ℓ-adic Representations Associated to Modular Forms over Imaginary Quadratic Fields

Abstract

Let π be a regular algebraic cuspidal automorphic representation of GL 2 over an imaginary quadratic number field K, and let ℓ be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible ℓ-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explicit finite set.

$p$-adic families and Galois representations for $GS_p(4)$ and $GL(2)$

“…where ι : C ∼ = Q , and rec is the local Langlands correspondence for GSp (4). If, moreover, π p is assumed to be tempered or generic, then…”

Mass equidistribution for automorphic forms of cohomological type on 𝐺𝐿₂

“…Theorem 1 proves the equidistribution of a measure μ π on Y which may be associated to any cuspidal automorphic form π of cohomological type on Γ\SL(2, C). If φ ∈ π is a vector of minimal K-type, we may define μ π to be the pushforward of |φ| 2 Theorem 1 is proven by combining the following two results, which summarise the extensions of Holowinsky and Soundararajan's respective approaches to proving the equidistribution of F k . Their statements are almost identical to those of the original theorems over Q, which are recalled in section 4.1, with the only significant difference being that we must impose our assumption on the uniform growth of the weight in Theorem 4.…”

Proving modularity for a given elliptic curve over an imaginary quadratic field

“…[HST93], [Tay94] and [BH07]) have proved that one can attach compatible families of two-dimensional Galois representations {ρ } to any regular algebraic cuspidal automorphic representation π of GL 2 (A K ), assuming that it has unitary central character ω with ω = ω c , where the superscript c denotes the action of the nontrivial automorphism of K. This is equivalent to saying that the central character is the restriction of a character of G Q . As in the case of classical modular forms "to be attached" means that there is a correspondence between the ramification loci of π and the representation ρ and also that, at unramified places p, the characteristic polynomial of ρ (Frob p ) agrees with the Hecke polynomial of π at p. However, since the method for constructing these Galois representations depends on using a theta lift to link with automorphic forms on GSp 4 (A Q ), it cannot be excluded that the representations ρ also ramify at the primes that ramify in K/Q.…”

“…The precise statement of the result, valid only under the assumption ω = ω c , is the following (cf. [Tay94], [HST93] and [BH07]): …”

“…Furthermore, the L-function Euler factors agree with that of the modular form outside a set of density 0 of prime ideals. The result was improved in [BH07] to agreement outside an explicit finite set of prime ideals. This now allows us to prove modularity, by checking directly the finitely many "bad" places.…”