The London-Paris Number Theory Seminar archive

Archive of past meetings (main page here)

37th meeting, London.

9-10 June, 2025, KCL. Automorphic forms.

Speakers: Lucile Devin (LMPA, Université du Littoral Côte d'Opale), Min Lee (Bristol), Asbjørn Nordentoft (Paris Saclay), Nicole Raulf (Lille), Alex Walker (UCL), Igor Wigman (KCL)

The webpage of the meeting, including schedule, titles, and abstracts, is here.

36th meeting, Paris.

November 4th and 5th, 2024, Jussieu. Special functions, transcendence and periods.

Speakers: Christopher Daw (Reading), Lucia di Vizio (Versailles), Peter Jossen (King's), Martin Orr (Manchester), Tanguy Rivoal (Grenoble), David Urbanik (IHES).

The webpage of the meeting, including schedule, titles, and abstracts, is here.

35th meeting, London.

June 13th and 14th, 2024, UCL. Algorithmic number theory.

Speakers: Lassina Dembélé, Tim Dokchitser, Elisa Lorenzo García, Chloe Martindale, Pascal Molin, Aurel Page.

The webpage of the meeting, including schedule, titles, and abstracts, is here.

34th meeting, Paris.

November 28th and 29th, 2023, Paris, Centre des Colloques du Campus Condorcet. Special values of L functions -- dedicated to the memory of John Coates.

Speakers: Mladen Dimitrov, Olivier Fouquet, Matteo Tamiozzo, Luis Garcia, Holly Green, Michael Harris.

The schedule, titles, and abstracts are available here.

33rd meeting, London.

June 12th, 13th and 14th, 2023, Imperial College London. To the memory and mathematical legacy of Yuri Manin.

Speakers: K. Cesnavicius, A. Morgan, A. Cadoret, D. Loughran, O. Wittenberg, S. Feyzbakhsh, Yu. Zarhin, L. Merel.

The schedule, titles, and abstracts are available here.

32ed meeting, Paris.

November 28th and 29th 2022, Institut de Math\'ematiques de Jussieu--Paris Rive Gauche. Higher Coleman theory and applications.

Speakers: George Boxer, Juan-Esteban Camargo, Giada Grossi, David Loeffler, Vincent Pilloni, Sarah Zerbes.

The schedule, titles, and abstracts are available here.

31st meeting, London.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from the Heilbronn Institute for Mathematical Research, ERC Advanced Grant AAMOT, and the ANR "CLap CLap".

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Steve Lester, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Daniel Fiorilli, Michael Harris, Marc Hindry, Stefano Morra, Matthew Morrow.

The 31st meeting of the LPNTS will take place in London, and is focussed on the representation theory of reductive groups.

The seminar will take place on 16th and 17th May 2022 in the Anatomy Lecture Theatre (K6.29) at Kings' College London. The schedule is:

Monday 16th May:

11:00 Welcome/coffee
11:30 Jessica Fintzen
12:30 Lunch
14:00 Rob Kurinczuk
15:00 Thomas Lanard
16:00 Coffee
16:30 Arthur-César Le Bras

Tuesday 17th May:

10:00 Simon Riche
11:00 Coffee
11:30 Beth Romano

More details are available on the KCL website.

[The schedule, titles, and abstracts are available here.]

30th meeting, Paris.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris (although it's been online at times in the Covid year). It is supported by grants from the Heilbronn Institute for Mathematical Research, ERC Advanced Grant AAMOT, and the ANR "CLap CLap".

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Steve Lester, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Daniel Fiorilli, Michael Harris, Marc Hindry, Stefano Morra, Matthew Morrow.

The 30th meeting of the LPNTS took place in Paris, and was focussed on elliptic curves, abelian varieties, torsion points and heights.

The seminar took place 29-30 November 2021 in Paris Nord, and also on Zoom. The schedule is available here.

[The speakers were Richard Griffon, Rachel Newton, Diego Izquierdo, Adam Morgan, Celine Maistret, and Marco D’Addezio. The schedule, titles, and abstracts are available here.]

29th meeting, Cambridge.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris (although it's been online at times in the Covid year). It is supported by grants from the Heilbronn Institute for Mathematical Research, ERC Advanced Grant AAMOT, and the ANR "CLap CLap".

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Steve Lester, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Daniel Fiorilli, Michael Harris, Marc Hindry, Stefano Morra, Matthew Morrow.

The 29th meeting of the LPNTS exceptionally took place neither in London nor Paris; it was a collaboration with the Isaac Newton Institute in Cambridge, and was dedicated to the memory of Professor Sir Swinnerton-Dyer.

The seminar took place 17th-20th May 2021 at the Newton Institute. The schedule is available here.

28th meeting, Zoom.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from the Heilbronn Institute for Mathematical Research, ERC Advanced Grant AAMOT, and the ANR "CLap CLap".

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Steve Lester, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Daniel Fiorilli, Michael Harris, Marc Hindry, Stefano Morra, Matthew Morrow.

The 28th meeting of the LPNTS was held online, and showcased the work of PhD students in London and Paris.

The seminar took place 23-25 November 2020 (Mon to Wed), 1230-1600 London time, 1330-1700 Paris time, with five 30 minute talks each day. The schedule is here.

[The speakers were Zhixiang Wu, Johannes Girsch, Amadou Bah, Andrew Graham, Sally Gilles, Ashwin Iyengar, Yichang Cai, Pol van Hoften, Juan Esteban Rodriguez, Hanneke Wiersema, Ning Guo, Omri Faraggi, Bart Michels, Petru Constantinescu, and Alexandre Lartaux.]

27th meeting, Paris.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from the Heilbronn Institute for Mathematical Research, ERC Advanced Grant AAMOT, and the ANR "CLap CLap".

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Olivier Fouquet, Michael Harris, Marc Hindry, Stefano Morra, Matthew Morrow.

The 27th meeting of the LPNTS: To the memory of Jean-Marc Fontaine and Jean-Pierre Wintenberger.

Dates: Monday Oct 14th and Tuesday Oct 15th, 2019.
URGENT: FOLLOWING THE CLOSURE OF THE ORSAY MATHEMATICS DEPARTMENT, THE SEMINAR WILL TAKE PLACE AT THE IHES (SIMONS AMPHITHEATRE)

If you are going to attend the XXVII Paris-London Number Theory Seminar, please send a confirmation email to Stefano Morra (morra@math.univ-paris13.fr).

Schedule:

Monday:
8.30-9.15: Welcome
9.30-10.30 Luc Illusie (Orsay): "Relèvements modulo p et filtration de Nygaard"
10.45-11.15 Pause Café
11.15-12.15 Pierre Colmez (CNRS Jussieu): Flânerie dans le programme de Fontaine.
12.15-14.00 Déjeuneur (Restaurant "Les Cèdres")
14.00-15.00 : Nathalie Wach (Strasbourg): "Le corps des normes de certaines extensions infinites de corps locaux"
15.00-15.30 Pause Café
15.45-16.45 : John Coates (Cambridge): "L-values and the exact Birch-Swinnerton-Dyer formula"
17.00-18.00 : Agnès David (Besançon): "Déformations galoisiennes et variétés de Kisin dans la conjecture de Breuil-Mézard"

Tuesday:
9.30-10.30 Bernadette Perrin-Riou (Orsay): "Promenade dans le jardin des symboles modulaires"
10.45-11.15 Pause Café
11.15-12.15 : Christophe Breuil (CNRS Orsay): "Espace de Drinfeld, complexe de de Rham et représentations localement analytiques de GL_3(Q_p)"
12.15-14.00 Déjeuneur (Restaurant "Les Cèdres")
14.00-15.00 : Toby Gee (Imperial College London): Lifting Galois representations
15.15-16.15 : Laurent Fargues (CNRS Jussieu): "La correspondance de Langlands locale: construction des paramètres semi-simples"

Abstracts can be viewed here.

A limit amount of funding is available for travel for UK participants to attend. To apply please contact Fred Diamond, fred.diamond at kcl.ac.uk. The meeting will benefit from funding from Institut Mathématiques de Jussieu and Département de Mathématiques d'Orsay.

26th meeting, London.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Olivier Fouquet, Michael Harris, Marc Hindry, Matthew Morrow, Jacques Tilouine.

The 26th meeting of the LPNTS will take place at King's College London. The theme is p-adic cohomology and integration.

Dates: 7th to 8th May.
Location: King's College London.
Lectures: Anatomy Lecture Theatre (K6.29), King’s Building
Tuesday coffees: the adjoining Anatomy Museum
Wednesday coffee: S5.20

Schedule:
Tuesday 7th May
11:00 Welcome/coffee
11:30 Jennifer Balakrishnan
12:30 Lunch
14:00 Netan Dogra
15:00 Veronika Ertl
16:00 Coffee
16:30 Wieslawa Niziol

Wednesday 8th May
10:00 Johannes Anschutz
11:00 Coffee
11:30 Andreas Langer

Titles and abstracts:

Jennifer Balakrishnan
Title: Explicit p-adic integration on curves.
Abstract: I will give a survey of some recent algorithms for p-adic (Coleman) integration on curves, including work of Best, Bianchi, and Tuitman. I will further describe applications concerning p-adic heights and computing rational points on curves.

Netan Dogra
Title: The Chabauty-Kim method at a prime of bad reduction.
Abstract: The Chabauty-Kim method uses p-adic integration, in the guise of a Frobenius action and Hodge filtration on fundamental groupoids, to determine rational points on curve. In my talk I will describe joint work in progress with Jan Vonk making this explicit and computable in the case where p is a prime of bad reduction for the curve. I will also discuss work with Alex Betts describing the information contained in the monodromy action on the fundamental groupoid.

Veronika Ertl
Title: Comparison of crystalline syntomic and rigid syntomic cohomology for strictly semistable schemes.
Abstract: We prove a comparison isomorphism between Nekovár and Nizioł's syntomic cohomology and log rigid syntomic cohomology for a strictly semistable scheme with a nice compactification. Key points are a generalization of Große-Klönne's log rigid cohomology and the compatibility of crystalline and rigid Hyodo-Kato maps on the Frobenius eigenspaces.
This is joint work with Kazuki Yamada.

Wieslawa Niziol
Title: Integral p-adic cohomology of Drinfeld half-spaces.
Abstract: I will compute the integral p-adic etale cohomology of Drinfeld half-spaces of any dimension. This refines the existing computation of the rational p-adic etale cohomology. The main tools are: the computation of the integral de Rham cohomology and the integral p-adic comparison theorems of Bhatt-Morrow-Scholze and Cesnavicius-Koshikawa which replace the quasi-integral comparison theorem of Tsuji used to compute the rational etale cohomology. This is a joint work with Colmez and Dospinescu.

Johannes Anschutz
Title: Prismatic Dieudonne theory
Abstract: Building on the theory of prismatic cohomology which was recently introduced by Bhatt and Scholze we want to present the definition of a prismatic Dieudonne functor over p-completely quasi-syntomic rings and explain how to use it to obtain a classification of p-divisible groups over such rings by a generalization of minuscule Breuil-Kisin(-Fargues) modules.
This is joint work with Arthur-Cesar Le Bras.

Andreas Langer
Title: Relative de Rham-Witt cohomology and applications to p-adic deformation theory .
Abstract: In my talk I will describe new results on the relative de Rham-Witt complex and its Nygaard filtration and then discuss two applications: one is a relative version of p-adic deformation theory of algebraic cycles due to Bloch/Esnault/Kerz; the other is a construction of higher Displays and a crystal of relative Displays for a class of smooth projective schemes satisfying some general assumptions.

25th meeting, Paris.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: Kevin Buzzard, Ana Caraiani, Fred Diamond, Vladimir Dokchitser, Yiannis Petridis, Sarah Zerbes.

Paris organizers: Olivier Fouquet, Michael Harris, Marc Hindry, Matthew Morrow, Jacques Tilouine.

The 25nd meeting of the LPNTS will take place in in Jussieu. The theme is Analytic Number Theory.

Dates: Afternoon of Monday 26th and morning of 27th November.
Location (coarse): Institut de Mathématiques de Jussieu-Paris Rive Gauche
Location (fine): corridor 15-16, floor 4, room 413

Schedule:
Monday 26 November
14.00-15.00 Joël Bellaïche
15.00-15.30 pause
15.30-16.30 Yiannis Petridis
16.30-17.00 pause
17.00-18.00 Cécile Dartyge

Tuesday 27 November
9.00-10.00 Nikolaos Diamantis
10.00-10.30 pause
10.30-11.30 Daniel Fiorilli
11.30-12.30 Florent Jouve

Titles and abstracts:

Joël Bellaïche (Brandeis University)
Divisibility of the coefficients of modular functions

Yiannis Petridis (University College London)
Arithmetic Statistics of Modular Symbols
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.

Cécile Dartyge (Université de Lorraine)
Exponential sums with reducible polynomials
Hooley proved that if f(X) is an irreducible polynomial of degree at least 2 with integer coefficients, then the fractions r/n, with 0 < r < n and f(r)=0 mod n, are well distributed in ]0, 1[. By Weyl's criterion, this question is connected with some exponential sums along these fractions. In this talk we consider such exponential sums where the polynomial f is reducible and of degree 2 or 3. This is a joint work with Greg Martin.

Nikolaos Diamantis (University of Nottingham)
Additive twists and a conjecture by Mazur, Rubin and Stein
A recent full proof of a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols will be discussed. This is a report on joint work with J. Hoffstein, M, Kiral and M. Lee.

Daniel Fiorilli (Université de Paris-Sud)
Chebyshev’s bias in Galois groups, I
(joint with Florent Jouve) In a 1853 letter, Chebyshev noted that there seems to be more primes of the form 4n+3 than of the form 4n+1. Many generalizations of this phenomenon have been studied. In this talk we will discuss Chebyshev’s bias in the context of the Chebotarev density theorem. For example, we will compare the number of primes p congruent to 1 modulo 3 for which 2 is a cube modulo p to the number of primes for which it is not. One of our goals will be to study extreme biases, that is we will state conditions on the implied Galois group that guarantee serious asymmetries. We will see that these questions are strongly linked with the representation theory of this group and the ramification data of the extension. During the talk we will focus on the S_n case, and take advantage of the rich representation theory of the symmetric group as well as bounds on characters due to Roichman, Féray, Sniady, Larsen and Shalev.

Florent Jouve (Université de Bordeaux)
Chebyshev’s bias in Galois groups, II
(joint with Daniel Fiorilli) In this second talk on the topic of inequities in the distribution of Frobenius elements in Galois groups our focus will be on particular families that either exhibit a surprising behavior as far as Chebyshev's bias is concerned or that are simple enough to enable a very precise computation of the group theoretic and ramification theoretic invariants that come into play in our analysis. Precisely the emphasis will be on some families of abelian, dihedral, or radical extensions of Q as well as families of Hilbert class fields H_d of quadratic fields K_d either seen as extensions of Q or of K_d.

24th meeting, London.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Matthew Morrow, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 24nd meeting of the LPNTS took place in London at UCL, on 29th and 30th of May, 2018. The theme was "mod p automorphic forms".

The schedule was

Tuesday 29th:
1000--1100: Coffee
1100--1200: Payman Kassaei
1200--1400: lunch
1400-1500: Marie-France Vigneras
1500-1530: coffee
1530-1600: David Helm
1630-1730: Stefano Morra

Wednesday 30th:
0900-1000: Julien Hauseux
1000-1030: coffee
1030-1130: Pascal Boyer
1130-1230: Jack Thorne
Titles and abstracts:

Payman Kassaei (King's College London)
Title: Stratifications on mod p Shimura varieties and applications to automorphic forms
Abstract: In this talk, I will describe in an introductory way stratifications on certain mod p Shimura varieties and present some past and recent applications to arithmetic of automorphic forms.

Marie-France Vigneras (Jussieu)
Title: On supersingularity
Abstract: There are explicit relations between: Elliptic curves over $Q_p$ with supersingular reduction,
Irreducible admissible representations of $GL_2(Q_p)$ with supersingular reduction modulo $p$,
Supersingular simple modules modulo $p$ of the pro-$p$ Iwahori Hecke algebra of $GL_2(Q_p)$,
Irreducible 2-dimensional representations modulo $p$ of the Galois group of $Q_p$.
How does this generalize to a finite extension $F/Q_p$, a positive integer $n$, a reductive $p$-adic group?

Pascal Boyer (Paris 13)
Title: Torsion or not Torsion
Abstract: About the $\mathbb Z_l$-cohomology of Shimura varieties, on can be interested by killing the torsion using for example localization at some well chosen maximal ideal of some Hecke algebra. On the opposite point of view, we can ask for the arithmetic meaning of torsion classes so that we are led to the problem of the construction of such classes. In this talk we will try to tackle these two aspects in the particular case of Shimura varieties of Kottwitz-Harris- Taylor type.

Julien Hauseux (Lille)
Title: Extensions between generalised Steinberg representations
Let G be a p-adic reductive group. We compute the extensions between mod p smooth generalised Steinberg representations of G. This is part of a work in progress with Colmez, Dospinescu, and Nizioł.

David Helm (Imperial College)
Title: Towards a local Langlands correspondence in families for split groups in depth zero
Abstract: The local Langlands correspondence in families identifies the integral Bernstein centre for the group GL_n(F) with a certain ring of functions on the moduli space of Langlands parameters for GL_n. I will describe a conjectural generalization of this result to depth zero representations of a split reductive group G over F, and explain how a key part of the proof of local Langlands in families for GL_n generalizes to this case. This is joint work with Jean-Francois Dat, Rob Kurinczuk, and Gil Moss.

Stefano Morra (Montpellier)
Title: Local models for Galois representations, and applications to automorphic forms
Deformation spaces of finite at group schemes are a central subject in p-adic Hodge theory, and in arithmetic questions of a global nature. The description of their singularities in specific situations led to a wide horizon of achievements, from the proof of the conjectures of Serre (as generalized by Buzzard-Diamond- Jarvis, Schein, Gee and others) to the establishment of the Shimura-Taniyama- Weil conjecture and many more cases of modularity lifting theorems in dimension 2. Nevertheless outside a limited number of situations (Barsotti-Tate, ordinary, Fontaine- Laffaille) the geometry of more general Galois deformation spaces remains mysterious. In this talk we introduce an algebraic variety, refinement with monodromy of the local model of Pappas-Rapoport, which controls the structure of generic potentially crystalline deformation spaces of Galois representations in arbitrary dimension. In particular we illustrate how its geometry is predicted by and predicts generalized geometric interpretations of the weight part of Serre and Breuil-Mézard conjectures as proposed by Emerton and Gee. This is ongoing joint work with Daniel Le, Bao Viet Le Hung and Brandon Levin.

Jack Thorne (Cambridge)
Title : Odd Galois Representations
Abstract : A popular slogan is that "all Galois representations which appear in the cohomology of Shimura varieties are conjugate self-dual up to twist". However, fine words butter no parsnips, and this is on the face of it not quite true. I will discuss what one can say in this direction. This is joint work with Christian Johansson.

23rd meeting, Paris.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Matthew Morrow, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 23rd meeting of the LPNTS took place in Paris (in Jussieu), on 27th and 28th November 2017. The topic was Periods. This was the schedule:

Monday 27/11/2017:

Tuesday 28/11/2017:

Abstracts:

Jean-Benoît Bost : Transcendence proofs and infinite-dimensional geometry of numbers.
Abstract : I will explain how some classical transcendence results concerning periods, notably the theorem of Schneider-Lang, may be given "natural proofs" based on the consideration of some infinite dimensional avatars of Euclidean lattices.

Javier Fresán : Gamma values as exponential periods
Abstract : As one can already guess from the Gaussian integral, the values of the gamma function at rational arguments are not expected to be periods, although suitable products of them do become periods of abelian varieties with complex multiplication. To deal with single gamma values, one needs to consider exponential periods instead. I will discuss how a joint work with Peter Jossen, in which we construct a Tannakian category of exponential motives over a subfield of the complex numbers, allows one to explain the transcendence conjectures for gamma values in a natural way.

Francis Brown : Invariants attached to periods
Abstract : Kontsevich and Zagier asked whether there exists an algorithm to determine if two periods are equal. I shall explain how, in some cases, a conjectural algorithm exists and is highly effective in practice. It enables one to discover identities between periods without any prior knowledge of relations.

Tony Scholl : Real plectic cohomology
Abstract : We will discuss joint work with Jan Nekovar on aspects of plectic cohomology and its connection with L-values.

Jie Lin : On factorization of automorphic periods
Abstract: The question on the factorization of automorphic periods was initiated by Shimura where periods refer to the Petersson inner products of algebraic forms. Essentially, he predicted that periods related to Hilbert modular forms, or more generally to algebraic forms on a division algebra, factorize as products of periods indexed by the split archimedean places of the division algebra. The initial conjecture was first proved by M. Harris and completed by H. Yoshida. However, their methods seem very difficult to generalize to higher ranks. In this talk, we will explain a new and simple proof for general rank. We will also explain how to read this factorization from the point of view of motives, and why it is important in the study of special values of L-functions.

22nd meeting, London.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Matthew Morrow, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 22nd meeting of the LPNTS will take place in London at UCL, on 5th and 6th June. The theme is "The Arthur trace formula, automorphic forms, arithmetic manifolds and their homology". The meeting will start at 1100 on the 5th (with coffee 1000-1100 beforehand) and finish on the afternoon of the 6th.

The talks on Monday 5th will be in room 505, 25 Gordon Street in the Department of Mathematics, UCL. The talks on Tuesday 6th will take place at Roberts, room 309. The location is about 10 minutes walk from the Eurostar terminal. You can look at the map of UCL and nearby area at this link.

The schedule:

Monday June 5th.
Location: UCL maths department, room 505 (google maps link)

1000-1100 Coffee
1100-1200 Olivier Taïbi (Imperial): "An introduction to the stabilisation of the trace formula and Arthur-Langlands packets."
1200-1400 Lunch
1400-1500 Colette Moeglin (Jussieu): "On some local aspects of Arthur's theory"
1500-1545 Coffee
1545-1645 Nicolas Bergeron (Jussieu): "On the cohomology ring of the universal K3 surface"
1700-1800 Drinks

Tuesday June 6th.
Location: Roberts 309 (google maps link)

1000-1100 Philippe Michel (Lausanne): "Sums of Kloosterman sums and sums of L-functions"
1100-1130 Coffee
1130-1230 Haluk Şengün (Sheffield): "Cohomology of arithmetic groups and Fermat's Last Theorem"
1230-1430 Lunch
1500-1700 Cultural activity

Some abstracts:

O. Taïbi: "An introduction to the stabilisation of the trace formula and Arthur-Langlands packets."
Abstract: Automorphic representations of reductive groups over number field generalise the classical notion of modular forms or Maass forms which are eigenvectors for the Hecke algebra. The Arthur-Selberg trace formula and its variants have proved extremely useful to study general automorphic representations, and their relation to Galois representations ("the Langlands programme"). In this introductory talk aimed at PhD students and non- specialists, I will explain what it means to stabilise trace formulae, and why it is crucial to understand this relation precisely: - locally (over local fields of characteristic zero), with the local Langlands correspondence and the introduction of Arthur-Langlands packets of representations, and - globally (over number fields), with Arthur's multiplicity formula for automorphic representations.

Colette Moeglin: "On some local aspects of Arthur's theory"
Abstract: I will first recall some basic facts about the spectral side of the trace formula to motivate the formulation of the local A-packet that I will give. After having given this formulation, I will explain what is known specially in the real case and what we expect to be true.

N. Bergeron: "On the cohomology ring of the universal K3 surface."
Abstract: The Deligne decomposition theorem makes it possible to reduce the study of the cohomology groups of the universal K3 surface, or more generally of universal families of polarized hyperkähler varieties, to the study of certain spaces of automorphic forms. This makes it possible to prove a cohomological version of the generalized Franchetta conjecture due to O'Grady but also to better understand the ring structure on the cohomology of these universal families. This is a joint work with Zhiyuan Li.

P. Michel: "Sums of Kloosterman sums and sums of L-functions."
Abstract: This talk is a review a recent series of works by V. Blomer, E. Fouvry, E. Kowalski, myself, D. Milicevic, W. Sawin as well as R. Zacharias. We will describe various estimates on sums of Kloosterman sums (or more generally trace functions) proven using methods from $\ell$-adic cohomology and some of their applications to the study of analytic properties of character twists of L-functions on average over the family of Dirichlet characters of some large prime modulus.

H. Şengün: "Cohomology of arithmetic groups and Fermat's Last Theorem."
Abstract: We use two fundamental reciprocity conjectures in the Langlands Programme that involve the cohomology of arithmetic groups and derive an algorithmically testable criterion for a number field K which, if satisfied, implies the truth of asymptotic Fermat's Last Theorem over K. Most imaginary quadratic fields satisfy the criterion. This is joint work with Samir Siksek (Warwick).

21st meeting, Paris.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Pierre Charollois, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 21st meeting of the LPNTS will take place in Paris, on 14th and 15th November. The theme is Perfectoid spaces/Espaces perfectoïdes. The start time is 11:15 on the 14th, giving London participants the possibility of catching an early Eurostar to Paris that morning.

The meeting will take place in Paris 6 (Jussieu), room 15-25 104 (this is on the 1st floor, between Towers 15 and 25, and note the late room change). From Eurostar-Gare du Nord, take metro line 4 to Odéon, then change to line 10 and exit at Jussieu.

For various administrative reasons, people will be asked to register. Registration is free. To register, please send an email to Benjamin Schraen (a link to his home page is above, and his email address can be found there).

Monday 14th (all talks in room 15-25 104 (1st floor between Towers 15 and 25) in Jussieu)

11h15 : Stroh (Jussieu) -- "Autour des tours"
12h15 : lunch (free for registered participants).
14h : André (Jussieu)
15h30 : Newton (Kings College London) -- "Galois representations and the completed homology of locally symmetric spaces"
16h45 : Ramero (Univ Lille) -- "Perfectoid spaces and log-regular rings"

Tuesday 15th Morning (same room)

9h : A. Vezzani (Paris 13) -- "The tilting equivalence and motivic Galois groups"
10h15 : Caraiani (University of Bonn/Imperial College London) -- "On the generic part of the cohomology of certain unitary Shimura varieties"
11h30 : Morrow (Jussieu) -- "Poincaré duality and Chern classes for A_inf cohomology"
12h30: lunch at the restaurant (free for all registered participants).

Tuesday afternoon (the room for the last talk is different : 16-26 113, 1st floor between Towers 16 and 26)

14h : Fargues (Jussieu) -- "Simple connectedness of the fibers of an Abel Jacobi morphism and local class field theory"

20th meeting, London.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Pierre Charollois, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 20th meeting of the LPNTS took place in London (at UCL), on 6th and 7th June 2016. The themes were representations of p-adic groups and arithmetic geometry.

Schedule:

06/06/16

Coffee/tea/pastries 10:00-11:00
Talk 1 11:00-12:00 D. Roessler
Lunch 12:00-14:00
Talk 2 14:00-15:00 L. Taelman
Coffee 15:00-15:45
Talk 3 15:45-16:45 A. Cadoret
Drinks 17:00-18:00

07/06/16

Talk 4 9:30-10:30 S. Stevens
Coffee 10:30-11:00
Talk 5 11:00-12:00 A.-M. Aubert
Lunch 12:00-14:00
Talk 6 14:00-15:00 A.-C. Le Bras
Coffee 15:00-15:45
Talk 7 15:45-16:45 K. Ardakov

Here are the titles and abstracts:

Shaun Stevens (UEA)
Title: Representations of p-adic groups and the local Langlands correspondence
Abstract: In this introductory talk, I will try to describe the some of the ideas, techniques, and questions in the representation theory of p-adic groups, motivated by the local Langlands correspondence, mostly just for complex representations. I will assume some familiarity with local fields and with the representation theory of finite groups.

Damian Roessler (Oxford)
Title: On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic
Abstract: Let K be a function field in one variable over a finite field of char. p>0. Let A be an ordinary abelian variety over K. We shall describe the proof of the following statement. Suppose that the group A(K^sep)[p^\infy] is finite. Then A(K^perf) is finitely generated. Here K^sep is the separable closure of K and K^perf=K^(-p^\infy) is the maximal purely inseparable extension of K. This has applications to the Mordell-Lang conjecture in positive characteristic.

Anna Cadoret (Ecole Polytechnique)
Title: Geometric monodromy - semisimplicity and maximality
Abstract: Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p, let f:Y-> X be a smooth proper morphism and x a geometric point on X. We show that the tensor invariants of bounded length d of the etale fundamental group \pi_1(X) acting on the \'etale cohomology groups H^*(Y_x,\F_l) are the reduction modulo-l of the tensor invariants of \pi_1(X,x) acting on H^*(Y_x,\Z_l) for l large enough depending on f:Y-> X, d. We use this result to discuss semisimplicity and maximality issues about the image of \pi_1(X,x) acting on H^*(Y_x,\Z_l). This is a joint work with Chun-Yin Hui and Akio Tamagawa.

Arthur-César Le Bras (Ecole Normale)
Title : Drinfeld’s coverings and the $p$-adic Langlands correspondence
Abstract : I will explain the proof of (a version of) a conjecture of Breuil-Strauch, which gives a purely geometric description of the $p$-adic local Langlands correspondence for $GL_2(\mathbf{Q}_p)$ for de Rham non trianguline Galois representations, using Drinfeld's coverings of the $p$-adic upper half-plane. It can be seen as a $p$-adic analogue of the realization of the (classical) local Langlands correspondence for supercuspidal representations in the $\ell$-adic cohomology of the Drinfeld tower. This is a joint work with Gabriel Dospinescu.

Lenny Taelman (Amsterdam)
Title: Complex multiplication and K3 surfaces over finite fields
Abstract: The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l-adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda-Tate: given a rational function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z?

Konstantin Ardakov (Oxford)
Title: Admissible representations of p-adic Lie groups, and D-modules
Abstract: The localisation theorem of Beilinson and Bernstein opened new doors in representation theory by relating representations of semisimple Lie algebras to D-modules on flag varieties. I will talk about an extension of this result, which gives an anti-equivalence between the category of admissible representations (locally analytic, with trivial infinitesimal central character) of a semisimple p-adic Lie group and the category of coadmissible equivariant D-modules on the rigid analytic variety.

Anne-Marie Aubert (Jussieu)
Title: Cuspidality and Hecke algebras for enhanced Langlands parameters
Abstract: Enhanced Langlands parameters for a p-adic group G are pairs formed by a Langlands parameter for G and an irreducible character of a certain component group attached to the parameter. We will introduce notion of cuspidality and of cuspidal support for these pairs, describe a map that attaches to an arbitrary enhanced Langlands parameter for G the inertial class of its cuspidal support, and associate to the latter a twisted affine Hecke algebra. It is joint work with Ahmed Moussaoui and Maarten Solleveld.

Note: some of these talks are introductory talks, aimed at PhD students.

19th meeting, Paris.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25) and the Heilbronn Institute for Mathematical Research.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Pierre Charollois, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The 19th meeting of the LPNTS took place in Paris, on 9th November 2015, at Paris 13.

It was at Amphitheater D, Institut Galilée, Université Paris 13 (see number 8 on the map here).

Schedule:
Coffee at 10h15
First talk at 10h30
Coffee,
Second talk at 11h45
Lunch around 13h
Last talk: 14h30, end around 15h45.

The speakers were:
P. Colmez : Faltings' height and Galois orbit of CM abelian varieties (Hauteur de Faltings et orbite galoisienne des variétés abéliennes à multiplication complexe)
A. Yafaev : Overview of the André-Oort conjecture
F. Andreatta : Sketch of proof

Eighteenth meeting, London.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It has been supported by grants from the London Mathematical Society, ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, and the Heilbronn Institute for Mathematical Research.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The next London meeting will be the Grothendieck-fest on 4th to 5th June, see here.

In addition there will be a special meeting of the London-Paris Number Theory Seminar in honour of the 150th Anniversary of the LMS at the British Mathematical Colloquium in Cambridge on 31st March - 1st April 2015. Registration is required in order to attend the BMC (now sold out), so priority for attendance will be given to registered participants in the unlikely event that there is not enough space in the lecture room. See here.

Seventeenth meeting, Paris, 10/11/14.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society, and ANR Projet ArShiFo ANR-BLAN-0114.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The schedule:

0900 Welcome and coffee in room 104
0930 Pierre Parent (Bordeaux): Heights on elliptic and modular curves.
1100 Sir Peter Swinnerton-Dyer (Cambridge): The effect of twisting on the 2-Selmer group.
1200--1400: lunch at L'Ardoise
1400 Tom Fisher (Cambridge): Visibility of 4-coverings of elliptic curves

1500 Coffee

1530 Jim Stankewicz (Bristol): Torsion points on CM elliptic curves over prime degree fields

Abstracts:

Parent: The determination of Galois representations mod p defined by elliptic curves over number fields can be translated in terms of rational points on modular curves. Among the most efficient tools to study rational points on curves in general are those provided by height approaches, as was illustrated by the proofs of Mordell's conjecture by Faltings and Vojta. But unfortunately these powerful methods are often non-effective, a key issue if one is interested not only in the finiteness but also in the mere existence of rational points. Modular curves are however quite special in geometric and arithmetic aspects, and that sometimes makes the question of effectivity easier. We will discuss this point of view through various recent and less recent applications of diophantine methods.

Swinnerton-Dyer: Let E be an elliptic curve whose 2-division points are rational, and let E_b be its quadratic twist by b. For any fixed E we study the distribution of the order of the 2-Selmer group of E_b as b varies. Doing this involves a sophisticated analysis of the process of 2-descent, which is joint work with Alexei Skorobogatov. If there is time, I shall also say something about the case when not all the 2-division points are rational.

Fisher: A smooth plane cubic C may naturally be viewed as a 3-covering of its Jacobian, with fibre above 0 the set of points of inflection on C. A family of 3-coverings, with constant fibre above 0, may be constructed by taking linear combinations of the equation for C and its Hessian. I will describe an analogous construction for 4-coverings of elliptic curves, and explain how this helps with the study of visibility of Tate-Shafarevich groups. This is joint work with Nils Bruin.

Stankewicz: In this talk, we answer the following question of Schuett: Let p be a prime, F a degree p number field, and E an elliptic curve over F with CM. As p grows, how does the torsion in E(F) grow? We find an answer to this question via a connection between real cyclotomic fields and torsion on CM elliptic curves. This is joint work with A. Bourdon and P. Clark.

Sixteenth meeting, London, 9/06/14.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society, and ANR Projet ArShiFo ANR-BLAN-0114.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The schedule:

1200-1300: Farrell Brumley (U. Paris 13): "Sup norms of automorphic forms"
1300-1500: Lunch
1500-1600: Harald Helfgott (CNRA, Ecole Normale Superieure): "The ternary Goldbach problem"
1600-1630: Coffee
1630-1730: Andrew Granville (U. de Montreal, Cambridge U): "The pretentious Riemann Hypothesis and the short proof of Linnik's Theorem"

Abstracts:

Brumley: There has been a lot of recent activity on the problem of estimating the sup norm of L^2 normalized automorphic forms. Most attention has been devoted to the compact setting, where there are firm conjectures to work with. In recent preprint, N. Templier and I have explored the question of what happens without the hypothesis of compacity. Cusp forms on non-compact finite volume locally symmetric spaces decay rapidly at infinity, but before doing so, they evince a sort of automorphic Gibbs phenomenon, attaining their largest value before dying. We are able to quantify this for SL_n(Z) and find bounds on a different scale in comparison with the compact setting.

Helfgott: The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the following decades, there was a succession of results reducing C, but only to levels much too high for a verification by computer up to C to be possible (C > 10^{1300}). (Works by Ramare and Tao solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas in the proof.

Granville: In this talk we will discuss several recent developments in our understanding of the basic questions of analytic number theory. Based on the work of many authors, Sound and I have been proposing, for the last few years, a coherent version of analytic number theory which makes no use of the zeros of L-functions. Until recently there were several lacunae in this theory, but now, based on ideas of Koukoulopoulos and Harper, we are able to achieve the same (or better) results than the "Riemann Theory". We shall present here a suitable reformulation of the Riemann Hypothesis, as well as indicate a new proof of Halasz's Theorem, leading to a surprisingly easy proof of Linnik's Theorem.

Fifteenth meeting, Paris, 18/11/13.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society, and ANR Projet ArShiFo ANR-BLAN-0114.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The schedule:

1000-1100 P. Kassaei: Analytic continuation of p-adic Hilbert modular forms and applications to the Artin conjecture-- The unramified case.
1130-1230 Y. Tian: Analytic continuation and modularity lifting in parallel weight one in the tamely ramified case.
1400-1500 C. Johansson: Proving control theorems for overconvergent forms using rigid cohomology.
1515-1615: V. Pilloni: Sur la conjecture de Fontaine-Mazur en poids 0.

Abstracts:

Kassaei:

We will present our proof of the Artin conjecture in the unramified case. In the course of doing so, we will also present an overview of the recent developments in analytic continuation of p-adic Hilbert modular forms, as well as generalizations of our work on the Artin conjecture by Pilloni, Stroh, Sasaki, and Tian.

Tian:

Let F be a totally real field, and p>2 be a prime unramified in F. Suppose given a 2-dimensional totally odd continuation p-adic Galois representation on F, which is residually modular and tamely ramified (up to twists by characters) at all places above p. We will show that, up to twist, such a Galois representation comes from a holomorphic Hilbert modular form in parallel weight one. A key ingredient in the proof is an analytic continuation result, which claims that any overconvergent Hilbert eigneform of finite slope can be extended to a very large area on the rigid analytic Hilbert modular variety of Iwahoric level. This is a joint work with Payman Kassaei and Shu Sasaki.

Johansson:

After the invention and use of the analytic continuation method by Buzzard and Taylor to prove icosahedral cases of the Artin conjecture, Kassaei realized that it might also be used (together with Kassaei's "gluing lemma") to reprove Coleman's theorem that an overconvergent U_p-eigenform of weight k>1 and slope <k-1 is classical. Coleman's original method used an analysis of the rigid cohomology of the ordinary locus in the modular curve. In this talk I will discuss recent work on extending this "cohomological" approach to higher dimensional Shimura varieties. Part of this is joint work with Vincent Pilloni.

Pilloni:

Nous démontrerons, sous les hypothéses techniques usuelles, que certaines représentations de Galois des corps totalement réels sont associées à des formes modulaires de Hilbert propre de poids 1. Le résultat clé est une caractérisation des espaces de formes modulaires classiques dans l'espaces des formes modulaires p-adique. Nous essayerons enfin de faire le lien avec le travail de Calegari-Geraghty sur ces questions. (Travail avec B. Stroh).

Fourteenth meeting, London, 3--4/6/13.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The schedule:

Monday 3 June, pm
2-3 F. Charles. On the twisted Lang-Weil estimates.
3.30-4.30 Y. Harpaz. The section conjecture for graphs and applications for singular curves
5-6 J.-L. Colliot-Thélène. Espaces homogènes sur les corps de fonctions de courbes sur un corps local

Tuesday 4 June, am
10-11 O. Wittenberg. On the kernel of the cycle class map for 0-cycles over local fields
11.30-12.30 D. Testa. Computability of Néron-Severi groups

Abstracts:

François Charles.

Given an algebraic variety over a finite field, the classical Lang-Weil estimates compute the number of intersection point of the diagonal with the graph of Frobenius. Hrushovski has proven a twisted version of these estimates, replacing the diagonal by a more general correspondence. While Hrushovski's proof relies on non-classical objects such as difference schemes, this has proven to yield a number of applications to algebraic geometry. In this talk, we will explain a self-contained intersection-theoretic proof of the twisted Lang-Weil estimates. This is joint work with Damien Rössler.

J.-L. Colliot-Thélène.

Over such a function field F, D. Harbater, J. Hartmann and D. Krashen have proved a local-global principle for the existence of rational points on principal homogeneous spaces under a connected linear algebraic group G over F when the underlying variety of G is F-rational, i.e. birational to affine space over the field F. In recent work with Parimala and Suresh, we show that this local-global principle may fail when the group G is not F-rational. The obstruction we use comes from the Bloch-Ogus complex for étale cohomology over an arithmetic surface extending the curve. One may then ask when this new obstruction is the only obstruction to the existence of rational points.

Y. Harpaz.

The connection between rational points and sections of the Grothendieck exact sequence has an analogue for topological spaces carrying an action of a finite group. The corresponding section conjecture in this setting is false in general (as for general varieties), but is true when the underlying space is a graph. In this lecture we will explain this result and show how it can be applied to prove that finite descent is the only obstruction to the Hasse principle for a certain class of singular curves.

D. Testa.

The Néron-Severi group of a variety is a finitely generated abelian group whose rank is the Picard number of the variety itself; it plays an important role in the determination of the algebraic part of the Brauer-Manin obstruction. I will begin with an introduction to Néron-Severi groups, and will discuss a few methods that have been used to determine Picard numbers in special cases. Then I will report on recent joint work with Bjorn Poonen and Ronald van Luijk on the computability of Picard numbers in general.

O. Wittenberg.

I will discuss the kernel of the cycle class map from the Chow group of 0-cycles to etale cohomology with finite or integral coefficients, in the case of surfaces defined over a p-adic field, including surfaces with nonzero geometric genus. (Joint work with H. Esnault.)

Thirteenth meeting, Paris, 22/10/2012

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society.

London organizers: David Burns, Fred Diamond, Alexei Skorobogatov, Andrei Yafaev, Kevin Buzzard.

Paris organizers: Michael Harris, Pierre Charollois, Marc Hindry, Ariane Mezard, Jacques Tilouine.

The 13th meeting took place in Jussieu on Monday 22nd October (following a seminaire Bourbaki week-end at the IHP). Room 104 Tour 15-25 (1st floor), Jussieu (map). Note that this means the corridor between towers #15 and #25, 1st floor.

The schedule:

9:30 Accueil/Welcome + coffee (corridor 15--16 room 417, 4th floor).

10:00 Sarah Zerbes (UCL), "Towards an Euler system for Rankin-Selberg convolutions".

11:00 Coffee.

11:30 David Vauclair (Univ. Caen), "The unramified Iwasawa main conjecture for abelian varieties over function fields of characteristic p".

Lunch break

14:00 David Burns (Kings College London), "Congruences between derivatives of twisted Hasse-Weil L-functions".

Note that Thomas de La Rochefoucauld, a student of Jan Nekovar, will defend his thesis entitled "Autour de la conjecture de parité" at 15:30 (my understanding of the French System is that this basically means that he will be giving a seminar on his work which we would be welcome to attend).

Twelfth meeting, London, 30/05/2012

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society.

London organizers: David Burns, Fred Diamond, Alexei Skorobogatov, Andrei Yafaev, Kevin Buzzard.

Paris organizers: Michael Harris, Guy Henniart, Marc Hindry, Ariane Mezard, Jacques Tilouine.

The 12th meeting will take place at University College London on May 30th, in Lecture Theatre 500 (5th floor, UCL, UCLU-Maths building, 25 Gordon Street). The theme is "Probabilistic Number Theory". The schedule is

1100--1200 Joel Rivat (Luminy): "On the digits of primes and polynomial sequences."

1200--1400 Lunch

1400--1500 Gerald Tenenbaum (Nancy): "On the distribution of divisors, a survey."

1500--1530 Coffee/Tea

1530--1630 Adam Harper (Cambridge): "Lower bounds for the sum of a random multiplicative function."

Abstracts:

Adam Harper: TITLE: Lower bounds for the sum of a random multiplicative function

ABSTRACT: A random multiplicative function is a probabilistic model for the Mobius function or for real Dirichlet characters. Because of the dependencies between its values, it is surprisingly difficult to give almost sure lower bounds for the size of the sum of a random multiplicative function. I will explain how this is connected to understanding random Dirichlet series, and how random Dirichlet series can be understood by developing sufficiently quantitative Gaussian process theory.

Joel Rivat:

Title: "On the digits of primes and polynomial sequences".

Abstract: "In 1968, A.O. Gelfond conjectured that the sum of the digits of the primes is equidistributed in residue classes. He also conjectured the same result for the sum of the digits of appropriate polynomial sequences. We will discuss the proof of these conjectures and some extensions and generalizations in recent joint works with Michael Drmota and Christian Mauduit."

Eleventh meeting, Paris, 21/11/2011

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society.

London organizers: David Burns, Fred Diamond, Alexei Skorobogatov, Andrei Yafaev, Kevin Buzzard.

Paris organizers: Michael Harris, Guy Henniart, Marc Hindry, Ariane Mezard, Jacques Tilouine.

The 11th meeting will take place on Monday 21st November 2011 in Paris on the subject of "ergodic methods in number theory".

The meeting will take place in Jussieu, Couloir 15--25, Salle 104, and the schedule is:

1100 Julia Wolf (Ecole Polytechnique, France): "Ergodic, analytic and combinatorial approaches to decomposition theorems for bounded functions"

1400 Alex Gorodnik (University of Bristol): "Quantitative ergodic theorems and Diophantine approximation"

1530 Ben Green (University of Cambridge): "Distribution of orbits on nilmanifolds and applications".

For more information (including abstracts of some of the talks) try Jussieu's seminar page here (look for 21 Nov).

Tenth meeting, London, 01/06/2011

The 10th meeting will take place on Wednesday 1st June 2011 at University College London in Lecture Theatre 500 (5th floor, UCL, Maths building).
The theme of the meeting is "Group Schemes and p-adic Hodge Theory."
The schedule of talks is as follows:

Ninth meeting, Paris, 08/11/2010

The London-Paris Number Theory Seminar is supported by a grant from the London Mathematical Society.

The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris.

London organizers: David Burns, Fred Diamond, Minhyong Kim, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Michael Harris, Guy Henniart, Marc Hindry, Ariane Mezard, Jacques Tilouine.

The 9th meeting will take place on Monday 8th November 2010 in Paris on the subject of "applications of model theory to number theory". More precise details are here.

Speakers are F.Loeser (ENS, Paris), J.Pila (Oxford), B.Zilber (Oxford).

Some travel funding is available for UK participants. Please contact one of the London organizers if you are interested in attending. Students should have their advisors email a brief supporting statement.

Further details will be announced in due course.

Eighth meeting, London, 02/06/2010

The 8th meeting took place on Wednesday 2nd June 2010 at King's College London in room 2B08 (=S-2.08) of the Strand Building. The schedule was

The London-Paris Number Theory Seminar is supported by an Alliance grant (British Council, French Ministries of Foreign Affairs and National Education).

Seventh meeting, Paris, 16/11/2009

The 7th meeting took place on Monday 16th November 2009 in Paris. The schedule was

Sixth meeting, London, 03/06-04/06/2009

The 6th meeting took place on Wednesday 3rd June and Thursday 4th June 2009 at King's College London in room 2B08 of the Strand Building. The theme was "p-adic modular forms" (in conjunction with the Paris-Nancy-London rotating workshop on p-adic analytic continuation). The schedule was

The London-Paris Number Theory Seminar is supported by an Alliance grant (British Council, French Ministries of Foreign Affairs and National Education).

Fifth meeting, Paris, 17/11/2008

The 5th meeting took place on Monday 17th November 2008 in the amphitheatre "Darboux" at the Insitut Henri Poincaré in Paris. The schedule was

Fourth meeting, London, 07/05/2008-08/05/2008

The 4th meeting took place on Wednesday 7th May and Thursday 8th May 2008 in Imperial College London. The theme was "Elliptic curves and abelian varieties". The schedule was

This meeting was sponsored by: The London Mathematical Society, Imperial College, King's College London, University College London

Third meeting, Paris, 12/11/2007

The 3rd meeting took place on Monday 12th November 2007 in the Institut Henri Poincaré in Paris. The schedule was

The talks took place in the Amphithéâtre Hermite, Institut Henri Poincaré.

The organizers of the meeting were: Michael Harris (harris@math.jussieu.fr), Guy Henniart (guy.henniart@math.u-psud.fr), Marc Hindry (hindry@math.jussieu.fr), Ariane Mezard (mezard@math.uvsq.fr), Jacques Tilouine (tilouine@math.univ-paris13.fr).

Second meeting, London, 02/05/2007

The 2nd meeting took place on Wednesday 2nd May 2007 in Imperial College London. The schedule was

The talks took place in the Institute for Mathematical Sciences, Imperial College London

The organizers of the meeting were: David Burns (david.burns@kcl.ac.uk), Alexei Skorobogatov (a.skorobogatov@imperial.ac.uk).

This meeting was supported by King's College London, Imperial College London and University College London.

First meeting, Paris, 13/11/2006

The 1st meeting took place on Monday 13th November 2006 in the Institut Henri Poincaré in Paris. The schedule was

The talks took place in the Amphithéâtre Darboux, Institut Henri Poincaré.

The organizers of the meeting were: Michael Harris (harris@math.jussieu.fr), Guy Henniart (guy.henniart@math.u-psud.fr), Marc Hindry (hindry@math.jussieu.fr), Ariane Mezard (mezard@math.uvsq.fr), Jacques Tilouine (tilouine@math.univ-paris13.fr).