Automorphic Forms and Galois Representations: The conjectural connections between automorphic representations and Galois representations

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We state conjectures on the relationships between automorphic representations and Galois representations, and give evidence for them.

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Families of L -Functions and Their Symmetry (2014, background): “algebraicity. For arithmetic considerations it is customary to twist Π by the 1−n 2 -th power of the modulus character in the conjecture. If so, one should replace “L-algebraic” by “C-algebraic”, cf. [20]. It is worth noting that Conjecture 4 suﬃces for our purpose in discussing geometric families. ”
Cohomology of local systems on the moduli of principally polarized abelian surfaces (2013, methodology): “As stated this conjecture is however false, and making the conjecture precise is a rather delicate matter, but see [Buzzard and Gee 2010].”
Ordinary representations of G(Q_p) and fundamental algebraic representations (2012, unknown): “Following [BG12], we define a twisting element (for G) as an element θ ∈ X(T ) such that for any α ∈ S one has 〈θ, α∨〉 = 1.”
The asymptotic growth of torsion homology for arithmetic groups (2010, background): “For a detailed treatment we refer the reader to the paper [14].”
Coherent sheaves on the stack of Langlands parameters (2020, methodology): “First recall the C-group of G introduced by Buzzard-Gee [BG11], following the construction in [Zhu, §1.”
A torsion Jacquet--Langlands correspondence (2012, background): “For a more precise formulation (including what is meant by “de Rham” for targets besides GLn) the reader should consult [13].”