The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,\omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.Comment: 41 page

A Modular Curve of Level 9 and the Class Number One Problem

In this note we give an explicit parametrization of the modular curve associated to the normalizer of a non-split Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem.Comment: 18 page

Torsion of rational elliptic curves over quadratic fields II

Let E be an elliptic curve defined over Q and let G=E(Q)_tors be the associated torsion group. In a previous paper, the authors studied, for a given G, which possible groups G\leq H could appear such that H=E(K)_tors, for [K:Q]=2. In the present paper, we go further in this study and compute, under this assumption and for every such G, all the possible situations where G\neq H. The result is optimal, as we also display examples for every situation we state as possible. As a consequence, the maximum number of quadratic number fields K such that E(Q)_tors\neq E(K)_tors is easily obtained.Ministerio de Economía y CompetitividadJunta de Andalucí

Chiral Four-Dimensional F-Theory Compactifications With SU(5) and Multiple U(1)-Factors

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP_2. We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)xU(1) and SU(5)xU(1)xU(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B=P^3. We determine the general G_4-flux that belongs to H^{(2,2)}_V of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G_4-flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New Chern-Simons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.Comment: 100 pages, 11 figures, 7 appendices; v3: minor changes requested by the referee, typos corrected, references added to the introductio

Checking complex networks indicators in search of singular episodes of the photochemical smog

A set of indicators derived from the analysis of complex networks have been introduced to identify singularities on a time series. To that end, the Visibility Graphs (VG) from three different signals related to photochemical smog (O3, NO2 concentration and temperature) have been computed. From the resulting complex network, the centrality parameters have been obtained and compared among them. Besides, they have been contrasted to two others that arise from a multifractal point of view, that have been widely used for singularity detection in many fields: the Hölder and singularity exponents (specially the first one of them). The outcomes show that the complex network indicators give equivalent results to those already tested, even exhibiting some advantages such as the unambiguity and the more selective results. This suggest a favorable position as supplementary sources of information when detecting singularities in several environmental variables, such as pollutant concentration or temperature

Accelerating zero knowledge proofs

Les proves de coneixement zero són una eina criptogràfica altament prometedora que permet demostrar que un predicat és correcte sense revelar informació addicional sobre aquest. Aquestes tipus de proves són útils en aplicacions que requereixen tant integritat computacional com privadesa, com ara verificar la correcció dels resultats d'una computació delegada a una altra entitat, on hi poden haver involucrats valors d'entrada confidencials. Tanmateix, té un impediment que obstaculitza la seva adopció pràctica: el procés potencialment lent de generació de les proves. Així doncs, aquest projecte explora la viabilitat d'accelerar les proves de coneixement zero mitjançant hardware, amb l'objectiu de superar aquest obstacle crític.Las pruebas de conocimiento cero representan una herramienta criptográfica altamente prometedora que permite demostrar la corrección de un predicado sin revelar información adicional. Estas pruebas son útiles en aplicaciones que requieren tanto integridad computacional como privacidad, como por ejemplo la validación de los resultados de una computación delegada a otra entidad, donde pueden estar involucrados valores de entrada confidenciales. Sin embargo, existe un desafío significativo que obstaculiza su adopción práctica: el proceso potencialmente lento de generación de pruebas. Como resultado, este proyecto explora la viabilidad de acelerar las pruebas de conocimiento cero utilizando hardware, con el objetivo de superar este obstáculo crítico.Zero-knowledge proofs represent a highly promising cryptographic tool that enables the validation of a statement's correctness without revealing any supplementary information. These proofs find utility in applications demanding both computational integrity and privacy, such as validating outsourced computation results, where confidential input values may be involved. However, a significant challenge hinders their practical adoption: the potentially time-consuming process of generating proofs. Consequently, this project investigates the feasibility of accelerating zero-knowledge proofs using hardware, aiming to overcome this critical hurdle.Outgoin

Doctor of Philosophy

dissertationAbstraction plays an important role in digital design, analysis, and verification, as it allows for the refinement of functions through different levels of conceptualization. This dissertation introduces a new method to compute a symbolic, canonical, word-level abstraction of the function implemented by a combinational logic circuit. This abstraction provides a representation of the function as a polynomial Z = F(A) over the Galois field F2k , expressed over the k-bit input to the circuit, A. This representation is easily utilized for formal verification (equivalence checking) of combinational circuits. The approach to abstraction is based upon concepts from commutative algebra and algebraic geometry, notably the Grobner basis theory. It is shown that the polynomial F(A) can be derived by computing a Grobner basis of the polynomials corresponding to the circuit, using a specific elimination term order based on the circuits topology. However, computing Grobner bases using elimination term orders is infeasible for large circuits. To overcome these limitations, this work introduces an efficient symbolic computation to derive the word-level polynomial. The presented algorithms exploit i) the structure of the circuit, ii) the properties of Grobner bases, iii) characteristics of Galois fields F2k , and iv) modern algorithms from symbolic computation. A custom abstraction tool is designed to efficiently implement the abstraction procedure. While the concept is applicable to any arbitrary combinational logic circuit, it is particularly powerful in verification and equivalence checking of hierarchical, custom designed and structurally dissimilar Galois field arithmetic circuits. In most applications, the field size and the datapath size k in the circuits is very large, up to 1024 bits. The proposed abstraction procedure can exploit the hierarchy of the given Galois field arithmetic circuits. Our experiments show that, using this approach, our tool can abstract and verify Galois field arithmetic circuits up to 1024 bits in size. Contemporary techniques fail to verify these types of circuits beyond 163 bits and cannot abstract a canonical representation beyond 32 bits

Computation of p-Adic Heights and Log Convergence

This paper is about computational and theoretical questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2 is log convergent.Mathematic