Primes represented by quadratic polynomials via exceptional characters

“This is a post-peer-review, pre-copyedit version of an article published in Archiv der Mathematik. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00013-021-01602-3”We estimate the number of primes represented by a general quadratic polynomial with discriminant Δ , assuming that the corresponding real character is exceptionalFernando Chamizo is partially supported by the MTM2017-83496-P Grant of the MICINN (Spain) and by “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). This latter grant supported the visit of Jorge Jiménez Urroz to the ICMAT where this work was completed. The second author is partially supported by the PID2019-110224RB-I00 Grant of the MICINN (Spain

Orders of CM elliptic curves modulo p with at most two primes

We prove the analog of Koblitz conjecture when replacing primes by almost prime numbers and considering elliptic curves with complex multiplication. In other words for infinitely many primes $p$, (given in a quantitative way describe in the paper), the reduction of an elliptic curve $E$ with complex multiplication has order of size an almost prime number

Extendable orthogonal sets of integral vectors

Motivated by a model in quantum computation, we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the quaternions and other hypercomplex numbersFernando Chamizo is partially supported by the PID2020-113350GB-I00 Grant of the MICINN (Spain) and by “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). This latter grant supported the visit of Jorge Jiménez-Urroz to the ICMAT where this work was completed. He is partially supported by the PID2019-110224RB-I00 Grant of the MICINN (Spain

A study of the separating property in Reed-Solomon codes by bounding the minimum distance

The version of record is available online at: http://dx.doi.org/10.1007/s10623-021-00988-zAccording to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.This work has been supported by the Spanish Government Grant TCO-RISEBLOCK (PID2019-110224RB-I00) MINECO .Peer ReviewedPostprint (published version

Primes represented by quadratic polynomials via exceptional characters

This is a post-peer-review, pre-copyedit version of an article published in Archiv der Mathematik. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00013-021-01602-3We estimate the number of primes represented by a general quadratic polynomial with discriminant ¿, assuming that the corresponding real character is exceptional.Peer ReviewedPreprin

Modular forms with large coefficient fields via congruences

We use the theory of congruences between modular forms to prove the existence of newforms with square-free level having a fixed number of prime factors such that the degree of their coefficient fields is arbitrarily large. We also prove a similar result for certain almost square-free levels.Peer ReviewedPostprint (published version

Cropping Euler factors of modular L-functions

According to the Birch and Swinnerton-Dyer conjectures, if A/Q is an abelian variety, then its L-function must capture a substantial part of the properties of A. The smallest number field L where A has all its endomorphisms defined must also play a role. This article deals with the relationship between these two objects in the specific case of modular abelian varieties Af =Q associated to weight 2 newforms for the group t1(N). Specifically, our goal is to relate ords=1 L(Af =Q, s), with the order at s D 1 of Euler products restricted to primes that split completely in L. This is attained when a power of Af is isogenous over Q to the Weil restriction of the building block of Af . We give separated formulae for the CM and non-CM cases.Postprint (published version

Revisiting cycles of pairing-friendly elliptic curves

A recent area of interest in cryptography is recursive composition of proof systems. One of the approaches to make recursive composition efficient involves cycles of pairing-friendly elliptic curves of prime order. However, known constructions have very low embedding degrees. This entails large parameter sizes, which makes the overall system inefficient. In this paper, we explore $2$-cycles composed of curves from families parameterized by polynomials, and show that such cycles do not exist unless a strong condition holds. As a consequence, we prove that no $2$-cycles can arise from the known families, except for those cycles already known. Additionally, we show some general properties about cycles, and provide a detailed computation on the density of pairing-friendly cycles among all cycles

Guía de gestión integrada de plagas: mango

Coordinadores: Ángel Martín Gil y Gregoria Aranda Aranda