Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

General Topology-I

The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and 'elementary' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices - on analysis, R-trees, and Berkovich's general theory of analytic spaces - are included to make the book as self-contained as possible. The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of p-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szego theorem and Bilu's equidistribution theorem.

Potential Theory and Dynamics on the Berkovich Projective Line

Let $\mathcal{O}$
 
be an orbit in ℤ
 n
 of a finitely generated subgroup Λ of GL
 n
 (ℤ) whose Zariski closure Zcl(Λ) is suitably large (eg isomorphic to SL2) We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$
 
at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$
 
 This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free

https://link.springer.com/content/pdf/10.1007%2Fs00222-009-0225-3.pdf

Affine linear sieve, expanders, and sum-product

I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil theorem XI. Faltings theorem XII. The ABC-conjecture XIII. Nevanlinna theory XIV. The Vojta conjectures Appendix A. Algebraic geometry Appendix B. Ramification Appendix C. Geometry of numbers Bibliography Glossary of notation Index.

/pdf/heights-in-diophantine-geometry-2nedf9b64d.pdf

Heights in Diophantine Geometry

Generalizing the construction from tropical algebraic geometry, we associate to every (irreducible d-dimensional) closed analytic subvariety of $\mathbb{G}_{m}^{n}$
 a tropical variety in ℝ
 n
 with respect to a complete non-archimedean place. By methods of analytic and formal geometry, we prove that the tropical variety is a totally concave locally finite union of d-dimensional polytopes. For an algebraic morphism f:X’→A to a totally degenerate abelian variety A, we give a bound for the dimension of f(X’) in terms of the singularities of a strictly semistable model of X’. A closed d-dimensional subvariety X of A induces a periodic tropical variety. A generalization of Mumford’s construction yields models of X and A which can be handled with the theory of toric varieties. For a canonically metrized line bundle L on A, the measures c
 1(L|
 X
 )∧d
 are piecewise Haar measures on X. Using methods of convex geometry, we give an explicit description of these measures in terms of tropical geometry. In a subsequent paper, this is a key step in the proof of Bogomolov’s conjecture for totally degenerate abelian varieties over function fields.

/pdf/tropical-varieties-for-non-archimedean-analytic-spaces-1dakopffkk.pdf

Tropical varieties for non-archimedean analytic spaces

Local heights of subvarieties over non-archimedean fields

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean valued elds. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis. MSC2010: 14T05, 14M25, 32P05 X) of polyhedra in R n . This process is called tropicaliza- tion and it can be used to transform a problem from algebraic geometry into a corresponding problem in convex geometry which is usually easier. If the toric co- ordinates are well suited to the problem, it is sometimes possible to use a solution of the convex problem to solve the original algebraic problem. Another strategy is to vary the ambient torus to compensate the loss of information due to the tropi- calization process. Tropicalization originates from a paper of Bergman (Berg) on logarithmic limit sets. The convex structure of the tropical variety Trop(X) was worked out by Bieri{Groves (BG) with applications to geometric group theory in mind. Sturmfels (Stu) pointed out that Trop(X) is a subcomplex of the Grobner complex. In fact, the polyhedral complex Trop(X) has some natural weights satisfying a balancing condition which appears rst in Speyer's thesis (Spe). This relies on the description

/pdf/a-guide-to-tropicalizations-3m15nutkca.pdf

A guide to tropicalizations

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang’s proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place v. As a corollary, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.

/pdf/the-bogomolov-conjecture-for-totally-degenerate-abelian-4j45ldlks6.pdf

Walter Gubler

Papers

Heights in Diophantine Geometry

Tropical varieties for non-archimedean analytic spaces

Local heights of subvarieties over non-archimedean fields

A guide to tropicalizations

The Bogomolov conjecture for totally degenerate abelian varieties