Charles Lamb and William Ireland

Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

“…When the first draft of this paper was written, our theorems about modularity of mod 5 representations of Gal(F /F ) came with conditions at 5; they also appeared in our main theorem about the strong Artin conjecture. At that time, [3] was not written up, and, in order to establish modularity of the mod 5 representation r of the absolute Galois group of a totally real field L with the image of proj r being A 5 (e.g. ρ L in the proof of the lemma above), it was necessary to assume either r is distinguished (with a view to making appeal to Ramakrishna/Taylor lifting argument), or the kernel of proj r does not fix L(ζ 5 ) (with a view to using Khare-Wintenberger 'finiteness of deformation rings' argument to lift r to a characteristic zero lifting that is modular).…”

“…• ρ E,5 : G M = Gal(Q/M ) → Aut(E [5]) is equivalent to a twist of ρ| G M by some character, • ρ E,3 : Gal(Q/M (ζ 3 )) → Aut(E [3]) is absolutely irreducible, • E has good ordinary reduction at every place of M above 3, and potentially good ordinary reduction at every place of M above 5.…”

“…By the Langlands-Tunnell theorem, there exists a weight 1 cuspidal Hilbert eigenform f 1 which gives rise to ρ E,3 . By 3-adic Hida theory, we may find a cuspidal Hilbert eigenform f 2 of weight 2 and of level prime to 3, ordinary at every prime of M above 3, which gives rise to ρ E, 3 . As E is ordinary at 3, f 2 renders ρ E,3 : G M → GL(T 3 E) 'strongly residually modular' in the sense of Kisin [23], and it follow from Theorem 3.5.5 in [23] that T 3 E is modular.…”

“…β is connected and lies entirely inside R Σ . By assumptions(2,3) in Definition 6.5, sp −1 (Q) ∩ R Σ is connected and deg β takes values arbitrarily close to 1 on it.Since sp −1 (Q) ⊂ sp −1 (V 0 ), it follows that sp −1 (Q) ∩ R Σ intersects U 0 [ǫ, 1] β ,and hence Q lies in C, the connected component of R Σ containing U 0 [ǫ, 1] β which clearly intersects sp −1 (W ϕ,η−{σ•β} ). The other statement in part (3) of the lemma can be proved in exactly the same way.…”

On the Distribution of Hecke Eigenvalues for Cuspidal Automorphic Representations for GL(2)

“…Using the known bounds towards the Ramanujan conjecture, we get Remark 2. One might ask why we do not also make use of L(s, π, Sym 9 ), given that we know (by Kim and Shahidi [7]) that for a cuspidal automorphic representation for GL(2) over a number field that is not of solvable polyhedral type, the L-function L(s, π, Sym 9 ) is holomorphic in the interval (1,2], and at s = 1 the L-function either has a simple pole, is invertible, or has a simple zero. The issue is that we do not know of the absolute convergence of the Euler product for Re(s) > 1 (it is only known for Re(s) > 2), which is required by this method.…”