**Contemporary Mathematics**

Volume: 614;
2014;
441 pp;
Softcover

MSC: Primary 11; 22; 14;

Print ISBN: 978-0-8218-9394-4

Product Code: CONM/614

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**Electronic ISBN: 978-1-4704-1658-4
Product Code: CONM/614.E**

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Individual Member Price: $108.80

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# Automorphic Forms and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro

Share this page *Edited by *
*James W. Cogdell; Freydoon Shahidi; David Soudry*

This volume contains the proceedings of the
conference Automorphic Forms and Related Geometry: Assessing the
Legacy of I.I. Piatetski-Shapiro, held from April 23–27, 2012, at
Yale University, New Haven, CT.

Ilya I. Piatetski-Shapiro, who passed away on 21 February 2009, was
a leading figure in the theory of automorphic forms. The conference
attempted both to summarize and consolidate the progress that was made
during Piatetski-Shapiro's lifetime by him and a substantial group of
his co-workers, and to promote future work by identifying fruitful
directions of further investigation. It was organized around several
themes that reflected Piatetski-Shapiro's main foci of work and that
have promise for future development: functoriality and converse
theorems; local and global \(L\)-functions and their periods;
\(p\)-adic \(L\)-functions and arithmetic geometry;
complex geometry; and analytic number theory. In each area, there were
talks to review the current state of affairs with special attention to
Piatetski-Shapiro's contributions, and other talks to report on
current work and to outline promising avenues for continued progress.

The contents of this volume reflect most of the talks that were
presented at the conference as well as a few additional contributions.
They all represent various aspects of the legacy of
Piatetski-Shapiro.

#### Readership

Graduate students and research mathematicians interested in automorphic forms, number theory, representation theory, and geometry.

# Table of Contents

## Automorphic Forms and Related Geometry: Assessing the Legacy of I.I. Piatetski-Shapiro

- Preface vii8 free
- On parameters for the group 𝑆𝑂(2𝑛) 110 free
- Piatetski-Shapiro’s Work on Converse Theorems 3140
- 1. 𝐿-functions for 𝐺𝐿_{𝑛}×𝐺𝐿_{𝑚} with 𝑚<𝑛 3342
- 2. Converse Theorems 3544
- 3. Limiting the rank of the twists 3645
- 4. Limiting the ramification of the twists 4049
- 5. Speculation on 𝐺𝐿₁ twists 4352
- 6. Converse theorems with poles 4453
- 7. Local converse theorems 4554
- 8. Final remarks 4655
- Acknowledgement 4857
- References 4857

- A 𝑝-adic integral for the reciprocal of 𝐿-functions 5362
- Harmonic analysis on symmetric spaces as complex analysis 6978
- 1. Horospherical reduction of complex symmetric spaces (geometrical picture) 7079
- 2. Horospherical reduction of complex symmetric spaces (analytic construction of intertwining operators) 7180
- 3. Dual horospherical Cauchy transform 7281
- 4. Cauchy formula on complex symmetric spaces 7382
- 5. Horospherical duality for real symmetric spaces. Compact spaces 7483
- 6. Horospherical duality for complex crowns of symmetric spaces 7584
- 7. Tube Stein manifolds on causal symmetric spaces 7887
- References 7988

- Testing rationality of coherent cohomology of Shimura varieties 8190
- Hecke fields of Hilbert modular analytic families 97106
- Structure of holomorphic unitary representations: the case of 𝐔_{2,2} 139148
- 1. Introduction 139148
- 2. Notation 140149
- 3. \Gln harmonics and invariants on \pvnkl 143152
- 4. Special highest weight vectors in \calh⊗𝐽 149158
- 5. The case 𝑛≥4, 𝑝=𝑞=2 150159
- 6. The case 𝑛=3, 𝑝=𝑞=2 155164
- 7. The case 𝑛=𝑝=𝑞=2 158167
- 8. The case 𝑛=1 161170
- 9. Application to holomorphic unitary representations 163172
- References 169178

- Mellin Transform of Whittaker functions 171180
- Automorphic Integral Transforms for Classical Groups I: Endoscopy Correspondences 179188
- 1. Introduction 179188
- 2. Arthur Parametrization of Discrete Spectrum 185194
- 3. Endoscopy Structures for Classical Groups 189198
- 4. Fourier Coefficients and Nilpotent Orbits 194203
- 5. Constructions of the Automorphic Kernel Functions 198207
- 6. Automorphic Descents and Automorphic Forms of Simple Type 219228
- 7. Theta Correspondence and (𝜒,𝑏)-Theory 224233
- 8. Endoscopy Correspondence and (𝜏,𝑏)-Theory 228237
- 9. Final Remarks 232241
- References 233242

- An inductive formula for 𝜖-factors 243252
- On a new functional equation for local integrals 261270
- 1. Introduction and statement of main result 261270
- 2. Basic estimates 266275
- 3. Model transition –first case 271280
- 4. Model transition –second case 277286
- 5. The functional equation 283292
- Appendix A. Convergence results 284293
- Appendix B. More functional equations 286295
- Acknolwedgement 292301
- References 293302

- Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands 295304
- 1. Introduction 295304
- 2. Notations et propriétés générales 298307
- 3. Le cas des représentations cuspidales 306315
- 4. Support cuspidal étendu et ensemble de blocs de Jordan 309318
- 5. Morphisme dans le L-groupe 319328
- 6. Morphisme de Langlands des paquets stables de séries discrètes 324333
- 7. 𝑅-groupe et cardinal des paquets stables de représentations tempérées 331340
- References 335344

- On a certain sum of automorphic 𝐿-functions 337346
- Analytic constructions of 𝑝-adic 𝐿-functions and Eisenstein series 345354
- 1. Complex and 𝑝-adic 𝐿-functions 348357
- 2. 𝑝-adic meromorphic continuation of the Siegel-Eisenstein series 349358
- 3. Pseudomeasures and their Mellin transform 356365
- 4. Application to Minkowski-Siegel Mass constants 358367
- 5. Link to Shahidi’s method for SL(2) and regular prime 𝑝 360369
- 6. Doubling method and Ikeda’s constructions 361370
- Appendix A. Appendix. On 𝑝-adic 𝐿-functions for 𝐺𝑆𝑝(4) 362371
- References 371380

- On stability of root numbers 375384
- CAP forms, Eisenstein series, and some arithmetic applications 387396
- Automorphic descent: an outgrowth from Piatetski-Shapiro’s vision 407416
- On the singularities of branch curves of 𝐾3 surfaces and applications 433442