In
mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional
Galois representation
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by
Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and
Jean-Pierre Wintenberger in 2008.
Formulation
The conjecture concerns the
absolute Galois group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' t ...
of the
rational number field
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
.
Let
be an
absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the intege ...
, continuous, two-dimensional representation of
over a finite field
.
:
Additionally, assume
is odd, meaning the image of complex conjugation has determinant -1.
To any normalized
modular eigenform
:
of
level ,
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
, and some
Nebentype character
:
,
a theorem due to Shimura, Deligne, and Serre-Deligne attaches to
a representation
:
where
is the ring of integers in a finite extension of
. This representation is characterized by the condition that for all prime numbers
,
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to
we have
:
and
:
Reducing this representation modulo the maximal ideal of
gives a mod
representation
of
.
Serre's conjecture asserts that for any representation
as above, there is a modular eigenform
such that
:
.
The level and weight of the conjectural form
are explicitly conjectured in Serre's article. In addition, he derives a number of results from this conjecture, among them
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
and the now-proven Taniyama–Weil (or Taniyama–Shimura) conjecture, now known as the
modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
(although this implies Fermat's Last Theorem, Serre proves it directly from his conjecture).
Optimal level and weight
The strong form of Serre's conjecture describes the level and weight of the modular form.
The optimal level is the
Artin conductor In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function.
Local Artin conductors
...
of the representation, with the power of
removed.
Proof
A proof of the level 1 and small weight cases of the conjecture was obtained in 2004 by
Chandrashekhar Khare and
Jean-Pierre Wintenberger, and by
Luis Dieulefait
Luis is a given name. It is the Spanish form of the originally Germanic name or . Other Iberian Romance languages have comparable forms: (with an accent mark on the i) in Portuguese and Galician, in Aragonese and Catalan, while is archai ...
, independently.
In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture, and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.
[ and .]
Notes
References
*
*
*{{Citation , last1=Stein , first1=William A. , last2=Ribet , first2=Kenneth A. , editor1-last=Conrad , editor1-first=Brian , editor2-last=Rubin , editor2-first=Karl , title=Arithmetic algebraic geometry (Park City, UT, 1999) , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, R.I. , series=IAS/Park City Math. Ser. , isbn=978-0-8218-2173-2 , mr=1860042 , year=2001 , volume=9 , chapter=Lectures on Serre's conjectures , pages=143–232
See also
*
Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Ferma ...
External links
Serre's Modularity Conjecture50 minute lecture by
Ken Ribet
Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Ferma ...
given on October 25, 2007
slidesPDF
other version of slidesPDF)
Lectures on Serre's conjectures
Modular forms
Theorems in number theory
Conjectures that have been proved