Galois representations

By Richard Taylor. 2002

139 citations recorded by the source corpus.

In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures.

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Citations in the corpus

On a theorem of Jordan (2003, methodology): “To go further, one needs a result of Tate (reproduced in [17],[12],[5]) which says that the field L has a quadratic extension L̃ having the following two properties: − L̃ is unramified over L (and hence also over Q( √ −283)); − L̃ is a Galois extension of Q. (An explicit construction of L̃, due to T”
Root Numbers of Non‐Abelian Twists of Elliptic Curves (2005, background): “The theorem follows from standard results from the theory of automorphic forms (see, for example, [25]), as we will now brieFy explain.”
COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES (2012, background): “2) ǫ := dimV − dimV IK is the codimension of the IK-invariant subspace and δ is the Swan conductor of V (see [23] § 2, or [26], § 3.”
Slope filtrations (2008, background): “6 and items (1e) and (2a) that ⊗-bounded slope filtrations are exactly the slope filtrations discussed in [5]; indeed, the latter were defined to be exact filtrations that satisfy (1e).”
On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q) (2009, methodology): “We would like to thank Ravi Ramakrishna for helpful feedback on the revision.”
Prismatic $F$-crystals and crystalline Galois representations (2021, methodology): “8 was proven independently by Zhiyou Wu [Wu21].”
Families of L -Functions and Their Symmetry (2014, methodology): “The reader is referred to [101] for an excellent survey of many topics discussed in this appendix.”
Functoriality and the inverse Galois problem (2006, background): “of an integrable discrete series representation π∞ of SO(m+1,m). • (c) Π is unramiﬁed outside {ℓ,q}, and σ(Πq) is isomorphic to the ρq ﬁxed above. The results of [Kot], [Cl], [HT], see Theorem 3.6 of [Tay] (applied to a twist of Π by the 1−n 2 power of the norm character), ensure that there is a con”