The theory of numbers, from Ancient Greece to the 21st century
(Slides for the 2024 Mahler public lecture)
pdf
The following papers are either submitted for publication, or in the process of being revised prior to being submitted. New drafts may appear from time to time.
An introduction to the categorical p-adic Langlands program
(with Toby Gee and Eugen Hellmann)
pdf
Explicit reciprocity laws and Iwasawa theory for modular forms
(with Robert Pollack and Tom Weston)
pdf
Localization of smooth p-power torsion representations of GL_2(Q_p)
(with Andrea Dotto and Toby Gee)
pdf
Moduli stacks of two-dimensional Galois representations
(with Ana Caraiani, Toby Gee, and David Savitt)
pdf
Local-global compatibility in the p-adic Langlands
programme for GL_2/Q
(Draft dated March 23, 2011)
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pdf
On a class of coherent rings, with applications to the
smooth representation theory of GL_2(Q_p) in characteristic p
(Draft dated May 19, 2008)
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pdf
The following papers have either appeared, or have been accepted to appear.
For those papers that have appeared in print, the
versions
presented here may differ slightly from the published
versions. (For example, they don't always incorporate revisions
made to proofs.)
Moduli stacks of etale (phi,Gamma)-modules: a survey
(with Toby Gee)
To appear in Proceedings of the International
Colloquium on Arithmetic Geometry, TIFR Mumbai, Janury 6-10, 2020.
pdf
Moduli stacks of etale (phi,Gamma)-module and existence
of crystalline lifts
(with Toby Gee)
Appeared in Annals of Math. Studies vol. 215, Princeton University Press,
2023.
pdf
Here is a list of errata.
Langlands reciprocity: L-functions, automorphic forms, and Diophantine equations
Appeared in The genesis of the Langlands program
(Julia Mueller, Freydoon Shahidi eds.),
London Mathematical Society Lecture Notes Series,
vol. 467, 2021, 301-386.
pdf
Globally realizable components of local deformation rings
(with Frank Calegari and Toby Gee)
Appeared in J. Institut Math. Jussieu 21 (2022), 533-602.
pdf
``Scheme-theoretic images'' of certain morphisms of stacks
(with Toby Gee)
Appeared in Alg. Geom. 8 (2021), 1-132.
pdf
On the density of supercuspidal points of fixed regular
weight in local deformation rings and global Hecke algebras
(with Vytas Paskunas)
Appeared in J. de l'Ecole Poly. Math. 7 (2020), 337-371.
pdf
Compatible systems of Galois representations associated to the
exceptional group E_6
(with George Boxer, Frank Calegari, Brandon Levin,
Keerthi Madapusi Pera, and Stefan Patrikis)
Appeared in Forum Math. Sigma 7 (2019), e4, 29 pages.
pdf
Patching and the p-adic Langlands program for GL_2(Q_p)
(with Ana Caraiani, Toby Gee, David Geraghty, Vytas Paskunas,
and Sug Woo Shin)
Appeared in Comp. Math. 154 (2018), no. 3, 503-548.
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pdf
Explicit Serre weights for two-dimensional Galois representations
(with Frank Calegari, Toby Gee, and Lambros Mavrides)
Appeared in Comp. Math 153 (2017), no. 9, 1893-1907.
pdf
Patching and the p-adic local Langlands correspondence
(with Ana Caraiani, Toby Gee, David Geraghty, Vytas Paskunas,
and Sug Woo Shin)
Appeared in Camb. J. of Math. 4 (2016), no. 2, 197-287.
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pdf
Homological stability for completed homology
(with Frank Calegari)
Appeared in Math. Ann. 364 (2016), no.3-4, 1025-1041.
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pdf
p-adic Hodge-theoretic properties of etale cohomology
with mod p coefficients, and the cohomology of Shimura varieties
(with Toby Gee)
Appeared in Algebra and Number Theory 9 (2015), no. 5, 1035-1088.
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pdf
Lattices in the cohomology of Shimura curves
(with Toby Gee and David Savitt)
Appeared in Invent. Math. 200 (2015), no. 1, 1-96.
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pdf
Completed cohomology and the p-adic Langlands program
Appeared in the Proceedings of the 2014 ICM.
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pdf
The local Langlands correspondence for GL(n) in families (with
David Helm)
Appeared in Ann. Sci. E.N.S. (4) 47 (2014), no. 4, 655-722.
pdf
A geometric perspective on the Breuil--Mezard conjecture
(with Toby Gee)
Appeared in J. Institut Math. Jussieu 13 (2014), no. 1, 183-223.
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pdf
Weight cycling and Serre-type conjectures for unitary groups (with
Toby Gee and Florian Herzig)
Appeared in Duke Math. J. 162 (2013), no. 9, 1649-1722.
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pdf
Completed cohomology --- A survey (with Frank Calegari)
Appeared in
Nonabelian Fundamental Groups and Iwasawa Theory
(John Coates, Minhyong Kim,
Florian Pop, Mohamed Saidi, Peter Schneider eds.),
London Mathematical Society Lecture Notes Series,
vol. 393, 2011, 239-257.
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pdf
Mod-p cohomology growth in p-adic analytic towers of 3-manifolds (with Frank Calegari)
Appeared in Groups, Geometry, and Dynamics
5 (2011), 355-366.
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pdf
p-adic families of modular forms [after Hida, Coleman, and Mazur]
Appeared in Seminaire Bourbaki, 2009/2010,
expose 1013, Asterisque 339 (2011), 31-61.
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pdf
On the effaceability of certain delta-functors (with Vytas Paskunas)
Appeared in Asterisque 331 (2010), 439--447.
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pdf
Ordinary parts of admissible representations of p-adic reductive
groups I. Definition and first properties
(Revised version, dated January 19, 2010)
Appeared in Asterisque 331 (2010), 335--381.
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pdf
Ordinary parts of admissible representations of p-adic reductive
groups II. Derived functors
(Revised version, dated January 19, 2010)
Appeared in Asterisque 331 (2010), 383--438.
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pdf
Jacquet modules of locally analytic representations
of p-adic reductive groups II. The relation to parabolic induction
(Draft dated October 10, 2007)
To appear in J. Institut Math. Jussieu.
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pdf
Bounds for multiplicities of unitary representations of
cohomological type in spaces of cusp forms (with Frank Calegari)
Appeared in Ann. of Math. 170 (2009), 1437--1446
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pdf
Representations p-adique ordinaires de GL_2(Q_p) et compatibilite
local-global (with Christophe Breuil)
Appeared in Asterisque 331 (2010), 235--295.
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pdf
Elliptic curves of odd modular degree (with Frank Calegari)
Appeared in Israel J. Math. 169 (2009), 417-444.
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pdf
Locally analytic representation
theory of p-adic reductive groups: A summary of some recent
developments
(Revised version, dated March 16, 2007)
Appeared in L-functions and Galois representations
(D. Burns, K. Buzzard and
J. Nekovar, eds.),
London Mathematical Society Lecture Note Series,
vol. 320, 2007, 407-437.
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pdf
A local-global compatibility conjecture in the p-adic Langlands
programme for GL_2/Q
Appeared in Pure and Applied Math. Quarterly 2 (2006), no. 2, 279-393.
(special issue in honour of John Coates' 60th birthday).
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pdf
Variation of Iwasawa invariants
in Hida families (with Robert Pollack and Tom Weston)
Appeared in Invent. Math. 163 (2006), no. 3, 523-580.
(The official electronic version can be found
here
on the SpringerLink website.)
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pdf
Jacquet modules of locally analytic
representations of p-adic reductive groups I. Construction
and first properties
Appeared in Ann. Sci. E.N.S. 39 (2006), no. 5, 775-839.
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pdf
p-adic L-functions and
unitary completions of representations of p-adic reductive groups
Appeared in Duke Math. J. 130 (2005), no. 2, 353-392.
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pdf
On the interpolation of systems of
eigenvalues attached to automorphic Hecke eigenforms
Appeared in Invent. Math. 164 (2006), no. 1, 1-84.
(The official electronic version can be found
here
on the SpringerLink website.)
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pdf
Locally analytic vectors in
representations of locally p-adic analytic groups
(Revised version, dated September 19, 2011)
To appear in Memoirs of the AMS.
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pdf
On the ramification of Hecke algebras
at Eisenstein primes (with Frank Calegari)
Appeared in Invent. Math. 160 (2005), no. 1, 97-144.
(The official electronic version can be found
here
on the SpringerLink website.)
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pdf
A geometric Jacquet functor (with David Nadler and Kari Vilonen)
Appeared in Duke Math. J. 125 (2004), no. 2, 267-278.
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pdf
An introduction to the Riemann-Hilbert correspondence
for unit F-crystals (with Mark Kisin)
Appeared in Geometric Aspects of Dwork Theory
(A. Adolphson etc. eds.), Vol. II, Walter de Gruyter, 2004, 677-700.
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pdf
The Riemann-Hilbert correspondence for unit F-crystals
(with Mark Kisin)
Appeared in Asterisque 293 (2004).
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pdf
The Hecke algebra has large index (with Frank Calegari)
Appeared in Math. Research Letters 11 (2004), no. 1, 125-137.
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pdf
Optimal quotients of modular Jacobians
Appeared in Math. Ann. 327 (2003), no. 3, 429-458.
(The official electronic version can be found
here on the SpringerLink website.)
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pdf
Supersingular elliptic curves, theta series and weight two modular forms
Appeared in J. Amer. Math. Soc. 15 (2002), no. 3, 671-714.
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pdf
Unit L-functions and a conjecture of Katz (with Mark Kisin)
Appeared in Ann. of Math. 153 (2001), no. 2, 329-354.
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pdf
The Eisenstein ideal in Hida's ordinary Hecke algebra
Appeared in IMRN 1999, No. 15, 793-802.
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pdf
A new proof of a theorem of Hida
Appeared in IMRN 1999, No. 9, 453-472.
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pdf
The following papers are ones which I do not intend to publish at the present time.
Formal algebraic stacks
pdf
On the vanishing of certain extensions of admissible
smooth representations in characteristic p
pdf
This very short note relates Gelfand--Kirillov dimensions,
duality, and extensions of representations.
Deformations of Galois representations and applications --- talk at the
IHP conference on the occassion of the Galois bicentennial
pdf
(These are more-or-less the notes of a talk I gave at the indicated
conference.
One thing I learned at the conference, from the talk of Bost,
and that I mentioned in my talk, but which is not included in the
notes, is that some of the Galois theoretic aspects of the Taylor--Wiles
auxiliary primes were already noted in a letter of Jacobi in the 19th
century. I find this thread, and other threads, of continuity in the
development of our subject very pleasing!)
A p-adic variational
Hodge conjecture and modular forms with complex multiplication
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pdf
Extensions of crystalline
representations (with Mark Kisin)
(Note that this paper is unfinished, and will likely remain so
in the future. The fourth section
in particular requires some corrections.)
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The following report gives an account of research undertaken by
a working group at the
American Insitute of Mathematics
investigating the mod p local Langlands correspondence, during the workshop
``p-adic representations, modularity, and beyond'', held at AIM
in February, 2006
pdf
In 1999 the IAS/Park City Mathematics Institute was
devoted to arithmetic geometry.
I gave an informal lecture
there introducing the subject of p-adic modular forms, a
write-up of which has been incorporated as an
appendix to Fernando Gouvea's lecture
notes from the Insititute.
Here is an early draft of this
appendix (it is missing figures
and some corrections).
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The final version is for sale
here .
Here is a gzipped tar file which contains the various
chapters (in postscript
format)
of my Harvard
doctoral thesis "2-adic modular forms of minimal slope".
emerton-thesis.tar.gz
If you are unsure what to do with a gzipped tar file, here are the directions.