On the sign of regular algebraic polarizable automorphic representations

Abstract

We remove a parity condition from the construction of automorphic Galois representations carried out in the Paris Book Project. We subsequently generalize this construction to the case of 'mixed-parity' (but still regular essentially self-dual) automorphic representations over totally real fields, finding associated geometric projective representations. Finally, we optimize some of our previous results on finding geometric lifts, through central torus quotients, of geometric Galois representations, and apply them to the previous mixed-parity setting.

A (3,6)-connected layer with an unprecedented adeninate nucleobase-derived heptanuclear disc

“…As explained in the end of the introduction, Question 3.3.2 (formulated in a more general setting) is studied in [Con,Pat13] and [Pat14] and some answers have been provided (see Propositions 5.3, 6.5 in [Con], Theorem 3.2.10, Corollary 3.2.12 in [Pat13], Proposition 5.5 in [Pat14]). As explained in the end of the introduction, Question 3.3.2 (formulated in a more general setting) is studied in [Con,Pat13] and [Pat14] and some answers have been provided (see Propositions 5.3, 6.5 in [Con], Theorem 3.2.10, Corollary 3.2.12 in [Pat13], Proposition 5.5 in [Pat14]).…”

“…More precisely, the trick that the automorphy of representation ρ of GO 4 -type can be reduced to the automorphy of 2-dimensional representations via tensor product and Ramakrishnan's theorem was also known and used in [Cal] and [Pat13]. Questions 3.3.1 in Section 3.3 is formulated differently (see Question 3.3.2) and in a general setting in [Con,Pat13,Pat14] and some answers to these questions are provided. These answers almost cover results obtained in Section 3.3 (see Remark 3.3.3 for details).…”

On automorphy of certain Galois representations of GO4-type

“…As explained in the end of the introduction, Question 3.3.2 (formulated in a more general setting) is studied in [Con,Pat13] and [Pat14] and some answers have been provided (see Propositions 5.3, 6.5 in [Con], Theorem 3.2.10, Corollary 3.2.12 in [Pat13], Proposition 5.5 in [Pat14]). The aim of this section is to provide answers to Question 3.3.1 via an elementary and self-contained argument, though many of our results here have been covered by Conrad and Patrikis's results in a more general settings (Proposition 3.3.4 is in [Con], Corollary 3.3.8 and Theorem 3.3.9 are proved in [Pat14]). …”

Variations on a Theorem of Tate

“…18 For some unconditional results in this direction, see [Pat13,§4]. 75 By assumption, N L/F • res(χ) = 1, so res(χ) = δ −i ∈ Gal(L/F) D for some integer i, unique modulo d. Then res(χδ i ) = 1, and we are done by exactness of the horizontal diagram.…”