Last edited by Mojin
Friday, May 1, 2020 | History

4 edition of arithmetic Riemann-Roch theorem for singular arithmetic surfaces found in the catalog.

arithmetic Riemann-Roch theorem for singular arithmetic surfaces

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  • 4 Currently reading

Published by American Mathematical Society in Providence, R.I .
Written in English

    Subjects:
  • Arithmetical algebraic geometry.,
  • Riemann-Roch theorems.

  • Edition Notes

    StatementWayne Aitken.
    SeriesMemoirs of the American Mathematical Society,, no. 573
    Classifications
    LC ClassificationsQA3 .A57 no. 573, QA242.5 .A57 no. 573
    The Physical Object
    Paginationviii, 174 p. ;
    Number of Pages174
    ID Numbers
    Open LibraryOL814792M
    ISBN 100821804073
    LC Control Number95052304

    I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of . In arithmetic geometry, the Mordell conjecture is the conjecture made by Mordell () that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational it was proved by Gerd Faltings (, ), and is now known as Faltings's conjecture was later generalized by replacing Q by any number fieldConjectured by: Louis Mordell.


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arithmetic Riemann-Roch theorem for singular arithmetic surfaces by Wayne Aitken Download PDF EPUB FB2

Finally, the author defines an intersection theory for arithmetic surfaces that includes a large class of singular arithmetic surfaces. This culminates in a proof of the arithmetic Riemann-Roch theorem.

Destination page number Search scope Search Text. The Riemann-Roch theorem has come a long way since its origins in the work of Bernhard Riemann years ago. Riemann was attempting to establish the existence of complex functions on multiply-connected surfaces with no by: An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces, Wayne Aitken, Memoirs of the AMS, Vol /, Seminaire de Theorie des Nombres (15th:Paris), Ed.

David, CUP Theory of Algebraic Integers by Richard Dedekind, Translated by John Stillwell, CUP This has facilitated the generalization of certain results, notably the Riemann-Roch theorem, to arithmetic varieties [7].

The Arakelov theory involves a particular choice of metric, leading to a. [Vo] Vojta, P.: Siegel's Theorem in the compact case. Ann. Math, – () Google Scholar [Z] Zhang, S.: Ample Hermitian line bundles on arithmetic by: In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order.

Our proof of the arithmetic Riemann-Roch theorem combines the classical technique of proof of the Grothendieck-Riemann-Roch theorem with deep re-sults of Bismut and his coworkers in local index theory.

The history of the previous work on this theorem and its variants is as follows. In [9] Faltings proved a variant of the theorem for surfaces. An arithmetic Hilbert–Samuel theorem for singular hermitian line bundles and cusp forms - Volume Issue 10 - Robert J.

Berman, Gerard Freixas i MontpletCited by: 6. For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces combined with the Noether formula. To see this, recall that for each divisor D on a surface there is an invertible sheaf L = O(D) such that the linear Field: Algebraic geometry.

The generalization of the Grothendieck-Riemann-Roch theorem to Arakelov geometry. References. Henri Gillet, Damian Rössler, Christophe Soulé, An arithmetic Riemann-Roch theorem in higher. degrees_ () pdf. of the classical Riemann-Roch theorem on Riemann surfaces and the Hirzebruch-Riemann-Roch theorem on compact complex manifolds.

In their development of arithmetic intersection theory, Gillet and Soul e were lead to extend the Grothendieck-Riemann-Roch theorem to the context of arith-metic.

The desingularization also allows a definition of the genus for singular curves, and the author devotes a few sentences alerting the reader to the fact that the Riemann-Roch theorem does hold for singular curves if the geometric genus is replaced by the arithmetic genus.5/5(3).

Atsushi Moriwaki, Numerical characterization of nef arithmetic divisors on arithmetic surfaces Henri Gillet, Damian Rössler, Christophe Soulé, An arithmetic Riemann-Roch theorem in higher degrees Damien Rössler, PréfaceCited by: The theorem is new even in the case that L is ample.

It is not a direct consequence of the arithmetic Riemann–Roch theorem of Gillet and Soul´e, due to difficulties on effectively estimating the analytic torsion and the contribution of H1(L). Theorem B is a special case of Theorem A under slightly weaker.

An arithmetic Riemann-Roch theorem for pointed stable curves ating families of compact surfaces. By a theorem of Burger [7, Th. ] we can replace the small eigenvalues by the lengths of the pinching geodesics.

Then Wolpert’s pinching expansion of the family hyperbolic metric [63, Exp. Topics covered include existence of meromorphic functions, the Riemann-Roch theorem, Abel's theorem, the Jacobi inversion problem, Noether's theorem, and the Riemann vanishing theorem.

A complete treatment of the uniformization of Riemann sufaces via Fuchsian groups, including branched coverings, is presented, as are alternate proofs for the 5/5(1).

An arithmetic Riemann-Roch theorem for pointed stable curves Article in Annales Scientifiques de l École Normale Supérieure 42(2) November with 15 Reads How we measure 'reads'.

The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry.

In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

Field: Algebraic geometry. Genre/Form: Electronic books: Additional Physical Format: Print version: Aitken, Wayne, Arithmetic Riemann-Roch theorem for singular arithmetic surfaces /.

We wish to prove the Grothendieck-Riemann-Roch theorem for non-singular quasi-projective varieties. This requires a great deal of preparatory theory: the construction of the Chow ring, a discussion of characteristic classes, and developing a K-theory for schemes.

After dispensing with these requisite components, we will prove the theorem, and Author: Thesis Advisor, Igor A. Rapinchuk. Riemann-Roch theorem for singular curves.

Ask Question Asked 6 years, 2 months ago. Ex S of Vakil's book. Riemann-Roch theorem on surfaces as generalization of Riemann-Roch on curves.

Adjunction formula and Riemann-Roch for surfaces. adelic Riemann–Roch theorem, which is also direct and relatively short. Combining with the relation between adelic and Zariski cohomology groups, this also implies the Riemann–Roch theorem for surfaces.

The adelic point of view in one-dimensional algebraic and arithmetic geometry not only. This development culminates in an arithmetic Riemann-Roch theorem for such arithmetic surfaces. The first part of the memoir gives a treatment of Deligne's functorial intersection theory, and the second develops a class of intersection functions for singular curves which behaves analogously to the canonical Green's functions introduced by Arakelov for smooth curves.

I found the famous Faltings book ``Lectures on arithmetic Riemann-Roch theorem". In the book, very analytic techniques such as Garding inequality or heat kernel are explained.

I have no idea where such analytic tools must come in to prove algebraic theorem. Get this from a library.

An arithmetic Riemann-Roch theorem for singular arithmetic surfaces. [Wayne Aitken]. The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the : Gerd Faltings.

The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof.

AN ARITHMETIC HILBERT-SAMUEL THEOREM FOR SINGULAR HERMITIAN LINE BUNDLES AND CUSP FORMS ROBERT BERMAN AND GERARD FREIXAS I MONTPLET Abstract. We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In partic-ular, the theorem applies to the log-singular metrics of Burgos-Kramer-Ku¨hn.

This book proves a Riemann-Roch theorem for arithmetic varieties, and the author does so via the formalism of Dirac operators and consequently that of heat kernels.

In the first lecture the reader will see the "classical" Riemann-Roch theorem in an even more general context then that mentioned above: that of smooth morphisms of regular schemes.4/5. This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves.

The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem.

A theorem expressing the Euler characteristic of a locally free sheaf on an algebraic or analytic variety in terms of the characteristic Chern classes of and (cf. Chern class).It can be used to calculate the dimension of the space of sections of (the Riemann–Roch problem).

The classical Riemann–Roch theorem relates to the case of non-singular algebraic curves and states that for any.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): is to develop and apply results bounding the height in finite coverings of a fixed arithmetic variety with a fixed branch locus.

It is to be foreseen that an important tool in the study of such coverings will be (a suitable version of) the arithmetic Riemann-Roch theorem. The theorem of Riemann-Roch and Abel’s theorem could be interpreted as answering the question: for which configuration of charges, dipoles, or multipoles on a compact Riemann surface of genus ≥ 1 would the flux functions (whose level curves are the flux lines and which are the harmonic conjugates of the electrostatic potential functions) in.

The first time I encountered the Riemann-Roch theorem was in Fulton's Algebraic Curves. The proof in this book is pretty mechanical and leaves much to be desired. The issue with the proof is that it doesn't really explain the how, as far as proofs go.

You will probably hear this more than once in your math career, but the truest way to. the arithmetic Riemann-Roch theorem of Gillet-Soul´e in its more general form [17].

The details will be worked out elsewhere. In arithmetic applications a weaker form of Theorem A may be sometimes enough: an arithmetic Hilbert-Samuel type formula. As an advantage, it provides a geometric interpretation of the arithmetic self-intersection number.

This new-in-paperback edition provides a general introduction to algebraic and arithmetic geometry, starting with the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem).5/5(1).

The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative arithmetic surfaces and we prove an arithmetic Riemann-Roch theorem in this setup.

In this paper, we give a new proof of an arithmetic analogue of the Riemann-Roch Theorem, due originally to Serge Lang. Lang's result was first proved using the lattice point geometry of Minkowski.

By contrast, our proof is completely adelic. It has the conceptual advantage that it uses a different analogue of the Riemann-Roch theorem proved by Tate in his thesis, in a manner similar to Author: Sam Mundy.

3. Arithmetic Chow groups and arithmetic Grothendieck–Riemann–Roch. In this section, which is intended primarily to set up notation, we briefly review the theory of arithmetic Chow groups of Gillet–Soulé, as well as the arithmetic Grothendieck–Riemann–Roch theorem; Author: Gerard Freixas i Montplet, Siddarth Sankaran.Riemann surfaces, Riemann-Roch?

Ask Question Asked 3 years, 2 months ago. Otherwise find out what text books is your course based on. $\endgroup$ – DonAntonio Feb 21 '17 at add a comment | 1 Answer Active Oldest Votes.

4 Riemann Roch theorem for surfaces. 0. Riemann Roch for Riemann Sphere.$\S$ The Riemann-Roch Theorem. The latter section gives a gentle exposition of the link between differential forms and their divisors before discussing RRT.

An upside of this book compared to Miranda's is the use of sheaf theoretic language, which I think is best introduced in a concrete context like the theory of Riemann surfaces.