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, Vladimir Dokchitser, 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 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.

Previous meetings:

20th meeting (UCL, 6--7/6/16)

21st meeting (Jussieu, 14--15/11/16)

22nd meeting (UCL, 5--6/6/17)

23rd meeting (Jussieu, 27--28/11/17)

24th meeting (UCL, 29--30/5/17)

This page is maintained by Kevin Buzzard.