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Harmonic Maass Forms and Mock Modular Forms: Theory and Applications [Hardback]

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  • Formāts: Hardback, 391 pages, height x width: 254x178 mm, weight: 825 g
  • Sērija : Colloquium Publications
  • Izdošanas datums: 30-Dec-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470419440
  • ISBN-13: 9781470419448
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Harmonic Maass Forms and Mock Modular Forms: Theory and ApplicationsPalielināt
  • Formāts: Hardback, 391 pages, height x width: 254x178 mm, weight: 825 g
  • Sērija : Colloquium Publications
  • Izdošanas datums: 30-Dec-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470419440
  • ISBN-13: 9781470419448
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470419448.html
Keywords:
Modular forms and Jacobi forms play a central role in many areas of mathematics. Over the last 10-15 years, this theory has been extended to certain non-holomorphic functions, the so-called "harmonic Maass forms''. The first glimpses of this theory appeared in Ramanujan's enigmatic last letter to G. H. Hardy written from his deathbed. Ramanujan discovered functions he called ``mock theta functions'' which over eighty years later were recognized as pieces of harmonic Maass forms. This book contains the essential features of the theory of harmonic Maass forms and mock modular forms, together with a wide variety of applications to algebraic number theory, combinatorics, elliptic curves, mathematical physics, quantum modular forms, and representation theory.

Recenzijas

This wonderful book is very exciting to me as it presents some very exotic and beautiful stuff that I, for one, had no notion of...I'm happy to be enlightened, if only in a preliminary way. But this book is tailor-made to lead wayward sons like me back home and to light the right kind of fire under us...Excellent." Michael Berg, MAA Reviews

Preface xi
Acknowledgments xv
Part 1 Background 1(58)
Chapter 1 Elliptic Functions
3(10)
1.1 Eisenstein series
4(1)
1.2 Weierstrass p-function
5(3)
1.3 Weierstrass ζ-function
8(2)
1.4 Eichler integrals of weight 2 newforms
10(3)
Chapter 2 Theta Functions and Holomorphic Jacobi Forms
13(36)
2.1 Jacobi theta functions
13(3)
2.2 Basic facts on Jacobi forms
16(4)
2.3 Examples of Jacobi forms
20(6)
2.3.1 The Jacobi theta function
21(1)
2.3.2 Jacobi-Eisenstein series
21(4)
2.3.3 Weierstrass p-function
25(1)
2.4 A structure theorem for Jk,m
26(1)
2.5 Relationship with half-integral weight modular forms
27(3)
2.5.1 Theta decompositions
27(2)
2.5.2 An isomorphism to Kohnen's plus space
29(1)
2.6 Hecke theory for Jk,m and the Jacobi-Petersson inner product
30(6)
2.6.1 Hecke theory of Jk,m
30(4)
2.6.2 The Jacobi-Petersson inner product
34(2)
2.7 Taylor expansions
36(7)
2.8 Related topics
43(6)
2.8.1 Siegel modular forms
44(2)
2.8.2 Skew-holomorphic Jacobi forms
46(3)
Chapter 3 Classical Maass Forms
49(10)
3.1 Definitions
49(1)
3.2 Fourier expansions
50(1)
3.3 General discussion
51(1)
3.4 Eisenstein series
52(1)
3.5 L-functions of Maass cusp forms
53(2)
3.6 Maass cusp forms arising from real quadratic fields
55(1)
3.6.1 Hecke characters
55(1)
3.6.2 Maass cusp forms from real quadratic fields
55(1)
3.7 Hecke theory on Maass cusp forms
56(1)
3.8 Period functions of Maass cusp forms
56(3)
Part 2 Harmonic Maass Forms and Mock Modular Forms 59(162)
Chapter 4 The Basics
61(6)
4.1 Definitions
61(2)
4.2 Fourier expansions
63(4)
Chapter 5 Differential Operators and Mock Modular Forms
67(16)
5.1 Maass operators and harmonic Maass forms
67(7)
5.2 The ξoperator and pairing of Bruinier and Funke
74(3)
5.3 The flipping operator
77(3)
5.4 Mock modular forms and shadows
80(3)
Chapter 6 Examples of Harmonic Maass Forms
83(30)
6.1 E*2 (τ) and Zagier's weight 3/2 Eisenstein series
83(4)
6.1.1 The Eisenstein series E*w (Τ)
83(2)
6.1.2 Zagier's weight 3/2 nonholomorphic Eisenstein series
85(2)
6.2 Weierstrass mock modular forms
87(4)
6.3 Maass-Poincare series
91(17)
6.4 p-adic harmonic Maass forms in the sense of Serre
108(5)
Chapter 7 Hecke Theory
113(20)
7.1 Basic facts
113(2)
7.2 Weakly holomorphic Hecke eigenforms
115(1)
7.3 Harmonic Maass forms and complex multiplication
116(1)
7.4 p-adic properties of integral weight mock modular forms
117(8)
7.4.1 Algebraicity
117(2)
7.4.2 p-adic coupling of mock modular forms with newforms
119(4)
7.4.3 Relationship with p-adic modular forms
123(2)
7.5 p-adic harmonic Maass functions
125(8)
Chapter 8 Zwegers' Thesis
133(26)
8.1 Zwegers' thesis I: Appell-Lerch series
133(15)
8.2 Zwegers' thesis II: indefinite theta series
148(11)
Chapter 9 Ramanujan's Mock Theta Functions
159(18)
9.1 Ramanujan's last letter to Hardy
159(2)
9.2 Work of Watson and Andrews
161(2)
9.3 Third order mock theta functions revisited
163(2)
9.4 Mock theta functions as indefinite theta series
165(2)
9.5 Universal mock theta functions
167(3)
9.6 The Mock Theta Conjectures
170(1)
9.7 The Andrews-Dragonette Conjecture
171(2)
9.8 Ramanujan's original claim revisited
173(4)
Chapter 10 Holomorphic Projection
177(6)
10.1 Principle of holomorphic projection
177(2)
10.2 Regularized holomorphic projection
179(1)
10.3 Kronecker-type relations for mock modular forms
180(3)
Chapter 11 Meromorphic Jacobi Forms
183(10)
11.1 Mock theta functions as coefficients of meromorphic forms
183(1)
11.2 Positive index Jacobi forms
183(5)
11.3 Negative index Jacobi forms
188(5)
Chapter 12 Mock Modular Eichler-Shimura Theory
193(14)
12.1 Classical Eichler-Shimura theory
193(5)
12.2 Period polynomials for weakly holomorphic modular forms
198(5)
12.3 Cycle integrals of weakly holomorphic modular forms
203(4)
Chapter 13 Related Automorphic Forms
207(14)
13.1 Introduction
207(1)
13.2 Mixed mock modular forms
208(3)
13.3 Polar harmonic Maass forms
211(7)
13.3.1 Divisors of modular forms
211(3)
13.3.2 Definitions of the functions in Theorem 13.4 and the proof of Theorem 13.5
214(1)
13.3.3 Green's functions
215(1)
13.3.4 Definition and construction of polar harmonic Maass forms
216(2)
13.4 Locally harmonic Maass forms
218(3)
Part 3 Applications 221(132)
Chapter 14 Partitions and Unimodal Sequences
223(22)
14.1 Asymptotic formulas for partitions
223(3)
14.2 Ramanujan's partition congruences
226(1)
14.3 Ranks and cranks
227(7)
14.3.1 Definition and generating functions
227(4)
14.3.2 Properties of the crank partition function
231(1)
14.3.3 Properties of the rank partition function
232(2)
14.4 Unimodal sequences
234(6)
14.5 Andrews' spt-function
240(5)
Chapter 15 Asymptotics for Coefficients of Modular-type Functions
245(18)
15.1 Prologue
245(1)
15.2 Asymptotic methods
246(1)
15.3 Classical holomorphic modular forms
247(4)
15.4 Weakly holomorphic modular forms and mock modular forms
251(2)
15.5 Coefficients of meromorphic modular forms
253(3)
15.6 Mixed mock modular forms
256(2)
15.7 The Wright Circle Method
258(5)
Chapter 16 Harmonic Maass Forms as Arithmetic and Geometric Generating Functions
263(20)
16.1 Zagier's work on traces of singular moduli
263(4)
16.2 Maass-Poincare series
267(2)
16.3 Relation to (theta) lifts
269(2)
16.4 Gross-Kohnen-Zagier and generalized Jacobians
271(3)
16.5 Cycle integrals and mock modular forms
274(4)
16.6 Weight one harmonic Maass forms
278(5)
Chapter 17 Shifted Convolution L-functions
283(8)
17.1 Rankin-Selberg convolutions
283(2)
17.2 Hoffstein-Hulse shifted convolution L-functions
285(1)
17.3 Special values of shifted convolution L-functions
285(6)
17.3.1 p-adic properties of special values
287(4)
Chapter 18 Generalized Borcherds Products
291(16)
18.1 The simplest Borcherds products
291(3)
18.2 Twisted Borcherds products
294(1)
18.3 Generalization to the mock modular setting
295(8)
18.3.1 The Weil representation
296(1)
18.3.2 The Γ0 (N) set-up
296(2)
18.3.3 Vector-valued harmonic Maass forms
298(1)
18.3.4 Twisted Siegel theta functions
299(1)
18.3.5 Twisted Heegner divisors
300(2)
18.3.6 Generalized Borcherds products
302(1)
18.4 Examples of generalized Borcherds products
303(4)
18.4.1 Twisted Borcherds products revisited
303(1)
18.4.2 Ramanujan's mock theta functions f (q) and w(q)
304(3)
Chapter 19 Elliptic Curves over Q
307(16)
19.1 The Birch and Swinnerton-Dyer Conjecture
307(5)
19.1.1 Rational points on elliptic curves
307(2)
19.1.2 The Birch and Swinnerton-Dyer Conjecture
309(3)
19.2 Quadratic twists of elliptic curves
312(2)
19.2.1 Quadratic twists
312(2)
19.3 The Shimura correspondence
314(1)
19.4 Central values of quadratic twist L-functions
314(3)
19.4.1 A theorem of Kohnen and Zagier
315(1)
19.4.2 A theorem of Waldspurger
315(2)
19.5 Harmonic Maass forms and quadratic twists of elliptic curves
317(6)
Chapter 20 Representation Theory and Mock Modular Forms
323(16)
20.1 Monstrous Moonshine
323(4)
20.2 Kac-Wakimoto characters
327(7)
20.2.1 The case with n = 1, m > or = to 2
327(3)
20.2.2 The case with m > n
330(2)
20.2.3 The case with m < n
332(1)
20.2.4 The case with m = n
333(1)
20.2.5 Additional supercharacters
334(1)
20.3 Umbral Moonshine
334(5)
Chapter 21 Quantum Modular Forms
339(14)
21.1 Introduction to quantum modular forms
339(1)
21.2 Quantum modular forms and Maass forms
340(1)
21.3 Quantum modular forms and Eichler integrals
341(3)
21.3.1 Kontsevich's function
341(1)
21.3.2 Eichler integrals and partial theta functions
342(2)
21.4 Quantum modular forms and radial limits of mock modular forms
344(4)
21.4.1 A unimodal rank generating function
344(1)
21.4.2 Radial limits and quantum modular forms
345(3)
21.5 Quantum modular forms and partial theta functions
348(5)
21.5.1 Connections with the Habiro ring
350(3)
Appendix A. Representations of Mock Theta Functions 353(14)
A.1 Order 2 mock theta functions
353(1)
A.2 Order 3 mock theta functions
354(2)
A.3 Order 5 mock theta functions
356(3)
A.4 Order 6 mock theta functions
359(3)
A.5 Order 7 mock theta functions
362(1)
A.6 Order 8 mock theta functions
363(2)
A.7 Order 10 mock theta functions
365(2)
Bibliography 367(20)
Index 387
Kathrin Bringmann, University of Cologne, Germany.

Amanda Folsom, Amherst College, MA.

Ken Ono, Emory University, Atlanta, GA.

Larry Rolen, Trinity College, Dublin, Ireland.