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Showing 1 to 20 of 22 for “"Galois representations"”.
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Deformations of Three-Dimensional Galois Representations
Having explicitly calculated universal deformations, the next logical question is whether we can calculate deformations with local restrictions. In fact, the ordinary deformation problem is representable, and we explain how to calculate the universal ordinary deformation. We proceed to perform this …
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Families of p̳-adic Galois representations
… and then apply it to the generic fibers of Galois deformation spaces. I study the finite slope deformation rings in details by computing the dimensions of their Zariski cotangent spaces via Galois cohomologies. It turns out that the Galois cohomologies tell us not only the formal smoothness …
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Galois representations of orthogonal rigid local systems
… two points. These correspond to continuous representations of the etale fundamental group, which can be specialized to compatible systems of Galois representations. This leads to the second maximally unipotent family. Because of the geometric origin, we can show using a theorem of …
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Lifting Galois Representations in a Conjecture of Figueiredo
… correspondence between degree 2 odd irreducible representations of the absolute Galois group of Q and modular forms. Letting M be an imaginary quadratic field, L.M. Figueiredo gave a related conjecture concerning degree 2 irreducible representations of the absolute Galois group of M and their …
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Totally Real Galois Representations in Characteristic 2 and Arithmetic Cohomology
… Ash, Doud, and Pollack. This conjecture involves Galois representations taking Gal(Q bar/Q) to the general linear group of 3 x 3 matrices in characterisic 2, and our examples are where complex conjugation is mapped to the identity. Since this case has not yet been examined, the results of this …
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Toward a deformation theory for Galois representations of function fields
… of describing the one-dimensional P-adic representations that lift a given representation over a finite field of the absolute Galois group of a function field K. In this case, the characterization of abelian p-power extensions of fields of characteristic p can be extended and refined to …
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Universal deformations of even Galois representations and relations to Maass wave forms
The whole area of deformations of Galois representations started in 1986 with a ground-breaking paper by Barry Mazur where he established the first existence theorem and some examples in the case of odd two-dimensional Galois representations. Such representations arise from elliptic curves and they …
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L-invariants and congruences for Galois representations of dimension 3, 4, and 8
The arithmetic of Galois representations plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint …
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Explicit Computations Supporting a Generalization of Serre's Conjecture
Serre's conjecture on the modularity of Galois representations makes a connection between two-dimensional Galois representations and modular forms. A conjecture by Ash, Doud, and Pollack generalizes Serre's to higher-dimensional Galois representations. In this paper we discuss an explicit …
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Proven Cases of a Generalization of Serre's Conjecture
… between modular forms and two-dimensional Galois representations. Ash, Doud, and Pollack have extended this conjecture to a correspondence between Hecke eigenclasses in arithmetic cohomology and n-dimensional Galois representations. We present some of the first examples of proven cases of …
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Components of the Emerton-Gee Moduli Stack of Galois Representations for GL2 and A Graph-Theoretic Approach to Computing Selmer Groups of Elliptic Curves over Q(i)
… as a moduli space for two-dimensional mod p representations of Gal(K/K). We determine precisely which irreducible components are smooth, which are normal, and which have Gorenstein normalizations. We prove that the normalizations of these components admit smooth–local covers by Cohen-Macaulay …
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The Taylor-Wiles method for reductive groups
… a local deformation problem for residual Galois representations ρ valued in an arbitrary reductive group Ĝ which we use to develop a variant of the Taylor–Wiles method. Our generalization allows Taylor–Wiles places for which the image of Frobenius is semisimple, a weakening of the regular …
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First explicit reciprocity law for unitary Friedberg—Jacquet periods
… on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin--Selberg, with applications to the Bloch--Kato conjecture and to Iwasawa theory. This thesis studies the case of Galois representations …
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Emerton-Gee Stacks, Serre Weights, and Breuil-Mézard Conjectures for GSp_4
… formulated by Gee-Herzig-Savitt for global Galois representations valued in GSp_4 satisfying Taylor--Wiles conditions, and a modularity lifting result for tamely potentially crystalline representations.
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L-functions in Number Theory
… attached to irreducible 4-dimensional S5-Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case.
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Counterexamples related to the Sato-Tate conjecture
… analytic methods. We also show that there are Galois representations $\rho\colon Gal(\overline{\mathbf{Q}} /\mathbf{Q}) \to GL_2(\mathbf{Z}_l)$, ramified at an arbitrarily thin (but still infinite) set of primes, whose Satake parameters can be made to converge at any specified rate to any fixed …
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Level raising for automorphic representations of GL(2n)
… isobaric sum of conjugate self-dual, cuspidal representations), we can attach a continuous $\ell$-adic Galois representation $r(\Pi)$ of the absolute Galois group of $E$. The residual Galois representation …
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