In
mathematics, the André–Oort conjecture is a problem in
Diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
, a branch of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, that can be seen as a non-abelian analogue of the
Manin–Mumford conjecture
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has b ...
, which is now a theorem (and is actually proven in several genuinely different ways).
The conjecture concerns itself with a characterization of the Zariski closure of sets of special points in
Shimura varieties.
A special case of the conjecture was stated by
Yves André in 1989 and a more general statement (albeit with a restriction on the type of the Shimura variety) was conjectured by
Frans Oort in 1995. The modern version is a natural generalization of these two conjectures.
Statement
The conjecture in its modern form is as follows. Each irreducible component of the
Zariski closure
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
of a set of special points in a
Shimura variety In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are no ...
is a special subvariety.
André's first version of the conjecture was just for one dimensional irreducible components, while Oort proposed that it should be true for irreducible components of arbitrary dimension in the moduli space of
principally polarised Abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
of dimension ''g''.
It seems that André was motivated by applications to transcendence theory while Oort by the analogy with the Manin-Mumford
conjecture.
Partial results
Various results have been established towards the full conjecture by Ben Moonen,
Yves André,
Andrei Yafaev,
Bas Edixhoven
Sebastiaan Johan Edixhoven (12 March 1962 – 16 January 2022) was a Dutch mathematician who worked in arithmetic geometry. He was a professor at University of Rennes 1 and Leiden University.
Education
Bas Edixhoven was born on 12 March 1962 ...
,
Laurent Clozel
Laurent Clozel (born October 23, 1953 in Gap) is a French mathematician. His mathematical work is in the area of automorphic forms, including major advances on the Langlands programme
Career and distinctions
Clozel was a student at the Écol ...
,
Bruno Klingler and
Emmanuel Ullmo
Emmanuel Ullmo (born 25 June 1965) is a French mathematician, specialised in arithmetic geometry. Since 2013 he has served as director of the Institut des Hautes Études scientifiques.
Biography
He wrote his thesis under Lucien Szpiro at the ...
, among others. Some of these results were conditional upon the
generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-function, ''L''-func ...
(GRH) being true.
In fact, the proof of the full conjecture under GRH was published by Bruno Klingler, Emmanuel Ullmo and Andrei Yafaev in 2014 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 t ...
.
In 2006,
Umberto Zannier
Umberto Zannier (born 25 May 1957, in Spilimbergo, Italy) is an Italian mathematician, specializing in number theory and Diophantine geometry.
Education
Zannier earned a Laurea degree from University of Pisa and studied at the Scuola Normale Su ...
and
Jonathan Pila
Jonathan Solomon Pila (born 1962) FRS One or more of the preceding sentences incorporates text from the royalsociety.org website where: is an Australian mathematician at the University of Oxford.
Education
Pila earned his bachelor's degree at ...
used techniques from
o-minimal geometry and
transcendental number theory
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendence
...
to develop an approach to the Manin-Mumford-André-Oort type of problems.
In 2009, Jonathan Pila proved the André-Oort conjecture unconditionally for arbitrary products of
modular curve
In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
s, a result which earned him the 2011
Clay Research Award __NOTOC__
The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research. The following mathematicians have received the award:
{, class=" ...
.
Bruno Klingler,
Emmanuel Ullmo
Emmanuel Ullmo (born 25 June 1965) is a French mathematician, specialised in arithmetic geometry. Since 2013 he has served as director of the Institut des Hautes Études scientifiques.
Biography
He wrote his thesis under Lucien Szpiro at the ...
and
Andrei Yafaev proved, in 2014, the functional transcendence result needed for the general Pila-Zannier approach and Emmanuel Ullmo has deduced from it a technical result needed for the induction step in the strategy. The remaining technical ingredient was the problem of bounding below the Galois degrees of special points.
For the case of the
Siegel modular variety
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally pola ...
, this bound was deduced by
Jacob Tsimerman in 2015 from the
averaged Colmez conjecture and the
Masser-Wustholtz isogeny estimates. The averaged Colmez conjecture was proved by
Xinyi Yuan and
Shou-Wu Zhang and independently by Andreatta, Goren, Howard and Madapusi-Pera.
In 2019-2020, Gal Biniyamini, Harry Schmidt and Andrei Yafaev, building on previous work and ideas of Harry Schmidt on torsion points in tori and abelian varieties and Gal Biniyamini's point counting results, have formulated a conjecture on bounds of heights of special points and deduced from its validity the bounds for the Galois degrees of special points needed for the proof of the full André-Oort conjecture.
In September 2021,
Jonathan Pila
Jonathan Solomon Pila (born 1962) FRS One or more of the preceding sentences incorporates text from the royalsociety.org website where: is an Australian mathematician at the University of Oxford.
Education
Pila earned his bachelor's degree at ...
, Ananth Shankar, and
Jacob Tsimerman claimed in a paper (featuring an appendix written by
Hélène Esnault and Michael Groechenig) a proof of the Biniyamini-Schmidt-Yafaev height conjecture, thus completing the proof of the André-Oort conjecture using the Pila-Zannier strategy.
Coleman–Oort conjecture
A related conjecture that has two forms, equivalent if the André–Oort conjecture is assumed, is the Coleman–Oort conjecture.
Robert Coleman conjectured that for sufficiently large ''g'', there are only finitely many smooth projective curves ''C'' of genus ''g'', such that the
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 var ...
''J''(''C'') is an
abelian variety of CM-type In mathematics, an abelian variety ''A'' defined over a field ''K'' is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(''A''). The terminology here is from complex multiplication theory, which was deve ...
. Oort then conjectured that the
Torelli locus – of the
moduli space of abelian varieties of dimension g – has for sufficiently large ''g'' no special subvariety of dimension > 0 that intersects the image of the
Torelli mapping in a dense open subset.
Generalizations
Manin-Mumford and André–Oort conjectures can be generalized in many directions, for example by relaxing the
properties of points being `special' (and considering the so-called `unlikely locus' instead) or looking at more general ambient varieties: abelian or semi-abelian schemes, mixed Shimura varieties etc.... These
generalizations are colloquially known as the
Zilber–Pink conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André–Oort, Manin–Mumford, and Mordell–Lang. For algebraic tori and semiabelian varieties it wa ...
s because problems of this type were proposed by Richard Pink and
Boris Zilber
Boris Zilber (russian: Борис Иосифович Зильбер, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a professor of mathematical logic at the University of Oxford.
H ...
.
[.]
Most of these questions are open and are a subject of current active research.
See also
*
Zilber–Pink conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André–Oort, Manin–Mumford, and Mordell–Lang. For algebraic tori and semiabelian varieties it wa ...
References
Further reading
*
*
{{DEFAULTSORT:Andre-Oort Conjecture
Diophantine geometry
Conjectures