Theta constants, Riemann surfaces, and the modular group: an by Hershel M. Farkas, Irwin Kra

By Hershel M. Farkas, Irwin Kra

There are exceedingly wealthy connections among classical research and quantity concept. for example, analytic quantity thought includes many examples of asymptotic expressions derived from estimates for analytic services, akin to within the facts of the best quantity Theorem. In combinatorial quantity thought, distinctive formulation for number-theoretic amounts are derived from relatives among analytic features. Elliptic services, specially theta capabilities, are a tremendous classification of such services during this context, which were made transparent already in Jacobi's Fundamenta nova. Theta services also are classically hooked up with Riemann surfaces and with the modular workforce $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which supply one other course for insights into quantity idea. Farkas and Kra, recognized masters of the speculation of Riemann surfaces and the research of theta capabilities, discover the following fascinating combinatorial identities via the functionality thought on Riemann surfaces with regards to the imperative congruence subgroups $\Gamma(k)$. for example, the authors use this method of derive congruences stumbled on via Ramanujan for the partition functionality, with the most aspect being the development of an analogous functionality in additional than a technique. The authors additionally receive a version on Jacobi's well-known outcome at the variety of ways in which an integer should be represented as a sum of 4 squares, changing the squares via triangular numbers and, within the method, acquiring a purifier consequence. the hot pattern of utilizing the information and strategies of algebraic geometry to the learn of theta features and quantity conception has ended in nice advances within the sector. even if, the authors decide to stick with the classical viewpoint. therefore, their statements and proofs are very concrete. during this ebook the mathematician acquainted with the algebraic geometry method of theta capabilities and quantity concept will locate many fascinating principles in addition to certain factors and derivations of latest and previous effects. Highlights of the booklet contain systematic stories of theta consistent identities, uniformizations of surfaces represented by way of subgroups of the modular staff, partition identities, and Fourier coefficients of automorphic features. necessities are an excellent realizing of complicated research, a few familiarity with Riemann surfaces, Fuchsian teams, and elliptic services, and an curiosity in quantity idea. The e-book includes summaries of a few of the mandatory fabric, quite for theta capabilities and theta constants. Readers will locate right here a cautious exposition of a classical viewpoint of study and quantity conception. awarded are a variety of examples plus feedback for research-level difficulties. The textual content is appropriate for a graduate direction or for self sustaining examining.

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9 Example Let G ˆ S4 and let B denote the basis v1 , v2 , v3 , v4 of V. 10 De®nition Let G be a subgroup of Sn . The FG-module V with basis v1 , . . , v n such that v i g ˆ v ig for all i, and all g P G, is called the permutation module for G over F. We call v1 , . . , v n the natural basis of V. Note that if we write B for the basis v1 , . . , v n of the permutation module, then for all g in G, the matrix [ g]B has precisely one nonzero entry in each row and column, and this entry is 1. Such a matrix is called a permutation matrix.

First, the fact that r is a homomorphism shows that v(( gh)r) ˆ v( gr)(hr) for all v P V and all g, h P G. Next, since 1r is the identity matrix, we have v(1r) ˆ v for all v P V. Finally, the properties of matrix multiplication give (ëv)( gr) ˆ ë(v( gr)), 38 FG-modules 39 (u ‡ v)( gr) ˆ u( gr) ‡ v( gr) for all u, v P V, ë P F and g P G. 2(1). Thus     1 0 0 1 X , br ˆ ar ˆ 0 À1 À1 0 If v ˆ (ë1 , ë2 ) P F 2 then, for example, v(ar) ˆ (Àë2 , ë1 ), v(br) ˆ (ë1 , Àë2 ), v(a3 r) ˆ (ë2 , Àë1 )X Motivated by the above observations on the product v( gr), we now de®ne an FG-module.

Ws is a basis of W. Show that V ˆ U È W if and only if u1 , . . , ur , w1, . . , ws is a basis of V. 5. (a) Let U1, U2 and U3 be subspaces of a vector space V, with V ˆ U1 ‡ U2 ‡ U3. Show that V ˆ U1 È U2 È U3 D U 1 ’ (U 2 ‡ U3 ) ˆ U 2 ’ (U 1 ‡ U3 ) ˆ U 3 ’ (U 1 ‡ U2 ) ˆ f0gX (b) Give an example of a vector space V with three subspaces U1, U2 and U3 such that V ˆ U1 ‡ U2 ‡ U3 and U1 ’ U 2 ˆ U1 ’ U3 ˆ U2 ’ U 3 ˆ f0g, but V Tˆ U1 È U2 È U3. 6. Suppose that U1, . . , Ur are subspaces of the vector space V, and that V ˆ U1 È .

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