London Number Theory Study Group
Previous Study Groups

The programme for

Summer 2017

The study groups were at KCL, and we had three: "My favourite conjecture" (run by James Newton), "p-divisible groups" (run by Wansu Kim and Joe Kramer-Miller) and "Topics in Sieve Theory".

Spring 2017

This term the study groups were on p-adic integration, organized by Jack Lamplugh and Joe Kramer-Miller, and p-adic Hodge theory over Lubin-Tate extensions (organised by Joaquin Rodrigues-Jacinto and Sarah Zerbes). They were at UCL.


Autumn 2016

This term the study groups were on "my favourite paper" (organised by Kevin Buzzard), an analytic study group, and log geometry (organised by Johannes Nicaise). They were at Imperial.


Summer 2016

This term the study groups were on Classification of mod p representations of p-adic groups (organised by Julien Hauseux), an analytic study group, and Scholze's Lubin-Tate tower paper (organised by Wansu Kim). They were at Kings.


Spring 2016

This term the study groups were on the Iwasawa theory of elliptic curves (organised by Guhan Harikumar) and "Pretentiousness and beyond" (organised by A. Granville). They were at UCL.


Autumn 2015

This term the study groups were on the p-adic Langlands programme for GL(2,Q_p) (organised by Toby Gee and James Newton), and the yoga of weights (organised by Wansu Kim and Toby Gee). They were at Imperial College.


Summer 2015

This term the study group was on the Arithmetic of Euler Systems. It was organised by Sarah Zerbes and was at Kings College.


Spring 2015

This term there were two study groups. The first was on p-adic modular forms and eigenvarieties, organised by James Newton and Sarah Zerbes; the second was on Deligne's proof of the Weil conjectures, organised by Olivier Taibi. The study groups were at UCL.


Autumn 2014

This term the study group was on Vincent Lafforgue's work on global Langlands conjectures; our main reference was this ArXiv paper of Lafforgue's. It was at Imperial College and was organised by Toby Gee.


Summer 2014

This term the study group was on Jared Weinstein's paper "Semistable models for modular curves of arbitrary level"; it was at Kings College and was organised by Toby Gee.


Spring 2014

This term there were two study groups: one on Bhargava's work on average ranks of elliptic curves (organized by Toby Gee), and one on perfectoid spaces (organized by Sarah Zerbes).


Autumn 2013

This term there was a study group on K3 surfaces, in particular on the Tate conjecture for K3 surfaces and recent work of Keerthi Madapusi Pera. It was organized by Toby Gee and Alexei Skorobogatov.


Summer 2013

This term there was a study group on the new approach to proving Langlands functoriality proposed by Langlands, Frenkel and Ngo using ideas from the geometric Langlands program. It was organized by Alex Paulin.


Spring 2013

This term there were two study groups, hosted at UCL. From 1300-1415 we had Modularity Lifting Theorems (run by Judith Ludwig and following Toby Gee's Arizona Winter School notes) and 1415-1530 we had Fargues--Fontaine (run by Fred Diamond and following the two Fargues--Fontaine papers).


Autumn 2012

This term there were two London Number Theory Study Groups, hosted by Imperial. From 1200 to 1330 there was a study group run by Christian Johansson, Tristan Kalloniatis, Judith Ludwig and Jack Shotton, on deformation theory. From 1400 to 1530 there was a study group on Arthur's Conjectures, run by Toby Gee.


Summer 2012

This term there were two London Number Theory Study Groups, hosted by Kings. From 1200 to 1330 there was a study group run by Fred Diamond on Breuil-Kisin modules and applications. From 1400 to 1530 there was a study group run on Shimura varieties, Hida Theory and Galois representations for unitary groups, run by Mahesh Kakde.


Spring 2012

This term there were two London Number Theory Study Groups, hosted by UCL. From 1300 to 1400 there was a study group run by Kevin Buzzard on Iwasawa theory and Wiles' proof of the Main Conjecture. From 1430 to 1530 there was a study group run by Ioannis Petridis on subconvexity in the theory of L-functions.


Autumn 2011

This term there were two Study Groups, held at Imperial. One was run by Kevin Buzzard and was on the Darmon-Dasgupta-Pollack paper, and the other was run by Fred Diamond and was an introduction to p-adic Hodge Theory.


Spring 2011

This term the study group was held at Kings, and was on Scholze's proof of the local Langlands conjectures for GL(n). It was organised by Fred Diamond and met 1400-1530.


Spring 2011

This term there were two Study Groups, held at UCL. "The Bernstein Centre" met 1200-1330, organised by Kevin Buzzard, and "Vanishing Cycles" met 1400-1530, organised by Wansu Kim.


Autumn 2010

This term there were two Study Groups, held at Imperial College. "L-functions of Shimura varieties" met 1200-1330, organised by Kevin Buzzard, and "Etale Homotopy" met 1400-1530, organised by Ambrus Pal.


Summer 2010

This term the Study Group was held at Kings College and was organised by Shu Sasaki. The subject was "Deformations of Galois representations over imaginary quadratic fields".

The study group schedule for Spring 2010

12 May Introduction (Shu Sasaki)
19 May Families of p-adic automorphic forms for GL_1 and GL_2 over imaginary quadratic fields (David Loeffler)
26 May The Strong Leopoldt conjecture (Edward Sanders)
2 June The London-Paris Number Theory Seminar
speakers: F. Brown, M. Hindry, M. Kakde
9 June Proof of the main theorem of Calegari-Mazur (Christian Johannson)
16 June Families of ordinary modular forms over imaginary quadratic fields (James Newton)


Spring 2010

This term the Study Group was held at University College and was organised by Mahesh Kakde. The subject was "Motivic L-functions".

The study group schedule for Spring 2010

13 January Overview (Minhyong Kim)
20 January Deligne's conjectures (Kevin Buzzard)
27 January Beilinson conjectures 1 (Andreas Holmstrom)
10 February Beilinson conjectures 2 (Andreas Holmstrom)
17 February Bloch Kato 1 (Cecilia Busuioc)
24 February Bloch Kato 2 (Wansu Kim)
3 March Bloch Kato 3 (Mahesh Kakde)
10 March Main conjecture (a la Kato) (Minhyong Kim)
17 March Noncommutative main conjecture (Mahesh Kakde)


Autumn 2009

This term the Study Group was held at Imperial College and was organised by Kevin Buzzard and Shu Sasaki. The subject was Ramakrishna's lifting theorems.

The study group schedule for Autumn 2009

7 October Introduction (Kevin Buzzard)
14 October Group cohomology etc. (Christian Johannson)
21 October local Tate duality etc. (Edward Sanders)
28 October Deformations of Galois representations: intro (Owen Jones)
4 November Deformations of Galois representations (Fred Diamond)
11 November Ramakrishna's theorem I (Shu Sasaki)
18 November Ramakrishna's theorem II (James Newton)
25 November Ramakrishna's theorem III (minimal lift) (Wansu Kim)
2 December Ramakrishna's theorem III (minimal lift) (continued) (Wansu Kim)
9 December Section 2 of Taylor's "On icosahedral Artin representations II" (Kevin Buzzard)


Summer 2009

This term the Study Group was held at King's College and was organised by Behrang Noohi. The subject was "Locally symmetric spaces and arithmetic groups".

The study group schedule for Summer 2009

29 April Introduction (Behrang Noohi)
Algebraic groups, part 1 (Owen Jones)
6 May Algebraic groups, part 2 (Kevin McGerty)
20 May Symmetric spaces (Behrang Noohi)
27 May
11:30am-1:00pm
Compactifications of locally symmetric spaces (Payman Kassaei)
3 June
  - 4 June
The London-Paris Number Theory Seminar
at King's College London, room 2B08
theme: p-adic modular forms
speakers: Buzzard, Fargues, Loeffler, Colmez, Mokrane, Dimitrov, Panchishkin
10 June Buildings (Kevin Buzzard)


Spring 2009

This term the Study Group was held at University College and was organised by Ambrus Pal. The subject was Ngo's recent proof of the fundamental lemma.

The study group schedule for Spring 2009

14 January Statement of the fundamental lemma (Kevin Buzzard)
21 January Passage to the equicharacteristic case (Ambrus Pal)
28 January Stacks and equivariant cohomology (Behrang Noohi)
4 February Neron models (Andrei Yafaev)
11 February Compactified Jacobians (Nick Shepherd-Barron)
18 February The Hitchin fibration (Tamas Hausel)
25 February
12:00-13:30
Perverse sheaves (Kevin McGerty)
During the week of March 9-13, instead of the usual study group Professor Ngo Bao Chau gave a series of lectures, followed by a question-and-answers session. The schedule was the following:
Monday 9 March
at 3:00 pm
room 706
Lecture 1:
Fundamental lemma and Hitchin fibration
Monday 9 March
at 4:30 pm
room 706
Lecture 2:
Symmetry of Hitchin fibration and endoscopy
Friday 13 March
at 4:00 pm
room 500
Lecture 3:
Decomposition theorem in the case of the Hitchin fibration


Autumn 2008

This term the Study Group was held in Imperial College and was organised by Ambrus Pal. The subject was the work of Ciperiani-Wiles on solvable points.

The study group schedule for Autumn 2008

8 October Introduction (Ambrus Pal)
15 October The modular curve X_0(N) (Kevin Buzzard)
22 October Complex multiplication (James Newton)
29 October Ramified over unramified principle (Alexei Skorobogatov)
5 November Kolyvagin classes (Minhyong Kim)
12 November The Andre-Oort conjecture (Andrei Yafaev)
19 November The Gross-Zagier formula, I (Fred Diamond)
26 November Solvable points on geometrically rational surfaces (Ambrus Pal)
3 December The Gross-Zagier formula, II (Richard Hill)
10 December The ex-Ax conjecture (Ambrus Pal)
17 December Brauer diagonalisation and solving in solvable extensions (Trevor Wooley)


Summer 2008

This term the Study Group was held in King's College and organised by Kevin Buzzard. The subject was the seminal Coleman-Mazur paper "The eigencurve".

The study group schedule for Summer 2008

23 April Introduction (Kevin Buzzard)
30 April Modular forms (James)
7 May
  - 8 May
The London-Paris Number Theory Seminar
14 May p-adic L-functions (Malcolm)
21 May p-adic modular forms, weight k (Payman)
28 May p-adic modular forms, weight kappa (Kevin)
4 June Compact operators on p-adic Banach spaces/modules (Owen)
11 June Eigencurve via Hecke algebras (David L)
18 June Eigencurve via deformation theory (Kevin/Fred)


Spring 2008

This term the Study Group was held in University College and organised by Ambrus Pal and Alexei Skorobogatov. The subject was arithmetic fundamental groups.

The study group schedule for Spring 2008

9 January Introduction
16 January Belyi's theorem (Owen Jones)
23 January Chabauty's theorem (Richard Thursby)
30 January Tannakian formalism (Alex Paulin)
6 February Deligne's paper on the fundamental group of the projective line without 3 points (Kevin Buzzard)
13 February Bloch-Kato and the definition of H^1_f of a Galois representation (David Loeffler)
20 February F-crystals (Sarah Zerbes)
27 February Crystalline fundamental group (Ambrus Pal)
5 March Global and local Selmer varieties (Minhyong Kim)
12 March Bloch-Kato/Fontaine-Mazur imply Mordell (Minhyong Kim)


Autumn 2007

This term the Study Group was held in Imperial College and organised by Sarah Zerbes. The subject was "Fontaine's theory of p-adic representations".

The study group schedule for Autumn 2007

3 October The l-adic case (Kevin)
10 October Overview of Fontaine's work (Sarah)
17 October Local fields (Owen)
24 October Witt vectors (James)
31 October General equivalence of categories (Richard)
7 November Fields of norms (Kevin)
14 November (phi,Gamma)-modules (Payman)
21 November B_dR + properties (Fred)
28 November B_cris/B_max + properties (Alex)
5 December B_st, example of the representation associated to an elliptic curve (tba)
12 December Relation to (phi,Gamma)-modules (outline) (David)


Summer 2007

This term the Study Group was held in University College and organised by Kevin Buzzard. The subject was Emerton's work on eigenvarieties.

The study group schedule for Summer 2007

25 April Constructing representations (Owen Jones, Imperial)
Travaux d'Emerton (Kevin Buzzard, Imperial)
2 May The London-Paris Number Theory Seminar
11 am - 4:30 pm, Imperial College London
9 May p-adic Banach spaces (Shu Sasaki, Imperial)
Shimura manifolds (Alex Paulin, Imperial)
16 May Emerton's spectral sequence (David Loeffler, Imperial; Richard Hill, UCL)
23 May Classical modular forms and locally algebraic representations (Payman Kassaei, KCL)
The case G(R) compact (Kevin Buzzard, Imperial)
30 May Jacquet functors and eigenvarieties (Kevin Buzzard, Imperial; Toby Gee, Imperial)


Spring 2007

This term the Study Group was held in King's College and organised by Payman Kassaei. The subject was "Representations of p-adic groups and the Local Langlands Conjecture".

The study group schedule for Spring 2007

17 January Introduction (Payman Kassaei, King's)
24 January Admissible representations of l-groups (Owen Jones, Imperial)
31 January The Jacquet functor and supercuspidal representations (Rebecca Torrey, King's)
7 February The classification theorem I (Shu Sasaki, Imperial)
14 February The classification theorem II (Brian Tyler, UCL)
21 February Whittaker and Kirillov models and strong multiplicity one (Payman Kassaei, King's)
28 February Supercuspidal representations of GL_n (David Loeffler, Imperial)
7 March The Satake isomorphism (Richard Hill, UCL)
14 March Local-global compatibility (Alex Paulin, Imperial)
21 March Henniart's approach to Local Langlands (Toby Gee, Imperial)


Autumn 2006

This term the Study Group was held in Imperial College and was organised by Richard Hill. The subject was "(g,K)-cohomology".

The study group schedule for Autumn 2006

4 October Introduction (Richard Hill, UCL)
11 October Structure of semisimple and reductive groups over R (Owen Jones, IC)
18 October Highest weight representations (David Loeffler, IC)
1 November Harish-Chandra isomorphism (Imma Galvez-Carrillo, London Metropolitan University)
8 November Classification of (g,K)-modules (Alex Paulin, IC)
15 November (g,k) and (g,K) and g cohomology and a vanishing theorem (Brian Tyler, UCL)
22 November Laplacian and Casimir elements and calculation of (g,k)-cohomology (Kevin Buzzard, IC)
29 November (g,k)-cohomology of principal series (Shu Sasaki, IC)
6 December (g,k)-cohomology of discrete series (Payman Kassaei, KCL)
13 December tba (Richard Hill, UCL)


Summer 2006

This term the Study Group was held in King's College and was organised by Manuel Breuning. The subject was quadratic forms.

The study group schedule for Summer 2006

3 May Introduction (Manuel Breuning)
10 May Witt's cancellation and decomposition theorems (James Barrett)
17 May Quadratic forms over completions of Q (Malcolm Bovey)
24 May Hasse-Minkowski (principle) (Andrew Parker)
31 May Witt rings (James Barrett)
7 June Clifford algebras (Andrew Parker)
14 June Pfister forms and function fields (Manuel Breuning)


Spring 2006

This term the Study Group was held in King's College and was organised by Manuel Breuning and Andrew Parker. The subject was "Classical K-theory of number fields".

The study group schedule for Spring 2006

18 January Overview (Andrew Parker and Manuel Breuning)
25 January Definitions and properties of K_0(R) and examples (Malcolm Bovey)
1 February Definitions and properties of K_1(R) and examples (Brian Tyler)
8 February Definitions and properties of K_2(R) and examples (David Mireles Morales)
15 February Long exact sequence of K-groups for Dedekind domains (James Barrett)
22 February K_2(Q) / Galois cohomology and Brauer Groups (David Mireles Morales / Andrew Parker)
1 March Merkurjev-Suslin Theorem (Andrew Parker)
8 March K_1 of the ring of integers of a number field and the congruence subgroup problem (Toby Gee)
15 March K_2 of the ring of integers of a number field (Manuel Breuning)
22 March Lichtenbaum Conjectures (David Burns)