Diophantine geometry over groups III: Rigid and solid solutions

This paper is the third in a series on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the third paper we analyze exceptional families of solutions to a parametric system of equations. The structure of the exceptional solutions, and the global bound on the number of families of exceptional solutions we obtain, play an essential role in our approach towards quantifier elimination in the elementary theory of a free group presented in the next papers of this series. The argument used for proving the global bound is a key in proving the termination of the quantifier elimination procedure presented in the sixth paper of the series. Partially supported by an Israel Academy of Sciences Fellowship.     

Diophantine geometry over groups V2: quantifier elimination II

This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of an AE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra of AE sets defined over a free group is invariant under projections, hence, show that every elementary set defined over a free group is in the Boolean algebra of AE sets. The procedures we use for quantifier elimination, presented in this paper, enable us to answer affirmatively some of Tarski’s questions on the elementary theory of a free group in the last paper of this sequence. Received (resubmission): January 2004 Revision: November 2005 Accepted: March 2006 Partially supported by an Israel Academy of Sciences Fellowship.     

Completions of certain modules over quantum groups 1

Completion is a procedure of constructing new representations for Lie algebras from given ones. This paper shows that a similar procedure exists for quantum groups.     

Diophantine geometry over groups I: Makanin-Razborov diagrams

Manucsrit re?u le 4 janvier 2000.     

Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.     

Hilbert symbol for Lubin-Tate formal groups. II

Explicit formulas are obtained for the generalized Hilbert symbol on the group of points of a formal Lubin-Tate group, simultaneously covering the cases of even and odd characteristics of the residue field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 85–96, 1983.     

Diophantine geometry over groups V1: Quantifier elimination I

This paper is the first part (out of two) of the fifth paper in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of anAE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra ofAE sets defined over a free group is invariant under projections, and hence show that every elementary set defined over a free group is in the Boolean algebra ofAE sets. The procedures we use for quantifier elimination, presented in this paper and its successor, enable us to answer affirmatively some of Tarski's questions on the elementary theory of a free group in the sixth paper of this sequence. Partially supported by an Israel Academy of Sciences Fellowship.     

The Hilbert pairing for formal groups over σ-rings

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 5–58.     

A note on extensions of algebraic and formal groups II

Partially supported by Grant-in-Aid for Scientific Research #05640063     

The formal translation equation for iteration groups of type II

We investigate the translation equation $$F(s+t, x) = F(s, F(t, x)),\quad \quad s,t\in{\mathbb{C}},\qquad\qquad\qquad\qquad({\rm T})$$ in ${\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ , the ring of formal power series over ${\mathbb{C}}$ . Here we restrict ourselves to iteration groups of type II, i.e. to solutions of (T) of the form ${F(s, x) \equiv x + c_k(s)x^k {\rm mod} x^{k + 1}}$ , where k ≥ 2 and c _k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions c _n (s) of $$F(s, x) = x + \sum_{n \ge q k}c_n(s)x^n$$ are polynomials in c _k (s). It is possible to replace this additive function c _k by an indeterminate. In this way we obtain a formal version of the translation equation in the ring ${(\mathbb{C}[y])\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczél–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character (depending on certain parameters, the coefficients of the infinitesimal generator H of an iteration group of type II) of these polynomials. Rewriting the solutions G(y, x) of the formal translation equation in the form ${\sum_{n\geq 0}\phi_n(x)y^n}$ as elements of ${(\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right])\left[\kern-0.15em\left[{y}\right]\kern-0.15em\right]}$ , we obtain explicit formulas for ${\phi_n}$ in terms of the derivatives H ^(j)(x) of the generator ${H}$ and also a representation of ${G(y, x)}$ as a Lie–Gröbner series. Eventually, we deduce the canonical form (with respect to conjugation) of the infinitesimal generator ${H}$ as x ^k + hx ^2k-1 and find expansions of the solutions ${G(y, x) = \sum_{r\geq 0} G_r(y, x)h^r}$ of the above mentioned differential equations in powers of the parameter h.     

Degeneration of the Hilbert pairing in formal groups over local fields

For an arbitrary local field K (a finite extension of the field Q_p) and an arbitrary formal group law F over K, we consider an analog c_F of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing c_F has a certain fundamental symbol property for all Lubin–Tate formal groups, then c_F = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring O_K of integers of K and have a fixed ring O₀ of endomorphisms, where O₀ is a subring of O_K. We prove that if the symbol c_F has the above-mentioned symbol property, then c_F = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.     

Triviality of universal norms on formal groups over a local field

A result of the local theory of class fields on the triviality of universal norms on a multiplicative group is generalized to the case of an arbitrary commutative multiparametric formal group defined over the ring of integers of a complete discretely normed field with a finite residue field.     

Convex polytopes and formal groups

Some constructions of commutative formal groups proceeding from convex polytopes and Laurent polynomials are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 87–90, 1978.     

Displays and formal p-divisible groups

Over p-adic Nagata rings, formal p-divisible groups are classified by nilpotent displays according to T. Zink. We extend this result to arbitrary p-adic rings. The proof uses the Grothendieck–Illusie deformation theory of truncated p-divisible groups.