On icosahedral Artin representations, II

By Richard Taylor. 2003

110 citations recorded by the source corpus.

We prove that some new infinite families of odd two dimensional icosahedral representations of the absolute Galois group of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] are modular and hence satsify the Artin conjecture. We also give an account of work of Ramakrishna on lifting mod l Galois representations to characteristic zero.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/419ace9f164b55f7e474ec6ccbd0cd49f7dc895e

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Elliptic curves over real quadratic fields are modular (2013, methodology): “ularity switching argument sketched in the previous section. In fact, similar arguments have been used to prove special cases of Serre’s modularity conjecture for GL 2(Fq) for q= 5, 7 and 9 by Taylor [80], Manoharmayum [51], [52] and Ellenberg [27]. Our argument in this section is closest in spirit ”
Root numbers and parity of ranks of elliptic curves (2009, unknown): “The two propositions below are known and are only included for completeness (see Taylor’s [28] proof of Thm.”
A geometric perspective on the Breuil–Mézard conjecture (2011, background): “N+. The ﬁrsttwo ofthese conditionsconcernthe splitting behaviourof aﬁnite number of places of N+, so the existence of an extension satisfying all but the third condition is guaranteed by Lemma 2.2 of [Tay03]. In order to satisfy the third condition as well, for each of the (ﬁnitely many) Galois inte”
On the Meromorphic Continuation of Degree Two L-Functions (2006, methodology): “Combining theorem 2.1 of [Tay4], theorem 5.7, theorem 3.3 and a standard descent argument (see for example the proof of theorem 2.4 of [ Tay3 ]) we obtain our main theorem.”
Serre’s modularity conjecture (II) (2009, background): “Many cases of Langlands’ conjecture, and hence Artin’s conjecture, were known earlier in the case when the projective image is A5 (using a different strategy) thanks to the work in [5] and [44].”
Modularity of rigid Calabi-Yau threefolds over Q (2003, methodology): “ prime to ℓ), and • ρ|GF 1 is unramiﬁed at all ﬁnite places not dividing ℓ. Here, we have used the fact one can ﬁnd solvable extensions with prescribed local extensions (see for instance Lemma 2.2 of [16]). It is now immediate that the result we want follows immediately from a higherweight versionof”
On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q) (2009, unknown): “and Taylor ([52]).”