left bank at the junction with the Tilemsi valley. For much of its history Gao was an impor ...

, and Habegger in a preprint in 2020 which has since appeared in the

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...

using the earlier work of Gao and Habegger on the geometric

Bogomolov conjecture In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proved by Emmanuel Ullmo ...

instead of Chabauty's method.

References

{{Reflist Conjectures Arithmetic geometry