Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d

For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2 (Nishiyama), but the formulas for the general (E,P) were not known, nor was the fact that these are also the minima for an elliptic curve of discriminant degree 12n over a function field of any genus. For n=3 both the minimal height (23/840) and the explicit curves are new. These (E,P) also have the property that that mP is an integral point (a point of naive height zero) for each m=1,2,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the three cases.Comment: 15 pages; some lines in the TeX source are commented out with "%" to meet the 15-page limit for ANTS proceeding

Data-Driven Analysis of Pareto Set Topology

When and why can evolutionary multi-objective optimization (EMO) algorithms cover the entire Pareto set? That is a major concern for EMO researchers and practitioners. A recent theoretical study revealed that (roughly speaking) if the Pareto set forms a topological simplex (a curved line, a curved triangle, a curved tetrahedron, etc.), then decomposition-based EMO algorithms can cover the entire Pareto set. Usually, we cannot know the true Pareto set and have to estimate its topology by using the population of EMO algorithms during or after the runtime. This paper presents a data-driven approach to analyze the topology of the Pareto set. We give a theory of how to recognize the topology of the Pareto set from data and implement an algorithm to judge whether the true Pareto set may form a topological simplex or not. Numerical experiments show that the proposed method correctly recognizes the topology of high-dimensional Pareto sets within reasonable population size.Comment: 8 pages, accepted at GECCO'18 as a full pape

L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. The main application is that for every prime p and every integer g>0 there are absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD conjecture holds and which have arbitrarily large rank.Comment: To appear in Inventiones Mathematica

Classification of Singular Fibres on Rational Elliptic Surfaces in Characteristic Three

We determine and list all possible configurations of singular fibres on rational elliptic surfaces in characteristic three. In total, we find that 267 distinct configurations exist. This result complements Miranda and Persson's classification in characteristic zero, and Lang's classification in characteristic two.Comment: 40 Pages. Minor typos correcte

Produção de sementes de coentro em função dos tipos de adubação.

O objetivo deste trabalho foi avaliar a influência de tipos de adubo na produção de sementes de coentro (Coriandrum sativum L.) nas condições de Botucatu-SP. Utilizou-se sementes Aglofora S.A, cv. Português

Magnetic-Field-Induced Antiferromagnetism in Two-Dimensional Hubbard Model: Analysis of CeRhIn5_5

We propose the mechanism for the magnetic-field-induced antiferromagnetic (AFM) state in a two-dimensional Hubbard model in the vicinity of the AFM quantum critical point (QCP), using the fluctuation-exchange (FLEX) approximation by taking the Zeeman energy due to the magnetic field $B$ into account. In the vicinity of the QCP, we find that the AFM correlation perpendicular to $B$ is enhanced, whereas that parallel to $B$ is reduced. This fact means that the finite magnetic field increases $T_N$, with the AFM order perpendicular to $B$. The increment in $T_N$ can be understood in terms of the reduction of both quantum and thermal fluctuations due to the magnetic field, which is caused by the self-energy effect within the FLEX approximation. The present study naturally explains the increment in $T_N$ in CeRhIn_5 under the magnetic field found recently.Comment: 5 page

Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19

It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence subgroups of quaternion algebras over Q. In a family of such K3 surfaces, a surface has rho=20 if and only if it corresponds to a CM point on X. We use this to compute equations for Shimura curves, natural maps between them, and CM coordinates well beyond what could be done by working with the curves directly as we did in ``Shimura Curve Computations'' (1998) = Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To appear in the proceedings of ANTS-VIII, Banff, May 200

Non-liftable Calabi-Yau spaces

We construct many new non-liftable three-dimensional Calabi-Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi-Yau space over some number field as introduced by the first author and D. van Straten.Comment: 16 pages, 5 tables; v2: minor corrections and addition

F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds

The Mordell-Weil group of an elliptically fibered Calabi-Yau threefold X contains information about the abelian sector of the six-dimensional theory obtained by compactifying F-theory on X. After examining features of the abelian anomaly coefficient matrix and U(1) charge quantization conditions of general F-theory vacua, we study Calabi-Yau threefolds with Mordell-Weil rank-one as a first step towards understanding the features of the Mordell-Weil group of threefolds in more detail. In particular, we generate an interesting class of F-theory models with U(1) gauge symmetry that have matter with both charges 1 and 2. The anomaly equations --- which relate the Neron-Tate height of a section to intersection numbers between the section and fibral rational curves of the manifold --- serve as an important tool in our analysis.Comment: 29 pages + appendices, 5 figures; v2: minor correction