Representation zeta functions of special linear groups

Zordan M. Representation zeta functions of special linear groups. Bielefeld: Universität Bielefeld; 2016.We introduce new methods from representation theory of algebraic groups into the study of representation zeta functions associated with compact $\mathfrak{p}$-adic analytic groups and arithmetic groups. We apply these new methods to compute the representation zeta functions of principal congruence subgroups of the groups $\mathrm{SL}_4(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic 0

Contributions to the theory of groups

A 1. The influence on a finite group of its proper abnormal structure. J. London Math. Soc. 40 (1965). 348-61; MR30#4838. • B 2. Abnormal depth and hypereccentric length in finite soluble groups, Math. Z. 90 (1965). 29-40; MR32#141. • C 3. On a splitting theorem of Gaschiitz. Proc. Edinburgh Math. Soc. (20 15 (1966). 57-60; MR33#5708. • A 4. Finite groups with prescribed Sylow tower subgroups. Proc. London Math. Soc. (3) 16 (1966). 577-89; MR33#5734. • B 5. Remarks on system normalizers and Carter subgroups. Proc. Intemat. Conf. Theory of Groups (Canberra I965) (1967). 303-5. • B 6. Finite soluble groups with pronormal system normalizers. Proc. London Math. Soc. (3) 17 (1967). 447-69; MR35#2967. • C 7. A natural setting for the extensions of a group with trivial centre by an arbitrary group. Enseignement Math. 13 (1967). 167-73; MR3871179. • B 8. Nilpotent subgroups of finite soluble groups. Math. Z. 106 (1968). 97-112; MR40#5736. • B 9. Absolutely faithful group actions. Proc. Cambridge Philos. Soc. 66 (1969). 231-7; MR40#1465. • CIO. On the splitting of extensions by a group of prime order. Math. Z. 117 (1970). 239-48; MR43#356. • Cll. Splitting properties of group extensions. Proc. London Math. Soc. (3) 22 (1971). 1-23; MR43#7515. • C12. Extensions by a free abelian group of rank 2. Proc. Roy. Irish Acad. 71A (1971). 19-26. MR44#4097 • BI3. A subnormal embedding theorem for finite groups. J. London Math. Soc. (2) 5 (1972) 253-9; MR47#326. • C14. Universal finite group extensions and a non-splitting theorem. Israel J. Math. 15 (1973) 375-83. • A15. Sufficient conditions for the existence of ordered Sylow towers in finite groups. J. Algebra 28 (1974) 116-26. • Cl6. Automorphism groups of groups with trivial centre. Proc. London Math. Soc. • C17. Frattini normal subgroups of finite groups. unpublished. • A18. On finite insoluble groups with nilpotent maximal subgroups. unpublished

Modularity of GL⁡2(Fp)\operatorname{GL}_2(\mathbb{F}_p)-representations over CM fields

We prove that many representations $\overline{\rho} : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_2(\mathbb{F}_3)$, where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p = 3$ is replaced by $p = 2$ or $p = 5$. As a consequence, we prove that a positive proportion of elliptic curves over any CM field not containing a 5th root of unity are modular.Comment: 94 page

Материалы конференции: "Алгебра и математическая логика: теория и приложения"

Сборник содержит тезисы докладов, представленных на международную конференцию "Алгебра и математическая логика: теория и приложения" ( г. Казань 2-6 июня 2014 год) и сопутствующую молодежную летнюю школу "Вычислимость и вычислимые структуры", посвященную 210-летию Казанского университета, 80-летию со дня основания кафедры алгебры (ныне кафедры алгебры и математической логики) Казанского университета Н.Г. Чеботаревым и 70-летию со дня рождения зав. кафедрой члена-корреспондента АН РТ М.М. Арсланова.17

Homological Properties of Structures in Commutative Algebra and Algebraic Combinatorics

The purpose of this work is to understand homological properties of structures appearing in commutative algebra and algebraic combinatorics, objects such as commutative rings and associated structures, such as ideals and modules, or simplicial complexes. In particular, we study vanishing conditions for Ext and Tor in connection with homological dimensions of the modules involved, the representation theory of maximal Cohen-Macaulay modules, and various homological properties of simplicial complexes though the lens of combinatorial commutative algebra. Specifically, we study when a Cohen-Macaulay local ring has only trivial vanishings of Ext or Tor, and provide sufficient numerical criterion under which these condition are satisfied. We apply these results to establish new cases of the famous Auslander-Reiten conjecture; other conditions on Ext and Tor are also explored in connection with this conjecture. We also study the connection between classifically studied representation types of the category of maximal Cohen-Macaulay modules of a Cohen-Macaulay local ring and newly introduced representation types which study those maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. We provide a classification theorem in dimension 1, and discuss partial results and obstacles in higher dimension. We also explore combinatorial constructions such as the nerve complex of a simplicial complex, and introduce the new notion higher nerve complexes. We explore their connection with order complexes of posets, in particular the face poset of a simplicial complex, and we prove that the depth and h-vector of the Stanley-Reisner ring of a simplicial complex can be computed in a nice way from the reduced homologies of these higher nerve complexes. We expand upon our study of these notions by studying balanced simplicial complexes, and using this abstraction we prove that, while one cannot characterize which of Serre's conditions are satisfied by a simplicial complex via the reduced homologies of higher nerve complexes, one can pin it down to one of two possible values. We also provide a depth formula for arbitrary balanced simplicial complexes and consider total Euler characteristics of links; using the latter, we provide some applications to the study of Gorenstein* complexes. Finally, we introduce the notion of minimal Cohen-Macaulay simplicial complexes and provide some necessary and sufficient conditions for this property. We conclude by showing that many recently introduced counterexamples to longstanding conjectures in the literature are minimal Cohen-Macaulay