## Number Theory Seminar

We are running a working seminar on things vaguely number-theoretic. For semester 2, we meet on Wednesdays at 1pm in room 107 (Richard Berry building).

You can subscribe to the mailing list here.

We will spend the foreseeable future discussing parts of Geometric Langlands implies Local Langlands.

#### Upcoming talks (chronological order)

**24 August 2016: Classical geometric Langlands, by Kari Vilonen**

#### Past talks: 2016

- 17 August 2016: Organisational meeting
- 10 August 2016: Reduction to weight $\leq p+1$ (after Edixhoven), by Alex Ghitza
- 3 August 2016: Weight optimisation for elliptic modular eigenforms (after Edixhoven), by Alex Ghitza
- 12 July 2016: The cohomology of arithmetic manifolds IV, by Arun Ram
- 6 July 2016: The cohomology of arithmetic manifolds III, by Arun Ram
- 28 June 2016: The cohomology of arithmetic manifolds II, by Arun Ram
- 21 June 2016: The cohomology of arithmetic manifolds I, by Arun Ram
- 7 June 2016: Elliptic modular forms IV (4.6-4.7), by Alex Ghitza
- 31 May 2016: Elliptic modular forms III (4.5), by Alex Ghitza
- 24 May 2016: Elliptic modular forms II (4.3-4.4), by Alex Ghitza
- 17 May 2016: Elliptic modular forms I (4.1-4.2), by Hanter Wibawanti
- 10 May 2016: Class field theory III (3.5-3.6), by Angus McAndrew
- 3 May 2016: Class field theory II (3.3-3.4), by Angus McAndrew
- 26 Apr 2016: Class field theory I (3.1-3.2), by Owen Colman
- 19 April 2016: Reciprocity laws - Fermat, Euler and Gauss, by Alex Ghitza
- 12 April 2016: Reciprocity laws - Introduction, by Alex Ghitza
- 22 March 2016: Modular curves as Riemann surfaces, by Owen Colman
- 15 March 2016: Eisenstein series and lattices, by Owen Colman
- 8 March 2016: Multiplicativity of coefficients for Siegel modular forms, by Alex Ghitza
- 1 March 2016: Multiplicativity of coefficients of Hecke eigenforms, by Alex Ghitza

#### Past talks: 2015

- 25 November 2015: The field generated by Hecke eigenvalues, by Alex Ghitza
- 18 November 2015: Numerical estimation of Hecke eigenvalues, by Alex Ghitza
- 11 November 2015: Pairs of eigenforms with the same eigenvalue II, by Alex Ghitza
- 4 November 2015: Pairs of eigenforms with the same eigenvalue I, by Alex Ghitza
- 28 October 2015: Fermat’s Last Theorem for regular primes, by Alex Ghitza
- 21 October 2015: Cyclotomic fields, by Jesse Fox
- 23 September 2015: Finiteness of the class group, by Sam Lyons
- 16 September 2015: Lattices and Minkowski’s theorem, by Angus McAndrew
- 9 September 2015: Factorisation of ideals in ring extensions, by Sam Lyons
- 2 September 2015: Unique factorisation of ideals and ideal class group, by Owen Colman
- 26 August 2015: Some commutative algebra, by Alex Ghitza
- 19 August 2015: Traces, norms and discriminants, by Jon Xu
- 12 August 2015: Rings of integers, by Dougal Davis
- 7 August 2015: Introduction to algebraic number theory, by Alex Ghitza
- 22 June 2015: Differential operators on modular forms, by Alex Ghitza
- 15 June 2015: Galois representations III, by Angus McAndrew
- 1 June 2015: Galois representations II, by Angus McAndrew
- 25 May 2015: Galois representations I, by Angus McAndrew
- 19 May 2015: Metric Diophantine approximation: the dependent theory, by Mumtaz Hussain (Newcastle)
- 11 May 2015: Hodge theory V, by Dougal Davis
- 4 May 2015: Hodge theory IV, by Dougal Davis
- 27 April 2015: Hodge theory III, by Dougal Davis
- 20 April 2015: Hodge theory II, by Dougal Davis
- 13 April 2015: Hodge theory I, by Dougal Davis
- 30 March 2015: Spin groups, by Alex Ghitza
- 23 March 2015: Everything is
**just super**, thanks for asking, by Alex Ghitza - 16 March 2015: Heisenberg groups a la Mumford III, by Arun Ram
- 9 March 2015: Heisenberg groups a la Mumford II, by Arun Ram
- 2 March 2015: Heisenberg groups a la Mumford I, by Arun Ram

#### Past talks: 2014

- 24 November 2014: Heisenberg groups and theta functions III, by Arun Ram
- 19 November 2014 at 1pm: Heisenberg groups and theta functions II, by Arun Ram
- 3 November 2014: Heisenberg groups and theta functions I, by Arun Ram
- 27 October 2014: error-free computation via p-adics: Hensel codes (12.8), by Neil Hoffman
- 20 October 2014: p-adic numbers: properties (some of 12.1-12.7), by Dougal Davis
- 13 October 2014: Equivalent numbers and Markov spectrum (5.5-5.7), by Yi Huang
- 6 October 2014: Continued fractions: quadratic irrationals (5.2-5.4), by Alex Ghitza
- 22 September 2014: Continued fractions: approximation (3.5-3.7), by Owen Colman
- 15 September 2014: Continued fractions: properties (3.1-3.4), by Ana Dow
- 8 September 2014: Farey sequences and Ford circles (2.7-2.8), by Yi Huang
- 1 September 2014: Uniform distribution in $[0,1]$ (2.6), by Alex Ghitza
- 25 August 2014: Irrationality criterion and Kronecker’s theorem, by Alex Ghitza
- 18 August 2014: Best rational approximations and Dirichlet’s theorem, by Alex Ghitza
- 11 August 2014: Recent progress on gaps between primes (after Zhang, Tao, Maynard, and others), by Alex Ghitza. Here is a link to Tao’s IHES talk
- 29 May 2014: Periods of non-cusp forms and the Eichler-Shimura isomorphism (Martin-Royer Sections 11 and 12), by Alex Ghitza
- 15 May 2014: Periods of cusp forms (Martin-Royer Section 10), by Alex Ghitza
- 8 May 2014: Hecke algebras for Siegel and paramodular forms, by Rob Sayer
- 1 May 2014: Jacobi forms and lifting theorems I, by Rob Sayer
- 17 April 2014: A peculiar family of modular forms mod p, by Alex Ghitza
- 10 April 2014: The classification of theta cycles, by Angus McAndrew
- 3 April 2014: Tomography by flat tori transform on Grassmannians and other symmetric spaces, by Eric Grinberg (UMass Boston)
- 27 March 2014: Homework as dialectic, by Rob Sayer
- 20 March 2014: Introduction to elliptic functions part III, by Arun Ram
- 13 March 2014: Introduction to elliptic functions part II, by Arun Ram
- 6 March 2014: Introduction to elliptic functions part I, by Arun Ram

#### Past talks: 2013

- 23 October 2013: Maeda’s conjecture on elliptic and Siegel modular forms, by Angus McAndrew
- 9 October 2013: Ramanujan-style congruences (after Dummigan and Fretwell), by Alex Ghitza
- 25 September 2013: Configurations of boundary components of Satake compactifications, by Alex Ghitza
- 16 September 2013: The Auslander bijections, by Claus Ringel (Bielefeld)
- 11 September 2013: A theta operator for Siegel modular forms (mod $p$), by Max Flander (Completion Seminar)
- 4 September 2013: Taking derivatives of (Siegel) modular forms, by Alex Ghitza
- 28 August 2013: Symmetric functions and Harder-type congruences, by Alex Ghitza
- 31 July 2013: Simultaneous non-vanishing of pairs of automorphic $L$-functions, by Jeff Hoffstein (Brown University)
- 10 July 2013: Alcove walks and Mirkovic-Vilonen cycles, by Arun Ram
- 26 June 2013: Local Langlands correspondence for $GL_2$ and Langlands reciprocity conjecture, by James Withers
- 5 June 2013: Automorphic representations of weight one, by Rob Sayer
- 29 May 2013: Modularity of $\tilde{\rho}_{E,3}$, by Lloyd Peters
- 22 May 2013: Explicit examples of Serre’s conjecture, by Rob Sayer
- 15 May 2013: Applications of Serre’s conjecture II, by Alex Ghitza
- 8 May 2013: Applications of Serre’s conjecture I, by Angus McAndrew
- 1 May 2013: The recipe for the weight $k$, by Alex Ghitza
- 17 April 2013: Statement(s) of Serre’s conjecture(s) and recipes for $N$, $\varepsilon$, $k$ mod $(p-1)$, by Alex Ghitza
- 10 April 2013: Galois representations mod $p$, by James Withers
- 27 March 2013: Absolute Galois group of $\mathbb{Q}$, decomposition, inertia, Frobenius elements, by Rob Sayer
- 20 March 2013: Modular forms mod $p$ and operators, by Max Flander
- 13 March 2013: A wide overview of modularity, by Alex Ghitza
- 6 March 2013: Organisational meeting

#### Past talks: 2012

- 19 December 2012: Flag varieties for $G_2$ and affine Hecke action, by Arun Ram
- 12 December 2012: Harder’s congruences for Siegel modular forms, by Alex Ghitza
- 5 December 2012, in room 213 (joint with AGT seminar): Invariants for representations (following Quillen’s equivariant cohomology), by Eric Friedlander
- 28 November 2012: Bump 1.8 – Artin L-functions and Langlands functoriality, by Rob Sayer
- 21 November 2012 at 3:20pm: Mirkovic-Vilonen cycles, by Alex Ghitza
- 14 November 2012: Bump 1.7 – Hecke characters and Hilbert modular forms, by Alex Ghitza
- 7 November 2012: The octonions and $G_2$, by Arun Ram
- 31 October 2012: Bump 1.6 – The Rankin-Selberg method, by James Withers
- 17 October 2012: Bump 1.5 – Twisting, by Jon Xu
- 10 October 2012: Bump 1.4 – Hecke operators, by Angus McAndrew
- 3 October 2012: Bump 1.3 – Modular forms for $\mathrm{SL}_2(\mathbb{Z})$, by James Withers
- 19 September 2012: Bump 1.2 – Modular group, by Yi Huang
- 12 September 2012: Bump 1.1 – Dirichlet L-functions, by Lloyd Peters
- 5 September 2012: Constructing theta maps over $(\mathbb{Z}_p^{\times}\ltimes\mathbb{Z}_p)$-extensions (continued), by Lloyd Peters
- 29 August 2012: we will take a break from our regular seminar so we can attend Sam Chow’s MPhil completion talk at 3pm in Alan Gilbert Theatre 3
- 22 August 2012: Constructing theta maps over $(\mathbb{Z}_p^{\times}\ltimes\mathbb{Z}_p)$-extensions, by Lloyd Peters
- 15 August 2012: Hecke characters, by Alex Ghitza
- 8 August 2012: Modular forms and complex multiplication, by Alex Ghitza
- 1 August 2012: Distinguishing Hecke eigenforms, by Sam Chow
- 25 July 2012: Compactifications of moduli spaces of abelian varieties, by Alex Ghitza
- 18 July 2012: Elliptic curves and polarizations, by Alex Ghitza
- 11 July 2012: The indexing of abelian varieties, by Arun Ram
- 4 July 2012: Hecke algebra for Baumslag-Solitar groups (continued), by Eddy Qiu (sorry, my notes for this are a terrible mess – I’m happy to put someone else’s here)
- 27 June 2012: Hecke algebra for Baumslag-Solitar groups by Eddy Qiu
- 20 June 2012: Hecke operators for Siegel modular forms by Max Flander
- 13 June 2012: Hecke algebras by Arun Ram

#### References for (past) talks on modularity

- video lectures from the MSRI workshop on Modularity for GL(2) and beyond.
- Diamond-Shurman: A first course in modular forms
- Edixhoven-FLT: Serre’s conjecture, in Modular forms and Fermat’s last theorem
- Edixhoven-weight: The weight in Serre’s conjectures on modular forms, Invent Math 109 (1992), no 3
- Gelbart: Modularity and the Langlands reciprocity conjecture, in Modular forms and Fermat’s last theorem
- Gross: A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math J 61 (1990), no 2
- Khare-app: Serre’s conjecture and its consequences, Japan J Math 5 (2010)
- Serre: Sur les representations modulaires de degre 2 de $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, Duke Math J 54 (1987), no 1. I am producing an English translation of this paper (work in progress)
- Taylor-ICM: Galois representations, Proceedings of the Beijing ICM
- Wiese: Galois representations (notes)

#### Some requests

Here are some suggestions for future talks:

- Atkin-Lehner theory for modular forms; this is in many of the books on modular forms
- The Satake isomorphism, see the paper by Nelsen and Ram and the paper by Dick Gross
- Half-integral weight modular forms, see Chapter 3 in Ken Ono’s book
*The web of modularity*, and Nick Ramsey’s thesis and subsequent paper - The Shimura correspondence linking modular forms of half-integral weight to modular forms of integral weight, see Bump’s short introduction for a start
- Number-theoretic aspects of partition functions, see Chapter 5 in Ken
Ono’s book
*The web of modularity* - Theta correspondence, see Prasad’s survey (this would very likely take a series of talks)
- Lehmer’s conjecture on coefficients of Delta, start with this mathoverflow question and ask me for more references
- Quasi-modular forms and differential structure, see Chapter III in the notes by Martin-Royer