Dihedral universal deformations

Analysis of the condition for universal deformations in compressible finite elasticity leads to the identification of two classes of strain energies that support universal irrotational deformations. These classes represent a significant generalization of the classes of strain energies for which universal deformation solutions have been hitherto available. Closed form solutions are presented for cylindrically and spherically radial irrotational deformation for all of these strain energies. Some questions about maximality of some classes of strain energies that support cylindrically radial universal deformations are resolved.

Let 𝒜 be the ring ℤ p [[t 0, t 1, t ∞]]/((t 0 + 1)(t 1 + 1)(t ∞ + 1) − 1)equipped with the non-trivial action of G ℚ ≔ Gal(ℚ¯/ℚ) described in the introduction. In Ihara (1986b), Ihara constructs a universal cocycle arising from the action of Gal(ℚ¯/ℚ) on certain quotients of the Jacobians of the Fermat curves for each n ≥ 1. This paper gives a different construction of part of Ihara's cocycle by considering the universal deformation of certain two-dimensional representations of Πℚ¯, the algebraic fundamental group of ℙ1(ℚ¯)\\{0, 1, ∞}. More precisely, we determine the universal deformation ring subject to certain deformation conditions arising from a residual representation Belyĭ's Rigidity Theorem is then used to extend each such universal deformation to a representation of Π K , where K is a finite cyclotomic extension of ℚ(μ p ∞ ). When ρ¯ is the representation arising from the p-division points of the Legendre family of elliptic curves, we give a geometric construction of one such extended universal deformation ρ, and show that part of Ihara's cocycle can be recovered by specializing ρ at infinity.

Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given SL_2-representation of a finitely generated group Pi over a field whose characteristic is not 2. We then show its connection with the character scheme for SL_2-representations of Pi when k is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.

Deformations and the rigidity method

We construct multiparameter quantizations of reductive Lie algebras which have the property of universality within a certain class of deformations. The universal deformations can be defined so that the algebra structure on each simple component is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear, as a special case of deformations of a semisimple algebra whose simple components remain classical. Deformations are defined as algebras over power series rings and it is essential to require them to be torsion free to secure the universality. The Poincare-Birkhoff-Witt theorem and the torsion freeness are established for the universal deformation on the basis of results on the representation theory of the deformed algebras.

It is shown that a static pure torsion can be imposed on a circular cylinder comprised of any homogeneous isotropic elastic material subject to suitable surface tractions alone if the material is incompressible, but such a pure torsion cannot generally be imposed if the material is compressible. Hence the static torsion is a universal deformation for an incompressible isotropic elastic material, but not for a compressible isotropic elastic material. The analysis of such a pure torsion is due to Rivlin (1947; 1948a; 1949a). A broader family of universal deformations of Ericksen & Rivlin (1954) is also discussed, where this larger family includes not only torsions but also extensions and inflations of a circular cylinder. The calculation of these and other universal exact solutions for finite elastic deformation represents a remarkable achievement in the mathematical theory of elasticity. The development here follows that of Truesdell & Noll (1965).

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

This collection of survey and research articles brings together topics at the forefront of the theory of L-functions and Galois representations. Highlighting important progress in areas such as the local Langlands programme, automorphic forms and Selmer groups, this timely volume treats some of the most exciting recent developments in the field. Included are survey articles from Khare on Serre's conjecture, Yafaev on the André-Oort conjecture, Emerton on his theory of Jacquet functors, Venjakob on non-commutative Iwasawa theory and Vigneras on mod p representations of GL(2) over p-adic fields. There are also research articles by: Böckle, Buzzard, Cornut and Vatsal, Diamond, Hida, Kurihara and R. Pollack, Kisin, Nekovář, and Bertolini, Darmon and Dasgupta. Presenting the very latest research on L-functions and Galois representations, this volume is indispensable for researchers in algebraic number theory.

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

Cohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the Arthur-Selberg trace formula, and with the proof in some cases of the Rmamanujan-Petersson conjecture and the global Langlands conjecture for function fields. It is based on graduate courses taught by the author, who uses techniques which are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. It is the first book on this subject and will be of much interest to all researchers in algebraic number theory and representation theory.

Deformations of Pseudorepresentations

This book contains conference proceedings from the 1996 Durham Symposium on 'Galois representations in arithmetic algebraic geometry'. The title was interpreted loosely and the symposium covered recent developments on the interface between algebraic number theory and arithmetic algebraic geometry. The book reflects this and contains a mixture of articles. Some are expositions of subjects which have received substantial attention, e.g. Erez on geometric trends in Galois module theory; Mazur on rational points on curves and varieties; Moonen on Shimura varieties in mixed characteristics; Rubin and Scholl on the work of Kato on the Birch-Swinnerton-Dyer conjecture; and Schneider on rigid geometry. Others are research papers by authors such as Coleman and Mazur, Goncharov, Gross and Serre.

The aim of this monograph is to demonstrate the application of algebra and algebraic number theory to the study of combinatorics problems. Three problems which will be discussed in this monograph are Group-Invariant Butson Hadamard Matrices, Unique Differences in Symmetric Subsets of F p and Upper Bounds for Cyclotomic Numbers. The main contribution of this thesis is the construction of new classes of the mentioned objects and better necessary conditions (than the known ones) for their existence. In some cases, our necessary conditions are also sufficient conditions or asymptotically best conditions.

Artin conjectured that certain Galois representations should give rise to entire L-series. We give some history on the conjecture and motivation of why it should be true by discussing the one-dimensional case. The first known example to verify the conjecture in the icosahedral case did not surface until Buhler's work in 1977. We explain how this icosahedral representation is attached to a modular elliptic curve isogenous to its Galois conjugates, and then explain how it is associated to a cusp form of weight 5 with level prime to 5. In 1917, Erich Hecke (10) proved a series of results about certain characters which are now commonly referred to as Hecke characters; one corollary states that one-dimensional complex Galois representations give rise to entire L-series. He re- vealed, through a series of lectures (9) at Princeton's Institute for Advanced Study in the years that followed, the relationship between such representations as gen- eralizations of Dirichlet characters and modular forms as the eigenfunctions of a set of commuting self-adjoint operators. Many mathematicians were inspired by his ground-breaking insight and novel proof of the analytic continuation of the L-series. In the 1930's, Emil Artin (1) conjectured that a generalization of such a result should be true; that is, irreducible complex projective representations of finite Ga- lois groups should also give rise to entire L-series. He came to this conclusion after proving himself that both 3-dimensional and 4-dimensional representations of the simple group of order 60, the alternating group on five letters, might give rise to L-series with singularities. It is known, due to the insight of Robert Langlands (16) in the 1970's relating Hecke characters with Representation Theory, that in order to prove the conjecture it suffices to prove that such representations are associated to cusp forms. This conjecture has been the motivation for much study in both Algebraic and Analytic Number Theory ever since. In this paper, we present an elementary approach to Artin's Conjecture by con- sidering the problem over Q. We consider Dirichlet's theorem which preceeded Hecke's results, and sketch a proof by introducing theta series. We then introduce Langland's program to exhibit cusp forms. We conclude by studying a specific ex- ample which is associated to an elliptic curve. We assume in the final sections that the reader is somewhat familiar with the basic properties of elliptic curves.

In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato’s zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato’s generalized Iwasawa main conjecture. Based on Kato’s original construction, we construct our zeta morphisms using many deep results in the theory of p-adic (local and global) Langlands correspondence for mathrm{GL}_{2/mathbb{Q}}. As an application of our zeta morphisms and the recent article (Kim et al. in On the Iwasawa invariants of Kato’s zeta elements for modular forms, 2019, arXiv:1909.01764v2), we prove a theorem which roughly states that, under some mu =0 assumption, Iwasawa main conjecture without p-adic L-function for f holds if this conjecture holds for one g which is congruent to f.