Lifting Galois representations of number fields

Producing geometric deformations of orthogonal and symplectic Galois representations

“…In particular, the formalism developed in [Tay03] (still in the case of GL 2 ) suggested that it should be possible to generalize the technique to algebraic groups beyonds GL 2 . Attempts were made in [Ham08] and [Man09] to generalize the technique to GL n , but ran into the obstruction that there were no odd representations for n > 2. The results in [Ham08] simply assume the existence of liftable local deformation conditions which probably do not exist, but do provide a nice model for generalizing Ramakrishna's method.…”

Higher congruence companion forms

“…Since Frob r lifts g we have χ(Frob r ) ≡ −1 mod p n , and consequently E 1 is a substantial deformation condition for ρ n . The rest is identical to the proof of Proposition 4.2, [9]: The dual Selmer group for E 1 has dimension one less than that of the dual Selmer group for E 0 . (Of course δ(E 1 ) = δ(E 0 ) = δ(D).…”

“…Note that ρ is equivalent to a representation of the form considered above only if p divides the order of ρ(I F ). (See Example 3.3 of [9].) Example 2.4.…”

“…We wish to consider global deformation conditions D for ρ with determinant ǫ such that ρ n is a deformation of type D. We shall abbreviate this and call D a deformation condition for ρ n . Except for a change in choice of lettering for primes of F we keep the notation of [9]. Thus D q is the local component at a prime q, t Dq is the tangent space there, and t ⊥ Dq ⊆ H 1 (G Fq , ad 0 ρ(1)) is the orthogonal complement of t Dq under the pairing induced by ad 0 ρ × ad 0 ρ(1)…”

“…Take g ∈ Gal(K/L) as in R1, consider pairs (M 1 , N 1 ), (M 2 , N 2 ) where {( 0 * * 0 )} = N 1 ⊂ M 1 = ad 0 ρ, {( * * 0 * )} = N 2 ⊂ M 2 = ad 0 ρ(1) and apply Proposition 2.2 of [9]. One can then find a prime r / ∈ S ∪ {q 0 } lifting g such that the restrictions of ξ, ψ to G r are not in H 1 (G r , N 1 ), H 1 (G r , N 2 ).…”