Deformations and elements of deformation theory (original) (raw)
A Superficial Working Guide to Deformations and Moduli
This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.
p-rigidity and Iwasawa μ-invariants
Algebra & Number Theory, 2017
Let F be a totally real field with ring of integers O and p be an odd prime unramified in F . Let p be a prime above p. We prove that a mod p Hilbert modular form associated to F is determined by its restriction to the partial Serre-Tate deformation space Gm ⊗ Op (p-rigidity). Let K/F be an imaginary quadratic CM extension such that each prime of F above p splits in K and λ a Hecke character of K. Partly based on p-rigidity, we prove that the µ-invariant of anticyclotomic Katz p-adic L-function of λ equals the µ-invariant of the full anticyclotomic Katz p-adic L-function of λ. An analogue holds for a class of Rankin-Selberg p-adic L-functions. When λ is self-dual with the root number -1, we prove that the µ-invariant of the cyclotomic derivatives of Katz p-adic L-function of λ equals the µ-invariant of the cyclotomic derivatives of Katz p-adic L-function of λ. Based on previous works of authors and Hsieh, we consequently obtain a formula for the µ-invariant of these p-adic L-functions and derivatives, in most of the cases. We also prove a p-version of a conjecture of Gillard, namely the vanishing of the µ-invariant of Katz p-adic L-function of λ.
On the Hida deformations of fine Selmer groups
Journal of Algebra, 2011
In this paper, we study the fine Selmer group of p-adic Galois representations and their deformations. We show that for an infinite family of elliptic cuspforms, if the μ-invariant of the dual fine Selmer group is zero for one member of the family, then the same holds for all the other members. Further the λ-invariants are equal for all but finitely many members in the family.
Deformations of Local Systems and Eisenstein Series
Geometric and Functional Analysis, 2008
Introduction 0.1. The goal of this paper is to realize a suggestion made by V. Drinfeld. To explain it let us recall the general framework of the geometric Langlands correspondence.
2010
Using first order deformation theory of pointed curves we show that the semigroup of a generic Weierstrass point whose semigroup has first nonzero element k consists only of multiples of k until after its greatest gap value, and that on the moduli space of curves two components of the divisor of points corresponding to curves possessing exceptional Weierstrass points intersect nontransversely. As usual let J( be the moduli space of curves of genus g. Many subloci of Jtg defined in terms of Weierstrass points have received much study. (1) Definition. For an integer k with 2 < k < g, Dk k = {[C] e Jfg. C possesses a Weierstrass point p with h°(C, kp) > 2). It is known that Dk k is irreducible of dimension 2g 3 + k and a general point in Dk k corresponds to a curve C with a Weierstrass point p with h°(C,kp) = 2. See Rauch [1], Lax [1, 2], Arbarello [1, 2]. Here we will prove: (2) Theorem. A generic point in Dk k corresponds to a curve with a Weierstrass point whose semigroup c...
2001
This work is the first part in a series of three dedicated to the foundations of integral aspects of Shimura varieties and of Fontaine's categories. It deals mostly with the unramified context of (arbitrary) mixed characteristic (0,p). Among the topics covered we mention: the generalization of the classical Serre-Tate theory of ordinary p-divisible groups and of their canonical lifts, the generalization of the classical Serre-Tate-Dwork-Katz theory of (crystalline) coordinates for ordinary abelian varieties, the strong form of the generalized Manin problem, global deformations in the generalized Shimura context, Dieudonn\'e's theories (reobtained, simplified and extended), the main list of stratifications of special fibres of integral canonical models of Shimura varieties of preabelian type, the uniqueness of such models in mixed characteristic (0,2), the existence (in many situations) of such models in mixed characteristic (0,2), steps towards the classification of S...
On the tangent space of the deformation functor of curves with automorphisms
Algebra & Number Theory, 2007
We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, p-cyclic covers of the affine line and to Lehr-Matignon curves.
Algebra & Number Theory, 2013
representations of elementary subgroups of Chevalley groups of rank ≥ 2. First, we extend the methods to analyze representations of elementary groups over arbitrary associative rings and, as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form SL n,D , where D is a finite-dimensional central division algebra over a field of characteristic 0. Second, we apply the previous results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank ≥ 2 over finitely generated commutative rings.
Integral and p-modular semisimple deformations for p-solvable groups of finite representation type
Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1991
We prove that the split integral group ring of a finite p-solvable group of finite representation type has a structure analogous to that of the p-modular semisimple deformation. The split integral deformation can be put in the same form as the p-modular deformation by an appropriate substitution for the parameter T. As an application we derive a simple formula for the matrix units in the semisimple group algebra over a nonmodular prime.