Hardy type derivations on fields of exponential logarithmic series

We consider the valued field \mathds{K}:=\mathbb{R}((\Gamma)) of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). We investigate how to endow \mathds{K} with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on \mathds{K}. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure \mathds{K}^{\rm{EL}} of (\mathds{K},l). Here we show how to extend such a log-compatible derivation on \mathds{K} to \mathds{K}^{\rm{EL}}.Comment: 25 page

Two variants of the support problem for products of abelian varieties and tori

Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem).Comment: 13 pages; v2 results generalized; v3 incorporated referee comments, final version to appear in Journal of Number Theor

Scaffolds and Generalized Integral Galois Module Structure

Let $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where $L/K$ was Galois and $A=K[G]$ for $G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in $A$. We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with $L/K$. We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois $p$-extensions in characteristic $p$. We also apply our results to the non-classical situation where $L/K$ is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power $K$-Hopf algebra.Comment: Further minor corrections and improvements to exposition. Reference [BE] updated. To appear in Ann. Inst. Fourier, Grenobl

Functional identities on matrix algebras

Complete solutions of functional identities $\sum_{k\in K}F_k(\bar{x}_m^k)x_k = \sum_{l\in L}x_lG_l(\bar{x}_m^l)$ on the matrix algebra $M_n(\mathbb{F})$ are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.Comment: 20 pages, comments welcome, version 2- applications have been adde

On the Tits alternative for some generalized triangle groups

One says that the Tits alternative holds for a finitely generated group G if G contains either a non abelian free subgroup or a solvable subgroup of finite index. Rosenberger states the conjecture that the Tits alternative holds for generalized triangle groups $T(k,l,m,R)=<a,b; a^k=b^l=R^m(a,b)=1>$. In the paper Rosenberger's conjecture is proved for groups T(2,l,2,R) with l=6,12,30,60 and some special groups T(3,4,2,R)