Curriculum Vitae (Sug Woo Shin)

Total Page:16

File Type:pdf, Size:1020Kb

Curriculum Vitae (Sug Woo Shin) Curriculum Vitae (Sug Woo Shin) Department of Mathematics, MIT 77 Mass Ave, building E18-426, Cambridge, MA 02139, USA email: swshin at math.mit.edu, web page: http://math.mit.edu/~swshin Education • Ph.D., Mathematics, Harvard University, Sep 2002-Jun 2007 thesis: Counting points on Igusa varieties, supervised by Richard Taylor • B. Sc., Mathematics, Seoul National University, Mar 1997-Feb 2000 (Military service, May 2000-Jul 2002) Position • Associate Professor (tenured), UC Berkeley, Jul 2014- • Assistant Professor (tenure-track), MIT, Jul 2011- (Associate Professor, Jul 2014-; on leave: Jul 2014-Jun 2015) • Member, Institute for Advanced Study, Sep 2010-Jun 2011 • Dickson Instructor, University of Chicago, Sep 2008-Aug 2010 (on leave: Sep-Dec 2008) • Member, Institute for Advanced Study, Sep 2007-Dec 2008 Visiting Position • KIAS scholar, Korea Institute for Advanced Study, 2011- • Associate member, Pohang Mathematics Institute, 2011- Research Interests number theory, Shimura varieties, Langlands functoriality, trace formula, automorphic forms Publication (1) (with Tasho Kaletha, Alberto Minguez, and Paul-James White) \Endoscopic classification of represen- tations: inner forms of unitary groups", preprint (2) (with Peter Sarnak and Nicolas Templier) \Families of L-functions and their symmetry", preprint (3) (with Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, and Vytautas Paˇsk¯unas)\Patching and the p-adic Langlands correspondence", preprint (4) (with Sara Arias-de-Reyna, Luis Dieulefait, and Gabor Wiese) \Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties", to appear in Math. Ann. (5) (with Junehyuk Jung) \More examples" (appendix to On sparsity of positive-definite automorphic forms in a family by Junehyuk Jung), preprint (6) (with Nicolas Templier) \On fields of rationality for automorphic representations", Compositio Math. 150 (2014), no.12, 2003{2053 (7) (with Nicolas Templier) \Sato-Tate theorem for families and low-lying zeros of automorphic L-functions", to appear in Invent. Math. (8) \Supercuspidal part of the mod l cohomology of GU(1; n − 1)-Shimura varieties", to appear in J. Reine Angew. Math. (9) (with Peter Scholze) \On the cohomology of compact unitary group Shimura varieties at ramified split places", J. Amer. Math. Soc. 26 (2013), no.1, 261-294 (10) \On the cohomological base change for unitary similitude groups" (appendix to the paper \Galois rep- resentations associated to holomorphic limits of discrete series I: Unitary Groups" by Wushi Goldring), Compositio Math. 150 (2014), no.2, 191{228 (11) \Geometric reductive dual pairs and a mod p theta correspondence", preprint (12) \Abelian varieties and the Weil representations", Algebra and Number Theory 6 (2012), no.8, 1719{1772 (13) \Automorphic Plancherel density theorem", Israel J. Math. 192 (2012), no.1, 83{120 (14) \Construction of Galois representations" (lecture notes for 2009 Clay summer school), preprint (15) \On the cohomology of Rapoport-Zink spaces of EL-type", Amer. J. Math 134 (2012), no.2, 407{452 2 (16) \Galois representations arising from some compact Shimura varieties", Ann. of Math. 173 (2011), no.3, 1645{1741 (17) \Stable trace formula for Igusa varieties", J. Inst. Math. Jussieu 9 (2010), no.4, 847{895. (18) \Counting points on Igusa varieties", Duke Math. J. 146 (2009), no.3, 509{568. Fellowships/Awards/Grants • Sloan Fellowship, 2013-2015 • NSF grant, 2012-2015 • MIT NEC Research funds (competitive internal fellowship), 2011-2013 • Clay Math Institute Liftoff Fellowship, 2007 • Harvard Merit Fellowship, 2006-2007 • Juan Carlos I scholarship (funding for travel to the ICM 2006) • Samsung Scholarship for graduate studies, 2003-2006 • Joseph Leonard Walsh Fund, 2002-2003 • Undergraduate Scholarship from Korea Foundation of Advanced Studies, 1998-2000 • International Mathematical Olympiad (2 golds, 1 bronze), 1994-1996 Conferences, Workshops, Schools, and Research Groups I have co-organized • Banff 5-day half workshop: Families of Automorphic Forms and the Trace Formula, Nov 30-Dec 5, 2014 • HIM Research in Groups: Families of automorphic forms and the trace formula, Jun 22-Jul 5, 2014 • Simons Symposium: Families of Automorphic Forms and the Trace Formula, Puerto Rico, Jan 26-Feb 1, 2014 • KIAS intensive lecture series on Shimura varieties and Rapoport-Zink spaces, Jan 13-16, 2014 • AIM Square: The p-adic Langlands Program for non-split groups, July 15-19, 2013; September 22-26, 2014 • Workshop on Modular Forms and Galois Representations, KIAS, Jan 2013 • Banff FRG: The p-adic Langlands Program for non-split groups, Aug 2012 • 2012 Conference on L-functions, Shineville Resort, Jeju island, Aug 2012 • Cohomology of Shimura varieties: Arithmetic Aspects and the Construction of Galois representations, Fields Institute, Mar 2012 • Rigid Geometry and the Local Langlands Correspondence, KIAS, Jan 2-5, 2012 • Theta Festival, Postech, South Korea, Aug 13-19, 2011 • Congruences of modular forms and Galois representations (summer school), Postech, South Korea, Jul 18-30, 2010 Invited Talks • \Endoscopic classification for inner forms of unitary groups", MSRI, Oct 2014 • \Cohomology of Shimura varieties and the Langlands program", Postech, PANT conference (ICM satel- lite), Aug 2014 • \Automorphic forms and cohomology", Bellairs Inst. in Barbados, May 2014 • \Asymptotic behavior of supercuspidal characters", Yale, Mar 2014; U. Minnesota, Apr 2014 • \Sato-Tate conjecture for families", Puerto Rico, Jan 2014 • \Patching and the p-adic Langlands correspondence", IMS (Singapore), Apr 2013; Paderborn, Oct 2013; UCSD, Dec 2013; Sanya, Dec 2013 • \Recent progress in the Langlands program", Seoul National Univ., Sep 2013; MIT, Nov 2013 • \Families of automorphic L-functions", Berkeley, Nov 2012; Northwestern, Dec 2012; Johns Hopkins, Jan 2013, Berkeley, Apr 2013; U. of Oklahoma, Oct 2013; Ajou Univ., Dec 2013 • \Non-emptiness of Newton strata", Banff, Jun 2012; Hokkaido, Jul 2012; UIC, Feb 2013; Harvard, Feb 2013 • \Uniform bounds on orbital integrals", Jussieu, Jul 2012 • \Fields of rationality for automorphic representations", ESI, Feb 2012; U of Michigan and Boston College, Apr 2012; U of Chicago and UCLA, May 2012 • \The Langlands-Kottwitz method for Shimura varieties with bad reduction", MIT, Nov 2011 • introductory lectures on reductive groups and automorphic representation theory, Postech, Aug 2011 • \Introduction to the trace formula", Durham, Jul 2011 • \Supercuspidal part of the mod l cohomology of GU(1; n − 1)-Shimura varieties", Kyoto, Jul 2011 • \Families of automorphic representations", IAS, Mar 2011; Brown, Nov 2011; UCLA, May 2012; Univ of Bonn, July 2012 3 • \A mod p theta correspondence", Kyoto, Sep 2010 • \A remark on the cohomology of some Shimura varieties", Taipei, Jul 2010 • \Cohomology of Shimura varieties and the Langlands philosophy" (Colloquium), U Minnesota, Mar 2010 • \Plancherel density theorem for automorphic representations", U of Michigan, Northwestern, Nov 2009, Caltech, MIT, Dec 2010. • \Theta functions from the geometric viewpoint (4 lectures)", POSTECH, Korea, Nov 2009 • \Automorphic representations of prescribed type", Paris Jussieu, Paris Orsay, Parid Nord, Cambridge(UK), Oct 2009. • \Finding automorphic representations of prescribed type", National Taiwan University, Taipei, Jul 2009 • \Construction of Galois representations" (short lectures), Clay summer school, Honolulu, Jun-Jul 2009 • \Cohomology of Rapoport-Zink spaces of EL-type", Univ. of Chicago, Apr 2009 • \Construction of Galois representations", Workshop on Shimura Varieties, Automorphic Representa- tions and Related Topics, Kyoto, Nov 2008 • \Stabilization I", \Points on special fibers of PEL Shimura varieties and introduction to vanishing cycles", \Construction of Galois representations", Banff, Aug 2008 • \Galois representations arising from some compact Shimura varieties", Harvard Univ., Mar 2008; Prince- ton Univ., Mar 2008; Caltech, May 2008; Northwestern Univ., May 2008; Univ. of Maryland, Sep 2008 • \Non-abelian class field theory", Seoul National Univ., Jul 2007 • \Endoscopy and Shimura varieties", Kyoto Mini-Workshop, May 2007; KIAS, Sep 2007 (conference) • \Representations in the cohomology of Rapoport-Zink spaces", MIT, Mar 2007 • \Local-global compatibility at ramified primes", Univ. of Connecticut, Nov 2006 • \Counting points on Igusa varieties", Harvard Univ., Nov 2006; Univ. of Chicago, Nov 2006; Univ. Paris 7, May 2007; Luminy, Jun 2007 (conference); Princeton Univ., Oct 2007 • \Shimura varieties and the Langlands correspondence", Korea Institute of Advanced Study, Aug 2005 Workshops and Conferences attended as a speaker • 2014 Pan Asian Number Theory Conference (ICM satellite), Postech (S. Korea), Aug 5-9, 2014 • The Bellairs workshop in Number Theory: The cohomology of arithmetic groups and the Langlands program, Barbados, May 2-9, 2014 • Simons Symposium: Families of Automorphic Forms and the Trace Formula, Puerto Rico, Jan 26-Feb 1, 2014 • The 4th conference of Tsinghua Sanya international mathematics forum, Sanya, Dec 17-22, 2013 • Conference on arithmetic algebraic geometry, Paderborn, Oct 7-11, 2013 • Modular Representation Theory of Finite and p-adic Groups, IMS (Singapore), Apr 1-26, 2013 • Arithmetic geometry of Shimura varieties, representation theory, and related topics, Hokkaido Univ, Jul 18-22, 2012 • Arithmetic geometry of orthogonal and unitary Shimura varieties, BIRS, Jun 3-8, 2012 • Cohomology of Shimura
Recommended publications
  • Arxiv:1401.5507V2
    FAMILIES OF L-FUNCTIONS AND THEIR SYMMETRY PETER SARNAK, SUG WOO SHIN, AND NICOLAS TEMPLIER Abstract. In [100] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [32], [118], [73] [49], [72] and especially [108] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [68] for 1 the distribution of the zeros near s = 2 . In this note we carry this out after introducing some basic invariants associated to a family. 1. Definition of families and Conjectures The zoo of automorphic cusp forms π on G = GLn over Q correspond bijectively to their standard completed L-functions Λ(s, π) and they constitute a countable set containing species of different types. For example there are self-dual forms, ones corresponding to finite Galois representations, to Hasse–Weil zeta functions of varieties defined over Q, to Maass forms, etc. From a number of points of view (including the nontrivial problem of isolating special forms) one is led to study such Λ(s, π)’s in families in which the π’s have similar characteristics. Some applications demand the understanding of the behavior of the L-functions as π varies over a family. Other applications involve questions about an individual L-function. In practice a family is investigated as it arises. For example the density theorems of Bombieri [13] and Vinogradov [113] are concerned with showing that in a suitable sense most Dirichlet L-functions have few violations of the Riemann hypothesis, and as such it is a powerful substitute for the latter.
    [Show full text]
  • Counting Points on Igusa Varieties of Hodge Type by Thomas a Mack
    Counting Points on Igusa Varieties of Hodge Type by Thomas A Mack-Crane A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Sug Woo Shin, Chair Professor Xinyi Yuan Professor Richard Bamler Professor Ori Ganor Spring 2021 Abstract Counting Points on Igusa Varieties of Hodge Type by Thomas A Mack-Crane Doctor of Philosophy in Mathematics University of California, Berkeley Professor Sug Woo Shin, Chair The Langlands-Kottwitz method seeks to understand Shimura varieties in terms of automorphic forms by deriving a trace formula for the cohomology of Shimura varieties which can be compared to the automorphic trace formula. This method was pioneered by Langlands [Lan77, Lan79], and developed further by Kottwitz in [Kot90, Kot92] in the case of Shimura varieties of PEL type with good reduction. Igusa varieties were introduced in their modern form by Harris-Taylor [HT01] in the course of studying the bad reduction of certain simple Shimura varieties. The relation between Igusa varieties and Shimura varieties was expanded to PEL type by Mantovan [Man04, Man05], and their study of the cohomology of Igusa varieties was expanded to PEL type and streamlined in the Langlands-Kottwitz style by Shin [Shi09, Shi10]. Following the generalization of Mantovan’s work to Hodge type by Hamacher and Hamacher-Kim [Ham19, HK19], we carry out the Langlands-Kottwitz method for Igusa varieties of Hodge type, generalizing the work of [Shi09]. That is, we derive a trace formula for the cohomology of Igusa varieties suitable for eventual comparison with the automorphic trace formula.
    [Show full text]
  • Construction of Automorphic Galois Representations: the Self-Dual Case
    CONSTRUCTION OF AUTOMORPHIC GALOIS REPRESENTATIONS: THE SELF-DUAL CASE SUG WOO SHIN This survey paper grew out of the author's lecture notes for the Clay Summer School in 2009. Proofs are often omitted or sketched, but a number of references are given to help the reader find more details if he or she wishes. The article aims to be a guide to the outline of argument for constructing Galois representations from the so-called regular algebraic conju- gate self-dual automorphic representations, implementing recent improvements in the trace formula and endoscopy. Warning. Although it is important to keep track of various constants (e.g. sign matters!), twists by characters, etc in proving representation-theoretic identities, I often ignore them in favor of simpler notation and formulas. Thus many identities and statements in this article should be taken with a grain of salt and not cited in academic work. Correct and precise statements can be found in original papers. I should also point out that significant progress has been made, especially in the case of non-self-dual automorphic representations ([HLTT16], [Sch15]) since the original version of these notes was written. I have kept updates minimal (mostly taking place in introduction, adding references to some recent developments) and do not discuss these exciting new results here. The interested readers are referred to survey papers such as [Wei16, Mor16, Scha, Sch16, Car]. Acknowledgments. I heartily thank the Clay Mathematics Institute for its generous support toward the 2009 summer school and choosing an exciting location as Hawaii. I am grateful to the participants for their interest and wonderful questions.
    [Show full text]
  • PETER SCHOLZE to RECEIVE 2013 SASTRA RAMANUJAN PRIZE the 2013 SASTRA Ramanujan Prize Will Be Awarded to Professor Peter Scholze
    PETER SCHOLZE TO RECEIVE 2013 SASTRA RAMANUJAN PRIZE The 2013 SASTRA Ramanujan Prize will be awarded to Professor Peter Scholze of the University of Bonn, Germany. The SASTRA Ramanujan Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The prize will be awarded in late December at the International Conference on Number Theory and Galois Representations at SASTRA University in Kumbakonam (Ramanujan's hometown) where the prize has been given annually. Professor Scholze has made revolutionary contributions to several areas at the interface of arithmetic algebraic geometry and the theory of automorphic forms. Already in his master's thesis at the University of Bonn, Scholze gave a new proof of the Local Langlands Conjecture for general linear groups. There were two previous approaches to this problem, one by Langlands{Kottwitz, and another by Harris and Taylor. Scholze's new approach was striking for its efficiency and simplicity. Scholze's proof is based on a novel approach to calculate the zeta function of certain Shimura varieties. This work completed in 2010, appeared in two papers in Inventiones Mathematicae in 2013. Scholze has generalized his methods partly in collaboration with Sug Woo Shin to determine the `-adic Galois representations defined by a class of Shimura varieties. These results are contained in two papers published in 2013 in the Journal of the American Mathematical Society. While this work for his master's was groundbreaking, his PhD thesis written under the direction of Professor Michael Rapoport at the University of Bonn was a more mar- velous breakthrough and a step up in terms of originality and insight.
    [Show full text]
  • Junehyuk Jung –
    Junehyuk Jung B [email protected] • Í sites.google.com/brown.edu/junehyuk/ Positions Brown University, Department of Mathematics Providence, RI Assistant Professor 7/2020– Texas A&M University, Department of Mathematics College Station, TX Assistant Professor 8/2017–6/2020 KAIST, Department of Mathematical Science Daejeon, South Korea Researcher 6/2013–7/2016 Visiting Positions Rice University, Department of Mathematics Houston, TX Adjunct Assistant Professor 7/2019–3/2020 Yale University, Department of Mathematics New Haven, CT Visitor 4/2017–6/2017 University of California Berkeley, Department of Mathematics Berkeley, CA Visitor 2/2017–3/2017 Northwestern University, Mathematics Department Evanston, IL Visitor 10/2016–11/2016 Institute for Advanced Study, School of Mathematics Princeton, NJ Member 9/2014–12/2015 Northwestern University, Mathematics Department Evanston, IL Visiting Postdoctoral Fellow 9/2013–12/2013 Education Princeton University Princeton, NJ Doctor of Philosophy 9/2008–6/2013 Mathematics Advisor: Peter C. Sarnak Thesis Title: “On the zeros of automorphic forms” University of Chicago Chicago, IL Master of Science 9/2006–8/2008 Mathematics University of Chicago Chicago, IL Bachelor of Arts 9/2005–8/2008 Mathematics with Honors Honors and Awards National Science Foundation grant Awarded, DMS-1900993 7/2019–6/2022 Sloan Research Fellowship Awarded, $ 35,000 per year 7/2019–6/2021 Posco TJ Park Science Fellowship Awarded, $ 35,000 per year 3/2014–2/2016 Samsung Scholarship for Graduate Studies Awarded, $ 50,000 per
    [Show full text]
  • Research Statement – Kevin Childers
    RESEARCH STATEMENT { KEVIN CHILDERS 1. Summary 1.1. L-functions. The first example of an L-function is the Riemann ζ-function. In his famous paper [Rie75], Riemann proves that the ζ-function, defined for complex numbers s with <(s) > 1 as X Y ζ(s) = n−s = (1 − p−s)−1; n≥1 p has meromorphic continuation to the complex plane and satisfies a functional equation. He also states the famous \Riemann hypothesis," and explains connections between ζ(s) and the density and pattern of prime integers. Since then, other L-functions have been attached to many arithmetically interesting objects X { such as algebraic varieties, Galois representations, and automorphic objects { and many deep conjectures have been made which encode arithmetic data of X into its L-functions. Arithmetic conjectures about L-functions often depend on first proving that they are analytic in a right half plane, have meromorphic continuation, and satisfy a desired functional equation. 1.2. Galois and automorphic representations. Proving analytic properties of L-functions associated to Galois representations is, in general, considered quite hard (e.g. Artin's conjecture). In [Chi19b], I construct Galois representations ρ with Zariski dense image in a p-adic Lie group of type E7 and prove that certain L-functions naturally attached to ρ are analytic in a right half plane, have meromorphic continuation to the complex plane, and satisfy the expected functional equation. These are the first examples of their kind in the literature, and in my current research I am extending these methods to produce other new examples of L-functions attached to Galois representations satisfying desired analytic properties.
    [Show full text]
  • Galois Representations for General Symplectic Groups 3
    Galois representations for general symplectic groups ArnoKret SugWooShin October 16, 2020 Abstract We prove the existence of GSpin-valued Galois representations correspond- ing to cohomological cuspidal automorphic representations of general sym- plectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard–Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G2 in the cohomology of Siegel modular va- rieties of genus three. Under some additional local hypotheses we also show automorphic multiplicity one as well as meromorphic continuation of the spin L-functions. Keywords. Automorphic representations, Galois representations, Langlands correspondence, Shimura varieties Contents 1 Conventions and recollections 14 arXiv:1609.04223v4 [math.NT] 15 Oct 2020 2 Arthur parameters for symplectic groups 18 3 Zariski connectedness of image 23 4 Weak acceptability and connected image 26 A. Kret: Korteweg-de Vries Institute Science Park 105 1090 GE Amsterdam, Netherlands; e- mail: [email protected] S. W. Shin: Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA // Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea; e-mail: [email protected] Mathematics Subject Classification (2010): Primary 11R39; Secondary 11F70, 11F80, 11G18 1 2 Arno Kret, Sug Woo Shin 5 GSpin-valued Galois
    [Show full text]
  • Arxiv:2102.10690V1 [Math.NT] 21 Feb 2021 A
    H0 OF IGUSA VARIETIES VIA AUTOMORPHIC FORMS ARNO KRET, SUG WOO SHIN Abstract. Our main theorem describes the degree 0 cohomology of Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well known analogue for complex Shimura varieties. As an application, we obtain a completely new approach to two geometric questions. (See §1.5 for a comparison with independent results by van Hoften and Xiao via a different approach.) Firstly, we verify the discrete part of the Hecke orbit conjecture, which amounts to irreducibility of central leaves, generalizing preceding works by Chai, Oort, Yu, et al. Secondly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extend- ing previous results by Igusa, Ribet, Faltings–Chai, Hida, and others. Our proof is based on a Langlands–Kottwitz-type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in repre- sentation theory of p-adic groups due to Howe–Moore and Casselman. Contents 1. Introduction 1 2. Preliminaries in representation theory and endoscopy 9 3. Jacquet modules, regular functions, and endoscopy 24 4. Asymptotic analysis of the trace formula 34 5. Shimura varieties of Hodge type 46 6. Igusa varieties 54 7. Cohomology of Igusa varieties 58 8. Applications to geometry 65 References 68 1. Introduction Igusa varieties were studied by Igusa [Igu68] and Katz–Mazur [KM85] in the case of modular curves.
    [Show full text]
  • On the Cohomology of Rapoport-Zink Spaces of El-Type
    ON THE COHOMOLOGY OF RAPOPORT-ZINK SPACES OF EL-TYPE SUG WOO SHIN Abstract. In this article, we are interested in determining the l-adic cohomology of Rapoport- Zink spaces associated to GLn over an unramified extension of Qp (called unramified \EL-type") in connection with the local Langlands correspondence for GLn. In fact, we compute (the alternating sum of) certain representation-theoretic functors defined in terms of their cohomology on the level of Grothendieck groups. In case Rapoport-Zink spaces parametrize p-divisible groups of dimension one, the above alternating sum is determined by [HT01] (without the unramifiedness assumption). For p-divisible groups of dim > 1, Fargues ([Far04]) has obtained an answer for the supercuspidal part when the p-divisible groups are basic (i.e. their associated Newton polygons are straight lines). Our main result is an identity among the cohomology of Rapoport-Zink spaces that appear in the Newton stratification of the same unitary Shimura variety. Our result implies a theorem of Fargues, providing a second proof, and reveals new information about the cohomology of Rapoport-Zink spaces for non-basic p-divisible groups. We propose an inductive procedure to completely determine the above alternating sum, assuming a strengthening of our main result along with a conjecture of Harris. 1. Introduction Rapoport-Zink spaces are moduli spaces of Barsotti-Tate groups (a.k.a. p-divisible groups) with additional structure ([RZ96]). They turn out to be closely related to the local geometry of Shimura varieties. This relationship may be thought of as a geometric counterpart of the interaction between the local and global Langlands correspondences.
    [Show full text]
  • Patching and the P-Adic Langlands Program for GL2(Qp)
    Patching and the p-adic Langlands program for GL2(Qp) Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paˇsk¯unas and Sug Woo Shin Abstract We present a new construction of the p-adic local Langlands correspondence for GL2(Qp) via the patching method of Taylor{Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author's local-global compatibility theorem, and relaxes a hypothesis on the local mod p representation in that theorem. 1. Introduction The primary goal of this paper is to explain how (under mild technical hypotheses) the patching + construction of [CEG 16], when applied to the group GL2(Qp), gives rise to the p-adic local Langlands correspondence for GL2(Qp), as constructed in [Col10b], and as further analyzed in [Paˇs13]and [CDP14]. As a by-product, we obtain a new proof of many cases of the local-global compatibility theorem of [Eme11] (and of some cases not treated there). 1.1 Background We start by recalling the main results of [CEG+16] and the role we expect them to play in the (hypothetical) p-adic local Langlands correspondence. Let F be a finite extension of Qp, and let GF be its absolute Galois group. One would like to have an analogue of the local Langlands correspondence for all finite-dimensional, continuous, p-adic representations of GF . Let E be another finite extension of Qp, which will be our field of coefficients, assumed large enough, with ring of integers O, uniformizer $ and residue field F.
    [Show full text]
  • Arxiv:2009.08476V2 [Math.NT] 29 Jun 2021
    CONGRUENCES OF ALGEBRAIC AUTOMORPHIC FORMS AND SUPERCUSPIDAL REPRESENTATIONS JESSICA FINTZEN AND SUG WOO SHIN, WITH APPENDIX C BY VYTAUTAS PASKˇ UNAS,¯ AND APPENDIX D BY RAPHAEL¨ BEUZART-PLESSIS Abstract. Let G be a connected reductive group over a totally real field F which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on G(AF ) and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of G. We illustrate how such congruences can be applied in the construction of Galois representations. Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL2 in [Sch18, §7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton–Paˇsk¯unas on density of supercuspidal points from definite unitary groups to general G as above. Contents Introduction 2 Guide for the reader 7 Notation and Conventions 8 1. Construction of supercuspidal types 10 1.1. Construction of supercuspidal types from 0-toral data 10 1.2. Construction of supercuspidal types from 1-toral data 12 2. Construction of a family of omni-supercuspidal types 15 2.1. Abundance of 0-toral data and 1-toral data 15 arXiv:2009.08476v2 [math.NT] 29 Jun 2021 2.2.
    [Show full text]
  • Cohomology of Igusa Curves – a Survey
    COHOMOLOGY OF IGUSA CURVES { A SURVEY SUG WOO SHIN Abstract. We illustrate the strategy to compute the `-adic cohomology of Igusa varieties in the setup of ordinary modular curves, with updates on the literature towards a genrealization. 1. Introduction Igusa curves were introduced by Igusa [Igu68] to understand the mod p geometry of modular curves when the level is divisible by p, for each prime p. As a generalization, we have Igusa varieties in the setup of Shimura varieties of Hodge type thanks to [HT01, Man05, Ham17, Zha, HK19] Igusa varieties shed light on the mod p and p-adic geometry of Shimura varieties via the so-called product structure. Moreover they play a vital role in the applications to the Langlands correspondence, vanishing results on the cohomology of Shimura varieties, and p-adic automorphic forms. We refer to [KS, §1] for a detailed introduction to Igusa varieties and further references. For an application to (an extension of) the Kottwitz conjecture on the cohomology of Rapoport{Zink spaces, see [Shi12, BM]. A fundamental problem on Igusa varieties is to compute their `-adic cohomology (with or without compact support) for primes ` 6= p. To this end, the Langlands{Kottwitz (LK) method for Shimura varieties has been adapted to Igusa varieties in [HT01, Shi09, MC21], at least when the level structure at p is hyperspecial. (In [HT01], one can go a little further.) While there are excellent exposotions1 on the LK method for modular curves (with good reduction mod p) by Clozel [Clo93, §3] and Scholze [Sch11, §5], in addition to Langlands's original papers [Lan73, Lan76], there is no counterpart for Igusa curves.
    [Show full text]