### Download Theta Constants, Riemann Surfaces and the Modular Group by Hershel M. Farkas and Irwin Kra PDF

By Hershel M. Farkas and Irwin Kra

There are exceedingly wealthy connections among classical research and quantity idea. for example, analytic quantity conception includes many examples of asymptotic expressions derived from estimates for analytic capabilities, akin to within the evidence of the top quantity Theorem. In combinatorial quantity conception, designated formulation for number-theoretic amounts are derived from kin among analytic features. Elliptic capabilities, in particular theta services, are a big type of such services during this context, which have been made transparent already in Jacobi's Fundamenta nova. Theta features also are classically attached with Riemann surfaces and with the modular crew $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which supply one other direction for insights into quantity conception. Farkas and Kra, famous masters of the speculation of Riemann surfaces and the research of theta services, discover right here fascinating combinatorial identities via the functionality idea on Riemann surfaces with regards to the valuable congruence subgroups $\Gamma(k)$. for example, the authors use this method of derive congruences stumbled on by means of Ramanujan for the partition functionality, with the most component 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 might be represented as a sum of 4 squares, exchanging the squares by means of triangular numbers and, within the approach, acquiring a cleanser end result. the new pattern of employing the guidelines and techniques of algebraic geometry to the examine of theta services and quantity thought has ended in nice advances within the zone. although, the authors decide to stick with the classical standpoint. consequently, their statements and proofs are very concrete. during this publication the mathematician accustomed to the algebraic geometry method of theta features and quantity conception will locate many fascinating principles in addition to distinct reasons and derivations of latest and previous effects. Highlights of the ebook comprise systematic stories of theta consistent identities, uniformizations of surfaces represented through subgroups of the modular crew, partition identities, and Fourier coefficients of automorphic capabilities. necessities are a fantastic realizing of complicated research, a few familiarity with Riemann surfaces, Fuchsian teams, and elliptic features, and an curiosity in quantity concept. The booklet includes summaries of a few of the mandatory fabric, rather for theta features and theta constants. Readers will locate right here a cautious exposition of a classical standpoint of research and quantity concept. provided are various examples plus feedback for research-level difficulties. The textual content is acceptable for a graduate direction or for self reliant studying.