Group Theory

Download E-books Theta Constants, Riemann Surfaces and the Modular Group PDF

There are exceptionally wealthy connections among classical research and quantity concept. for example, analytic quantity concept includes many examples of asymptotic expressions derived from estimates for analytic features, akin to within the evidence of the top quantity Theorem. In combinatorial quantity conception, targeted formulation for number-theoretic amounts are derived from kinfolk among analytic services. Elliptic features, specifically theta services, are a tremendous classification of such capabilities during this context, which have been made transparent already in Jacobi's Fundamenta nova. Theta services also are classically hooked up with Riemann surfaces and with the modular team $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which supply one other course for insights into quantity conception. Farkas and Kra, recognized masters of the idea of Riemann surfaces and the research of theta features, discover right here attention-grabbing combinatorial identities via the functionality concept on Riemann surfaces concerning the crucial congruence subgroups $\Gamma(k)$. for example, the authors use this method of derive congruences came across via Ramanujan for the partition functionality, with the most element being the development of an analogous functionality in additional than a technique. The authors additionally receive a variation on Jacobi's well-known consequence at the variety of ways in which an integer may be represented as a sum of 4 squares, exchanging the squares by way of triangular numbers and, within the procedure, acquiring a purifier outcome. the new development of using the guidelines and techniques of algebraic geometry to the learn of theta services and quantity conception has ended in nice advances within the zone. although, the authors decide to stick with the classical viewpoint. for this reason, their statements and proofs are very concrete. during this publication the mathematician acquainted with the algebraic geometry method of theta services and quantity conception will locate many fascinating principles in addition to unique motives and derivations of latest and previous effects. Highlights of the booklet contain systematic stories of theta consistent identities, uniformizations of surfaces represented by means of subgroups of the modular staff, partition identities, and Fourier coefficients of automorphic capabilities. must haves are a fantastic figuring out of complicated research, a few familiarity with Riemann surfaces, Fuchsian teams, and elliptic services, and an curiosity in quantity idea. The e-book comprises summaries of a few of the mandatory fabric, really for theta capabilities and theta constants. Readers will locate the following a cautious exposition of a classical perspective of research and quantity conception. provided are quite a few examples plus feedback for research-level difficulties. The textual content is appropriate for a graduate path or for self reliant examining.

Show description

Read Online or Download Theta Constants, Riemann Surfaces and the Modular Group PDF

Similar Group Theory books

Statistical Applications of Jordan Algebras (Lecture Notes in Statistics)

This monograph brings jointly my paintings in mathematical facts as i've got seen it throughout the lens of Jordan algebras. 3 technical domain names are to be visible: purposes to random quadratic kinds (sums of squares), the research of algebraic simplifications of maxi­ mum chance estimation of patterned covariance matrices, and a extra huge­ open mathematical exploration of the algebraic enviornment from which i've got drawn the implications utilized in the statistical difficulties simply pointed out.

Representation Theory: A First Course (Graduate Texts in Mathematics)

The first target of those lectures is to introduce a newbie to the finite­ dimensional representations of Lie teams and Lie algebras. on the grounds that this objective is shared through numerous different books, we should always clarify during this Preface how our technique differs, even though the capability reader can most likely see this larger through a brief flick through the ebook.

Group Theory and Its Application to Physical Problems (Dover Books on Physics)

"A remarkably intelligible survey . . . good geared up, good written and extremely transparent all through. " — Mathematical ReviewsThis first-class textual content, lengthy certainly one of the best-written, so much skillful expositions of crew concept and its actual functions, is directed essentially to complex undergraduate and graduate scholars in physics, in particular quantum physics.

Extra resources for Theta Constants, Riemann Surfaces and the Modular Group

Show sample text content

Rated 4.44 of 5 – based on 16 votes