Modular Forms on Schiermonnikoog

doi:10.1017/CBO9780511543371

Modular forms are functions with an enormous amount of symmetry that play a central role in number theory, connecting it with analysis and geometry. They have played a prominent role in mathematics since the 19th century and their study continues to flourish today. Modular forms formed the inspiration for Langlands' conjectures and play an important role in the description of the cohomology of varieties defined over number fields. This collection of up-to-date articles originated from the conference 'Modular Forms' held on the Island of Schiermonnikoog in the Netherlands. A broad range of topics is covered including Hilbert and Siegel modular forms, Weil representations, Tannakian categories and Torelli's theorem. This book is a good source for all researchers and graduate students working on modular forms or related areas of number theory and algebraic geometry.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-vii
- By Bas Edixhoven, Gerard van der Geer, Ben Moonen
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List of contributors
pp viii-x
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1 - Modular Forms
pp 1-12
- By Bas Edixhoven, Gerard van der Geer, Ben Moonen
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2 - On the basis problem for Siegel modular forms with level
pp 13-28
- By Siegfried Böcherer, Hidenori Katsurada, Rainer Schulze-Pillot
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3 - Mock theta functions, weak Maass forms, and applications
pp 29-56
- By Kathrin Bringmann
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4 - Sign changes of coefficients of half integral weight modular forms
pp 57-66
- By Jan Hendrik Bruinier, Winfried Kohnen
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5 - Gauss map on the theta divisor and Green's functions
pp 67-78
- By Robin de Jong
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6 - A control theorem for the images of Galois actions on certain infinite families of modular forms
pp 79-84
- By Luis Dieulefait
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7 - Galois realizations of families of Projective Linear Groups via cusp forms
pp 85-92
- By Luis Dieulefait
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8 - A strong symmetry property of Eisenstein series
pp 93-106
- By Bernhard Heim
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9 - A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences
pp 107-144
- By Tomoyoshi Ibukiyama
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10 - Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms
pp 145-188
- By Andrew Knightly, Charles Li
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11 - Modular shadows and the Lévy—Mellin ∞—adic transform
pp 189-238
- By Yuri I. Manin, Matilde Marcolli
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12 - Jacobi forms of critical weight and Weil representations
pp 239-266
- By Nils-Peter Skoruppa
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13 - Tannakian Categories attached to abelian varieties
pp 267-274
- By Rainer Weissauer
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14 - Torelli's theorem from the topological point of view
pp 275-284
- By Rainer Weissauer
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15 - Existence of Whittaker models related to four dimensional symplectic Galois representations
pp 285-310
- By Rainer Weissauer
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16 - Multiplying Modular Forms
pp 311-342
- By Martin H. Weissman
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17 - On projective linear groups over finite fields as Galois groups over the rational numbers
pp 343-350
- By Gabor Wiese
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Modular Forms on Schiermonnikoog

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-vii

List of contributors
pp viii-x

1 - Modular Forms
pp 1-12

2 - On the basis problem for Siegel modular forms with level
pp 13-28

3 - Mock theta functions, weak Maass forms, and applications
pp 29-56

4 - Sign changes of coefficients of half integral weight modular forms
pp 57-66

5 - Gauss map on the theta divisor and Green's functions
pp 67-78

6 - A control theorem for the images of Galois actions on certain infinite families of modular forms
pp 79-84

7 - Galois realizations of families of Projective Linear Groups via cusp forms
pp 85-92

8 - A strong symmetry property of Eisenstein series
pp 93-106

9 - A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences
pp 107-144

10 - Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms
pp 145-188

11 - Modular shadows and the Lévy—Mellin ∞—adic transform
pp 189-238

12 - Jacobi forms of critical weight and Weil representations
pp 239-266

13 - Tannakian Categories attached to abelian varieties
pp 267-274

14 - Torelli's theorem from the topological point of view
pp 275-284

15 - Existence of Whittaker models related to four dimensional symplectic Galois representations
pp 285-310

16 - Multiplying Modular Forms
pp 311-342

17 - On projective linear groups over finite fields as Galois groups over the rational numbers
pp 343-350

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