We conduct a parallel Hunter search in order to find a degree 7 number field K ramified at primes q and p with discriminant d(K)=q^6 p^2 where q=11 and 2

The theme of this thesis is the study of moduli stacks of representations of an associative algebra, with an eye toward continuous representations of profinite groups such as Galois groups. The central object of study is the geometry of the map \(\bar{\psi}\) from the moduli stack of representations to the moduli scheme of pseudorepresentations. The first chapter culminates in showing that \(\bar{\psi}\) is very close to an adequate moduli space of Alper. In particular, \(\bar{\psi}\) is universally closed. The second chapter refines the results of the first chapter. In particular, certain projective subschemes of the fibers of \(\bar{\psi}\) are identified, generalizing a suggestion of Kisin. The third chapter applies the results of the first two chapters to moduli groupoids of continuous representations and pseudorepresentations of profinite algebras. In this context, the moduli formal scheme of pseudorepresentations is semi-local, with each component Spf \(B_\bar{D}\) being the moduli of deformations of a given finite field-valued pseudorepresentation \(\bar{D}\). Under a finiteness condition, it is shown that \(\bar{\psi}\) is not only formally finite type over Spf \(B_\bar{D}\), but arises as the completion of a finite type algebraic stack over Spec \(B_\bar{D}\). Finally, the fourth chapter extends Kisin's construction of loci of coefficient spaces for p-adic local Galois representations cut out by conditions from p-adic Hodge theory. The result is extended from the case that the coefficient ring is a complete Noetherian local ring to the more general case that the coefficient space is a Noetherian formal scheme. / Mathematics

Hida a étudié l'image de la représentation galoisienne associée à une famille p-adique de Hida de formes automorphes. Il a montré que l'image d'une famille non CM de formes modulaires classiques ordinaires contient un sous-groupe de congruence. Il a aussi lié le niveau optimal du groupe de congruence à l'idéal de congruence entre la famille de Hida non-CM et des familles CM. Cette thèse se divise en deux parties. La première partie est à généraliser ce genre de résultats dans le cas ordinaire pour les familles de Hida sur les groupes réductifs sous les hypothèses techniques. La deuxième partie se consacre à étudier les cas concrets. On montre que les hypothèses techniques sont satisfaites. Donc le même type de résultats est établi automatiquement. / Hida has studied the image of Galois representation associated to a p-adic Hida family of automorphic forms. He has proved that the image of a non-CM family ofordinary classic modular forms contains a congruence subgroup. He also related the optimal level of congruence subgroup to the congruence ideal between the non-CM Hida family and the CM ones. This thesis is divided into two parts. In the first part,we generalize this type of results to ordinary Hida families over reductive groups under some technical hypothesis. In the second part, we consider concrete cases. We prove that the technical hypothesis are satised for these cases. Hence the same type of results is established automatically.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24389号 / 理博第4888号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 並河 良典, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Soient 1 -> N -> H -> H' -> 1 une suite exacte centrale de groupes algébriques sur Q_p^alg et F un corps de nombres. Etant donnée une représentation Galoisienne r' : Gal_F -> H', on s'intéresse à ses relèvements à valeurs dans H à travers le morphisme H -> H'. Un relèvement r : Gal_F -> H sera dit minimal, s'il est non-ramifié aux places où r' est non-ramifiée et est de Rham/semi-stable/cristalline aux places divisant p si r' l'est. Dans cette thèse, nous montrons l'existence de relèvements minimaux dans certains cas. / Let 1 -> N -> H -> H' -> 1 be an exact sequence of algebraic groups over Q_p^alg and F be a number field. Given a Galois representation r' : Gal_F -> H', we are interested in its lifts with values in H through the morphism H -> H'. We say a lift r : Gal_F -> H is minimal, if it is unramied at places where r' is unramified and is de Rham/semi-stable/crystalline at p-adic places if r' is so. In this thesis, we prove the existence of such minimal lifts in some cases.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18769号 / 理博第4027号 / 新制||理||1580(附属図書館) / 31720 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 小野 薫, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM