This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, Washington. ...Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics. The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic K3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\mathbb F_p(t)$, when $p$ is prime and ...$r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb F_q(t^1/d)$.
For thirty years, the biennial international conference AGC$^2$T (Arithmetic, Geometry, Cryptography, and Coding Theory) has brought researchers to Marseille to build connections between arithmetic ...geometry and its applications, originally highlighting coding theory but more recently including cryptography and other areas as well.This volume contains the proceedings of the 16th international conference, held from June 19-23, 2017.The papers are original research articles covering a large range of topics, including weight enumerators for codes, function field analogs of the Brauer-Siegel theorem, the computation of cohomological invariants of curves, the trace distributions of algebraic groups, and applications of the computation of zeta functions of curves. Despite the varied topics, the papers share a common thread: the beautiful interplay between abstract theory and explicit results.
This volume contains the proceedings of the 11th International Conference on Finite Fields and their Applications (Fq11), held July 22-26, 2013, in Magdeburg, Germany.Finite Fields are fundamental ...structures in mathematics. They lead to interesting deep problems in number theory, play a major role in combinatorics and finite geometry, and have a vast amount of applications in computer science.Papers in this volume cover these aspects of finite fields as well as applications in coding theory and cryptography.
This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, ...Washington.Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics.The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic $K$3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
Diophantine geometry has been studied by number theorists for thousands of years, since the time of Pythagoras, and has continued to be a rich area of ideas such as Fermat's Last Theorem, and most ...recently the ABC conjecture. This monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The authors provide a clear path through the subject for graduate students and researchers. They have re-examined many results and much of the literature, and give a thorough account of several topics at a level not seen before in book form. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time. The book concludes with a comprehensive bibliography. It is destined to be a definitive reference on modern diophantine geometry, bringing a new standard of rigor and elegance to the field.
Abstract
This paper is a self-independent continuation of my article on Comparative geometry between the plane and the sphere that was presented at the previous edition of ICon-MaSTEd Conference ...2020. Below I discuss the possibility of adding a third geometry to the plane and the sphere, namely, the hyperbolic geometry on the hemisphere. I describe my own path to the subject, then the content of the syllabus which contains basic concepts of hyperbolic geometry for future preschool, elementary school, and secondary school teachers. Finally, I give reasons to introduce the subject into primary and secondary schools, not just for the “talented” but also for the “average” students.
This is the first of two volumes dedicated to the centennial of the distinguished mathematician Selim Grigorievich Krein. The companion volume is Contemporary Mathematics, Volume 734.Krein was a ...major contributor to functional analysis, operator theory, partial differential equations, fluid dynamics, and other areas, and the author of several influential monographs in these areas. He was a prolific teacher, graduating 83 Ph.D. students. Krein also created and ran, for many years, the annual Voronezh Winter Mathematical Schools, which significantly influenced mathematical life in the former Soviet Union.The articles contained in this volume are written by prominent mathematicians, former students and colleagues of Selim Krein, as well as lecturers and participants of Voronezh Winter Schools. They are devoted to a variety of contemporary problems in functional analysis, operator theory, several complex variables, topological dynamics, and algebraic, convex, and integral geometry.
The author constructs and studies a scheme theoretical version of the Tits vectorial building, relates it to filtrations on fiber functors, and uses them to clarify various constructions pertaining ...to affine Bruhat-Tits buildings, for which he also provides a Tannakian description.
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