Lafforgue's theorem
Encyclopedia
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue
, completes the Langlands program
for general linear group
s over function field
s, by giving a correspondence between automorphic form
s on these groups and representations of Galois group
s.
The Langlands conjectures were introduced by and describe a correspondence between representations of the Weil group
of a function field
and representations of algebraic group
s over the function field, generalizing class field theory
of function fields from abelian Galois groups to non-abelian Galois groups.
. More precisely the Artin map gives a map from the idele class group to the abelianization of the Weil group.
Lafforgue's theorem states that there is a bijection σ between:
that preserves the L-function at every place of F.
The proof of Lafforgue's theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π. The idea of doing this is to look in the ℓ-adic cohomology of the moduli stack of shtukas of rank n that have compatible level N structures for all N. The cohomology contains subquotients of the form
which can be used to construct σ(π) from π. A major problem is that the moduli stack is not of finite type, which means that there are formidable technical difficulties in studying its cohomology.
Lafforgue's theorem implies the conjecture of that an irreducible finite-dimensional l-adic representation of the absolute Galois group with determinant character of finite order is pure of weight 0.
Laurent Lafforgue
Laurent Lafforgue is a French mathematician.He won 2 silver medals at International Mathematical Olympiad in 1984 and 1985....
, completes the Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
for general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
s over function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
s, by giving a correspondence between automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s on these groups and representations of Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s.
The Langlands conjectures were introduced by and describe a correspondence between representations of the Weil group
Weil group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is a profinite group generally denoted WF...
of a function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
and representations of algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s over the function field, generalizing class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
of function fields from abelian Galois groups to non-abelian Galois groups.
Langlands conjectures for GL1
The Langlands conjectures for GL1(K) follow from (and are essentially equivalent to) class field theoryClass field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
. More precisely the Artin map gives a map from the idele class group to the abelianization of the Weil group.
Automorphic representations of GLn(F)
The representations of GLn(F) appearing in the Langlands correspondence are automrphic representations.Lafforgue's theorem for GLn(F)
Here F is a global field of some positive characteristic F, and ℓ is some prime not equal to p.Lafforgue's theorem states that there is a bijection σ between:
- Equivalence classes of cuspidal representations π of GLn(F), and
- Equivalence classes of irreducible ℓ-adic representations σ(π) of dimension n of the absolute Galois group of F
that preserves the L-function at every place of F.
The proof of Lafforgue's theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π. The idea of doing this is to look in the ℓ-adic cohomology of the moduli stack of shtukas of rank n that have compatible level N structures for all N. The cohomology contains subquotients of the form
- π⊗σ(π)⊗σ(π)∨
which can be used to construct σ(π) from π. A major problem is that the moduli stack is not of finite type, which means that there are formidable technical difficulties in studying its cohomology.
Applications
Lafforgue's theorem implies the Ramanujan–Petersson conjecture that if an automorphic form for GLn(F) has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1.Lafforgue's theorem implies the conjecture of that an irreducible finite-dimensional l-adic representation of the absolute Galois group with determinant character of finite order is pure of weight 0.