The dilogarithm function in geometry and number theory pdf

Geometry number theory calculus probability basic mathematics logic classical mechanics electricity and magnetism computer science quantitative. Dilogarithm function for numeric and symbolic arguments. Dilogarithm identities, partitions and spectra in conformal field theory, i article pdf available january 1993 with 35 reads how we measure reads. Nonetheless, it has many intriguing properties and has appeared in various branches of mathematics and physics such as number theory the study of asymptotic behaviour of partitions, e. The dilogarithm function in geometry and number theory 5. Frontiers in number theory, physics, and geometry ii pdf. Algebra, geometry and number theory msc leiden university.

Introduction to modern number theory and arithmetic geometry. Relations between crystal bases, branching functions b. The result is an algebraic integral with parameters, and the physical problem of e cient evalua. Two related special functions are referred to as spences function, the dilogarithm itself. The dilogarithm function in geometry and string theory. Frontiers in number theory, physics, and geometry ii on. Pdf dilogarithm identities, partitions and spectra in. Bergman 6 introduction to the theory of algebraic functions of one variable, c. Linear operators and automorphic forms 241257 14 t. It will be observed that the notion of an open set is rather important here. Naturally some of the convex bodies involved can though in practice arent always too bad get somewhat complicated or defy simple computations of volume, which of course is a.

Compute the dilogarithm function for these numbers. Number theoryalgebraic geometry representation theory. Pdf doc del clipboard journal article mr20586182005g. Number theory immediately available upon purchase as print book shipments may be delayed due to the covid19 crisis. Rogers ramanujans type identities, asymptotic behavior of. The survey 33 is an excellent introduction to some aspects of this function. The research field number theory and geometry brings together people in the department with interests in arithmetic and various aspects of geometry, especially arithmetic and diophantine geometry. In analysis and number theory it is a quantum version of the veterm identity for the dilogarithm function, see 5, 21, 22 and references therein. The union of any number of open subsets is an open set. After a century of neglect the dilogarithm appeared twenty years ago in works of gabrielovgelfandlosik ggl on a combinatorial formula for the rst pontryagin class, bloch on k theory and regulators bl1 and wigner on lie groups. Number theory immediately available upon purchase as print book. The dilogarithm function in geometry and number theory1 the dilogarithm function is. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. The dilogarithm function for complex argument leonard c.

Frontiers in number theory, physics, and geometry ii. The notion of functions has been vastly generalized and their special values are the subject of the celebrated conjectures of birchswinnertondyer. The number of rational approximations to algebraic numbers and the number of solutions of norm form equations 232240. Essentially, a set xis open if every point xinside xhas points round it that are also in x.

There is a strong emphasis on algebraic methods and a direct interest in the computational aspects of the theory. We show that all of these equations follows from the case of functions of degree three and antisymmetry relation. The prime number theorem for arithmetic progressions ii 2 38 16. The dilogarithm function also appears in hyperbolic geometry, conformal eld theory and the theory of cluster algebras. Dilogarithm identities progress of theoretical physics.

Some exponential diophantine equations ii 258273 15. Don zagier, the dilogarithm function, in frontiers in number theory. The geometry of knot complements abhijit champanerkar. Zagier, the dilogarithm function in geometry and number theory, number theory and related topics, tata inst. Geometry and string theory the dilogarithm function. Contributions to automorphic forms, geometry, and number theory, 521560, johns hopkins univ. The algebra, geometry and number theory masters specialisation focuses on number theory in the broadest sense, ranging from discrete dynamical systems to arithmetic geometry and motives. Msri model theory, arithmetic geometry and number theory. Rogersramanujans type identities, asymptotic behavior of. Schilling 5 the kernel function and conformal mapping, s. K2 and lfunctions of elliptic curves euler in 1735 discovered that and dirichlet in 1839 proved that we begin by reinterpreting these sums as special values of functions of number fields. The dilogarithm function also appears in hyperbolic geometry.

Frontiers in number theory, physics, and geometry ii on conformal field theories, discrete groups and renormalization. In geometry the common value of the two sides is called the donaldsonthomas dt invariant associated with the a 2 quiver, or it is interpreted as the simplest wallcrossing for. Donaldson branes, calibrations and supergravity 79 jerome p. Zagier iii discuss the conjecture that the values at s m of the dedekind zeta function fs for an arbitrary number field f can be expressed in terms of values of d. Liu tong introduction to modern number theory and arithmetic geometry classical number theory. The dilogarithm function in geometry and number theory. The blochwignerramakrishnan polylogarithm function don zagier maxplanckinsfitut fiir mathematik, gottfriedclarenstrasse 26, d5300 bonn 3, federal republic of germany to hans grauert the polylogarithm function 9 co n appears in many parts of mathematics and has an extensive literature 2. The group organizes the number theory seminar and the annual number theory days, jointly with epf lausanne and university of basel. Number theory and geometry department of mathematics.

In this note, we give a survey of the in nitesimal version of the above theory. In recent years it has become much better known, due to its appearance in hyperbolic geometry and in algebraic k theory on the one hand and in. The polylogarithm function is an important function for integration, and finding seemingly complicated sum. Traditionally the dilogarithm function li2zis defined by the taylor series espansion. The geometry of knot complements city university of new york. Conformal field theory and torsion elements of the bloch group. Because these numbers are not symbolic objects, dilog returns floatingpoint results. Gauntlett mtheory on manifolds with exceptional holonomy 127 sergei gukov special holonomy and beyond 159 nigel hitchin. The inner structure of the dilogarithm in algebraic fields. Conformal field theory prime discrete groups number theory.

The dilogarithm function, defined in the first sentence of chapter i, is a function which has been known for more than 250 years, but which for a long time was familiar only to a few enthusiasts. In the padic analytic number theory of number elds, elliptic curves, and modular forms, \coleman maps provide the corresponding padic bridge. Connection between dilogarithm identities and algebraic ktheory torsion in k 3 r is discussed. Now enters geometry in 1980s, william thurstons seminal work established a strong connection between hyperbolic geometry and knot theory, namely that most knot complements are hyperbolic.

In fact, the lobachevsky function can be considered as an imaginary part of the dilogarithm function. The dilogarithm function for complex argument 2809 the transformation 3. Zagier, the dilogarithm function in geometry and number theory,number theory and related topics, tata inst. Polylogarithm is connected to the infinite geometric progression sum. Tata institute of fundamental research and oxford university press. Thurston introduced tools from hyperbolic geometry to study knots that led to new geometric invariants, especially hyperbolic volume. Number theory and geometry department of mathematics eth. Lewin department of electrical and computer engineering, university of colorado, boulder, colorado 80309 communicated by w. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. The appearance of the dilogarithm and logarithms at oneloop order is predictable from unitarity considerations since this reproduces the behaviour of the oneloop scattering amplitude under single, or double twoparticle cuts in four dimensions. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. How important is differential geometry for number theory. Douglas m theory, g 2manifolds and four dimensional physics 31 bobby s. In mathematics, spences function, or dilogarithm, denoted as li 2 z, is a particular case of the polylogarithm.

Minkowski theory is a rich area of number theory that uses geometry extensively to study solutions to diophantine problems. In number theory and algebraic geometry, london math. Geometry and string theory the dilogarithm function skip the navigation links home page all pages recently revised. The dilogarithm function in geometry and number theory 274. The intersection of a nite number of open subsets is an open set. Then we have fxconst lx we continue the function lx on all real axisr r1 f1g by the following rules.

Bloch that elliptic dilogarithm is subject of big bunch of socalled steinberg functional equation parametrized by rational functions on an elliptic curve. Journal of number theory 19, 345373 1984 the inner structure of the dilogarithm in algebraic fields l. Depending on its arguments, dilog returns floatingpoint or exact symbolic results. The prospect of a gon proof for ternary hasseminkowski. Larsens alternative, moments, and the monodromy of lefschetz pencils. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. 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.