A survey on recent p-adic integration theories and arithmetic applications

"Vegeu el resum a l'inici del document del fitxer adjunt."

Exact algorithms for p-adic fields and epsilon constant conjectures

We develop several algorithms for computations in Galois extensions of p-adic fields. Our algorithms are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to p-adic numbers. As an application we describe an algorithmic approach to prove or disprove various conjectures for local and global epsilon constants.

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Z(p). This is done by defining an index preserving morphism s: Br(<(K(X))over cap>)` --> Br(K(X))` which splits res : Br(K (X)) --> Br(<(K(X))over cap>), where <(K(X))over cap> is the completion of K (X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over <(K(X))over cap>. (C) 2010 Elsevier Inc. All rights reserved.

p-adic L functions and trivial zeroes

The following is adapted from the notes for the lecture. It announces results and conjectures about values of the p-adic L function of the symmetric square of an elliptic curve.

On Selmer groups and factoring p-adic L-functions

Thesis (Ph.D.)--University of Washington, 2016-06

Improved Cryptoanalysis of the Self-shrinking P-adic Cryptographic Generator

The Self-shrinking p-adic cryptographic generator (SSPCG) is a fast software stream cipher. Improved cryptoanalysis of the SSPCG is introduced. This cryptoanalysis makes more precise the length of the period of the generator. The linear complexity and the cryptography resistance against most recently used attacks are invesigated. Then we discuss how such attacks can be avoided. The results show that the sequence generated by a SSPCG has a large period, large linear complexity and is stable against the cryptographic attacks. This gives the reason to consider the SSPSG as suitable for critical cryptographic applications in stream cipher encryption algorithms.

Pseudo-randomness of round-off errors in discretized linear maps on the plane

We analyze the sequences of round-off errors of the orbits of a discretized planar rotation, from a probabilistic angle. It was shown [Bosio & Vivaldi, 2000] that for a dense set of parameters, the discretized map can be embedded into an expanding p-adic dynamical system, which serves as a source of deterministic randomness. For each parameter value, these systems can generate infinitely many distinct pseudo-random sequences over a finite alphabet, whose average period is conjectured to grow exponentially with the bit-length of the initial condition (the seed). We study some properties of these symbolic sequences, deriving a central limit theorem for the deviations between round-off and exact orbits, and obtain bounds concerning repetitions of words. We also explore some asymptotic problems computationally, verifying, among other things, that the occurrence of words of a given length is consistent with that of an abstract Bernoulli sequence.

Stochastic volatility jump-diffusions for European equity index dynamics

Major research on equity index dynamics has investigated only US indices (usually the S&P 500) and has provided contradictory results. In this paper a clarification and extension of that previous research is given. We find that European equity indices have quite different dynamics from the S&P 500. Each of the European indices considered may be satisfactorily modelled using either an affine model with price and volatility jumps or a GARCH volatility process without jumps. The S&P 500 dynamics are much more difficult to capture in a jump-diffusion framework.

Forma traço sobre algumas extensões galoisianas de corpos p-Ádicos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Dynamics of quantum-dot mode-locked lasers with optical injection

We present the dynamics of quantum-dot passively mode-locked semiconductor lasers under optical injection. We discuss the benefits of various configurations of the master source including single, dual, and multiple coherent frequency sources. In particular, we demonstrate that optical injection can improve the properties of the slave laser in terms of time-bandwidth product, optical linewidth, and timing jitter.

Dynamics of quantum-dot mode-locked lasers with optical injection

We present the dynamics of quantum-dot passively mode-locked semiconductor lasers under optical injection. We discuss the benefits of various configurations of the master source including single, dual, and multiple coherent frequency sources. In particular, we demonstrate that optical injection can improve the properties of the slave laser in terms of time-bandwidth product, optical linewidth, and timing jitter.

Tree automorphisms and quasi-isometries of Thompson's group F

We prove that automorphisms of the infinite binary rooted tree T2 do not yield quasi-isometries of Thompson's group F, except for the map which reverses orientation on the unit interval, a natural outer automorphism of F. This map, together with the identity map, forms a subgroup of Aut(T2) consisting of 2-adic automorphisms, following standard terminology used in the study of branch groups. However, for more general p, we show that the analgous groups of p-adic tree automorphisms do not give rise to quasiisometries of F(p).

Spaces with Noetherian cohomology

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? This note provides, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements, with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.