Hilbert irreducibility sequences and nonstandard arithmetic

We give sufficient conditions for a sequence of integers to be a Hilbert irreducibility sequence and give such a sequence explicitly.     

Transfer principles in nonstandard intuitionistic arithmetic

 Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. Received: 7 January 2000 / Revised version: 26 March 2001 / Published online: 12 July 2002     

Interpreting weak Kőnig's lemma in theories of nonstandard arithmetic

We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.     

Fast algorithms for computing isogenies between elliptic curves

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree ( different from the characteristic) in time quasi-linear with respect to . This is based in particular on fast algorithms for power series expansion of the Weierstrass -function and related functions.

     

Distinguished isogenies for formal groups in local fields with small ramification

The existence of distinguished isogenies is proved for formal groups defined over the ring of integers of a local field with ramification index less than the characteristic of the residue field. This generalizes the result of Demchenko for Honda formal groups. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 93–100.     

Poonen's question concerning isogenies between Smart's genus 2 curves

We describe a method for proving that two explicitly given genus two curves have isogenous jacobians. We apply the method to the list of genus 2 curves with good reduction away from 2 given by Smart. This answers a question of Poonen.

     

A definable nonstandard enlargement

This article establishes the existence of a definable (over ZFC), countably saturated nonstandard enlargement of the superstructure over the reals. This nonstandard universe is obtained as the union of an inductive chain of bounded ultrapowers (i.e. bounded with respect to the superstructure hierarchy). The underlying ultrafilter is the one constructed by Kanovei and Shelah [10]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

One parameter subgroups and nonstandard analysis

We present here a discussion from a nonstandard viewpoint of some aspects of the theory of locally compact groups without small subgroups, as relevant to the solution of Hilbert's fifth problem. For background on nonstandard analysis we refer to Robinson [4].     

Harrison pseudo-hamiltonian and nonstandard matching conditions

Justification is given for nonstandard matching conditions for solutions of the Schrödinger equation for quasiparticles with different effective masses. A method of differential transformations of functions is used for the Schrödinger equation. The method makes it possible to obtain the mass-resonance condition and to analyze the characteristic features of the conditions of the total internal reflection and the electron emission.Translated from Matematicheskie Melody i Fiziko-mekhanicheskie Polya, No. 26, pp. 101–105, 1987.     

Residual spectrum of split reductive groups supported in a Borel subgroup under isogenies

The purpose of this note is to find a relationship between residual spectrum of groups in a given isogeny class. We show that for residual spectrums associated with Borel subgroups, it is enough to consider just one group in a given isogeny class.     

Local microcontinuity of nonstandard polynomials

For nonstandard polynomials the monadic concept of microcontinuity is supplemented with a typically polynomial absolute microcontinuity. It is examined how these notions are interrelated, and related to the coefficients and to the standard notion of convergent power series. It is found that (absolute) microcontinuity is a genuine nonstandard concept, either nonexistent or trivial for standard data.     

A nonstandard Riemann existence theorem

We study elementary extensions of compact complex spaces and deduce that every complete type of dimension is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.

     

Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic

The problem of computing an explicit isogeny between two given elliptic curves over Fq, originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovtsev & Stolbunov.While the large characteristic case is well understood, only suboptimal algorithms are known in small characteristic; they are due to Couveignes, Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the differences between them and run some comparative experiments. We also present the first complete implementation of Couveignes' second algorithm and present improvements that make it the algorithm having the best asymptotic complexity in the degree of the isogeny.