First, i will describe joint works with anand deopurkar that describes the geometry of the moduli of almost k3 stable log surfaces degenerations of smooth quadrics with a genus 4 curve by relating to the moduli of curves of genus 4. Homogeneous flows, moduli spaces and arithmetic clay. That is, the points of a moduli space correspond to solutions of geometric problems. On arithmetic curves in the moduli spaces of curves. This setting provides new geometric tools for interpreting elements of the brauer group. Elliptic modular curves to study points of order non elliptic curves, one looks at the set of pairs y1n. The broad themes of this conference are hodge theory, algebraic cycles. Moduli spaces and arithmetic geometry from 9 nov 2015 through nov 2015.
Msri dynamics on moduli spaces of geometric structures. In arithmetic geometry, rather than studying one variety, it is fruitful to look at the space of all varieties. Geometry and arithmetic of moduli spaces of coverings 2008 and geometry and arithmetic around galois theory 2009. Moduli spaces and arithmetic geometry, advanced studies pure mathamatics, vol. Even within algebraic geometry, this discussion leaves out important recent work of farkas, gibney. Lorentz center moduli spaces and arithmetic geometry from 9. In particular, we obtain the notion of arithmetic degree dege. With this close relation in mind, the rims conference moduli spaces and arithmetic geometry was held at kyoto university during september 815, 2004 as the th international research institute of the mathematical society. Sometimes the analogy is quite direct, sometimes less so, and there are parts of arithmetic geometry that still lack dynamical analogues. Icerm cycles on moduli spaces, geometric invariant theory.
Source moduli spaces and arithmetic geometry kyoto, 2004, s. Actually the intermediate step, namely from analytic to algebraic geometry, will turn out to be a source of beautiful and difficult problems. The contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. If a moduli problem can be expressed in terms of algebraic geometry then in many. Saito, moduli of stable parabolic connections, riemannhilbert correspondence and geometry of painleve equation of type vi, ii in moduli spaces and arithmetic geometry, adv. Arithmetic properties of moduli spaces and topological string partition functions of some calabiyau threefolds. For example, we compute picard group, chow ringand chern classes ofvector bundles over spec\z, and construct the moduli space of such vector bundles. In this program we aim to bring together key researchers in these and related areas to highlight the recent exciting progress and to explore future.
As is well known, y1n has a natural structure as an a ne curve, and e. If one can show that a collection of interesting objects e. Moduli spaces in topology are often referred to as classifying spaces. Moduli spaces and arithmetic geometry advanced studies. New approach to arakelov geometry nikolai durov february 1, 2008. In 2001, the proof of the local langlands conjectures for gl n was based on the geometry of certain shimura varieties. In particular, it covers the geometry of homogeneous varieties, the delignemumford moduli spaces of stable curves and the kontsevich moduli spaces of stable maps using intersection theory. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. Effective geometry and arithmetic of curves and their moduli. Such spaces frequently arise as solutions to classification problems. Instances of these moduli spaces are the ones related to some versions of the fundamental lemma. Eis an elliptic curve, and p2etors is a point of order n. It is an expanded version of the notes for a series of lectures delivered at a workshop on moduli spaces and the arithmetic of dynamical systems at the bellairs research institute, barbados, in 2010.
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic of moduli stack of curves department of mathematics. Canonical metrics in the moduli spaces of riemann surfaces ii. There are many very famous classical metrics on the teichmu. Shingtung yau geometry of the moduli space of curves. Deformation theory, patching, quadratic forms, and the brauer group aimpl. Periods, moduli spaces and arithmetic of algebraic varieties school in arithmetic geometry. Highlights included a complete proof of gottsches conjecture, a proof of rationality of a moduli space constructed via git quotient using reduction modulo p, and a. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected by eichlershimura, igusa, and delignerapoport. And lastly, i do take your point that most moduli spaces we care about are quot or hilbert schemes. This led in particular to the creation of grothendieckteichmuller theory.
Arithmetic of moduli stack of curves department of. Lorentz center moduli spaces and arithmetic geometry. Arithmetic geometry on the moduli space of algebraic curves. The authors goal is to provide an overview, with enough details and pointers to the existing literature, to give. This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. With a view towards future arithmetic applications, the. Effective geometry and arithmetic of curves and their moduli spaces. More precisely, one studies the space of isomorphism classes of varieties having a speci ed structure, i.
Minicourses by fabrizio andreatta, ana caraiani and eugen hellmann. The resulting moduli spaces pervade geometry of many sortsdi. Everything i have said is said more eloquently in those references. Arithmetic dynamics and dynamics on moduli spaces mathoverflow. Arithmetic geometry and number theory series on number. A moduli space parameterizes geometric objects with alike structures and encodes in itself the geometry of all possible families of such objects. Pdf document information annals of mathematics fine hall washington road princeton university princeton, nj 08544, usa phone. C4 be the epsmoduli space of the twisted cubics in p3 con structedin1. The geometry of some special arithmetic quotients book also available for read online, mobi, docx and mobile and kindle reading. Examples of the dynamics considered are the action of. Geometry of moduli spaces and representation theory. Here different solutions are identified if they are isomorphic that is, geometrically the same. Most of the results can be found in 5, 6, 7 and 18. Jacobi cohomology, local geometry of moduli spaces, and hitchin.
The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Brauer groups and obstruction problems moduli spaces and. An introduction to moduli spaces of curves and its. Mathematical society of japan, 2006, dates received. This lecture notes volume is the fruit of two researchlevel summer schools jointly organized by the gtem node at lille university and the team of galatasaray university istanbul. Arithmetic properties of moduli spaces and topological. Fundamental ofs teichmuller and moduli spaces we briefly recall the fundamental theory of the geometry of teichmiiller and moduli spaces of hyperbolic riemann surfaces in this section. But despite their ubiquity and, in some cases, our substantial understanding. These arithmetic surfaces are moduli stacks over specz. We usually assume that g 2 so that each riemann surface gadmits a canonical metric conformal to the complex structure, and g is considered as a hyperbolic metric so that hyperbolic geometry can be used. Lastly, since representation theory is a mathematical formulation of the notion of symmetry, the last two articles are devoted to some aspects of enumerative algebraic geometry that are touched upon by geometric representation theory.
These are notes for a series of ve lectures on \moduli and degenerations of algebraic curves via tropical geometry delivered at the cimpacimatictp school on moduli of curves, february 29march 4, 2016 in guanajuato, mexico. Moduli spaces are spaces of solutions of geometric classification problems. Popp, moduli theory and classification theory of algebraic varieties, springer 1977 mr0466143 zbl 0359. With this close relation in mind, the rims conference moduli spaces and arithmetic geometry was held at kyoto university during september 815, 2004 as the th international research institute of. Moduli spaces of covers constitute an appropriate tool for certain arithmetic problems involving algebraic curves and rational functions.
Topology and geometry of the moduli space of curves pdf. Pdf download the geometry of some special arithmetic. A coherent system on c consists of a pair e,v, where e is an algebraic vector bundle over c of rank n and. An explicit construction usually gives deep information on the arithmetic and geometry of x. This workshop will focus on three aspects of moduli spaces. Homogeneous flows, moduli spaces and arithmetic, pisa, italy. Geometric aspects of the moduli spaces of riemann surfaces. Room 420, research institute for mathematical sciences. Since their inception in, hitchin systems have been a rich source of examples of algebraically completely integrable systems. Since you are interest in perfectoid moduli spaces, i suggest we follow scholze iii. To read about moduli of sheaves, i recommend the the geometry of moduli spaces of sheaves by huybrechts and lehn their preliminary section on quot schemes is good too. Let mg,k te the moduli space of riemann surfaces of genus g with k punctures. More concretely a cover consists of an irreducible curve xde. In the 2010s, peter scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over padic fields with application to galois representations and certain cases of the weightmonodromy conjecture.
Moduli spaces for dynamical systems brown university. The th mathematical society of japan international research institute moduli spaces and arithmetic geometry date. Arithmetic geometry is centered around diophantine geometry, the study of rational points of algebraic varieties in more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. We apply the above notions to arakelov geometry as well. This monograph studies moduli problems associated to algebraic dynamical systems. The text will be of interest to all involved in the study of moduli spaces with symmetries, and contains in addition a. So we first have to agree on some uniform setting for your questions. One of the most wellknown examples is when xis an elliptic curve over a number eld, one can construct the heegner point on x.
Geometric topology of moduli spaces of curves lizhen ji university of michigan uc irvine january 16, 2012. Their curvatures and boundary behaviors are studied in details. The aim of this talk is to describe some of the moduli spaces that come up in dynamics, and to see how. Arithmetic and geometry around galois theory pierre. Vector bundles on curves over x p a werner absolute cmperiods complex and padic h yoshida special zeta values in positive characteristic j yu automorphic forms, eisenstein series and spectral decompositions l weng geometric arithmetic. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Lectures on tropical curves and their moduli spaces melody chan abstract. Concerning the geometry of moduli spaces of sheaves of higher rank, there. Moduli spaces in algebraic geometry 5 abstracts polarized k3 surfaces of gnenus 16 shigeru mukai let t g2,3. Download the geometry of some special arithmetic quotients in pdf and epub formats for free.
Deligne pairings over moduli spaces of punctured riemann surfaces k obitsu et al. Mock modular forms in combinatorics and arithmetic geometry pdf. In these notes, we present a connection between the complex dynamics of a family of rational functions f t. Seshadri, spaces of unitary vector bundles on a compact riemann surface ann. Symplectic geometry of a moduli space of framed higgs bundles. Polygons in symmetric spaces and euclidean buildings, geometry and functional analysis 192009. Convex functions on symmetric spaces and geometric invariant theory for spaces of weighted configurations on flag manifolds, journal of diff, geom. Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Particularly the modern study of moduli spaces and arithmetic geometry have many important techniques and ideas in common. The key point is the understanding of the ricci and the perturbed ricci metric.
Nevertheless, something is known about the type of singularities they can attain. These are notes for a series of ve lectures on \ moduli and degenerations of algebraic curves via tropical geometry delivered at the cimpacimatictp school on moduli of curves, february 29march 4, 2016 in guanajuato, mexico. Arithmetic dynamics, arithmetic geometry, and number theory. The school was designed to serve as a comprehensive introduction to the theory of. The modern study of moduli spaces and arithmetic geometry demonstrates that these two areas have many important techniques and ideas in common. Moduli spaces and arithmetic dynamics, crm monograph series 30, ams, 2012. In mathematics, in particular algebraic geometry, a moduli space is a geometric space usually a scheme or an algebraic stack whose points represent algebrogeometric objects of some fixed kind, or isomorphism classes of such objects. Effective geometry and arithmetic of curves and their. Moduli spaces of sheaves of rank 2 on a surface are not smooth, unless we consider sheaves with special invariants on special surfaces.
Spaces and topological string partition functions of some calabiyau threefolds the harvard community has made this article openly available. The following papers of joe with mumford and eisenbud developed. Birational geometry and moduli spaces are two important areas of algebraic geometry that have recently witnessed a flurry of activity and substantial progress on many fundamental open questions. Moduli spaces can be thought of as giving a universal space of parameters for the problem. Tyurin, the geometry of moduli of vector bundles russian math. The arithmetic of dynamical systems, springerverlag, gtm 241, 2007, theres also the following monograph that discusses dynamicalrelated moduli spaces from an algebraic and arithmetic viewpoint. P1, parameterized by t in a riemann surface x, and the arithmetic dynamics of f t on rational points p1k where k cx or qx.
Cohomology of local systems on moduli spaces of curves and of abelian varieties s. Recent results on arithmetic geometry of the moduli space of algebraic curves. We introduce moduli spaces of smooth and stable curves, the tauto. Minicourses by fabrizio andreatta, ana caraiani and eugen hellmann, plus a couple of research talks. The moduli spaces of curves, via the arithmetic of the knudsenmumford stratification and their etale fundamental groups representation, have a long tradition in the study of the absolute galois group of rationals. Arithmetic dynamics and arithmetic geometry 8 arithmetic geometry. Moduli spaces in algebraic geometry mfo, oberwolfach. Moduli spaces of elliptic curve and abelian varieties geometry of moduli spaces, e.
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