## 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).

#### Upcoming talks (chronological order)

#### Past talks: 2016

- 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