In mathematics, the Arthur conjectures are some conjectures about

automorphic representation In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

s of

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

s over the

adele Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a rec ...

s and

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...

s of reductive groups over

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

s made by , motivated by the

Arthur–Selberg trace formula In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur (mathematician), James Arthur in a long series of pape ...

. Arthur's conjectures imply the generalized Ramanujan conjectures for cusp forms on general linear groups.

References

* * * Automorphic forms Representation theory Conjectures {{Algebra-stub