London Number Theory Seminar

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL.

KINGS This term (Summer 2016), the seminar will be hosted by Kings College, and will be held on Wednesdays starting at 4pm. The talks will be in room S4.23 of the Department of Mathematics, starting on Wed 27th April and finished on 29th June. The talks will be preceded by tea at 3:30 in S5.21.

A preliminary list of speakers is below; for full details including up to date titles and abstracts please go to Mahesh's web page here.

27/4/16 Fernando Shao (Oxford)
Title: Vinogradov's three primes theorem with almost twin primes
Abstract: The general theme of this talk is about solving linear equations in sets of number theoretic interest. Specifically I will discuss the problem with the linear equations being N = x+y+z (for a fixed large N) and the set being "almost twin primes". The focus will be on the underlying ideas coming from both additive combinatorics and sieve theory. This is joint work with Kaisa Matomaki.

4/5/16 No seminar because of conference RandomWavesInLondon.

11/5/16 Oleksiy Klurman (Universite de Montreal/UCL)
Title: Correlations of multiplicative functions and applications.
Abstract: A deep problem in analytic number theory is to understand correlations of general multiplicative functions. In this talk, we derive correlations formulas for so-called bounded "pretentious" multiplicative functions. This has a number or desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\to \{-1,1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. If time permits, we discuss some further applications to the related problems.

18/5/16 Giovanni Rosso (Cambridge)
Title: Trivial zero for a p-adic L-function associated with Siegel forms
Abstract: We shall begin with an introduction to L-functions in arithmetic, their p-adic interpolation and trivial zero. We shall state a conjecture of Greenberg and Benois which predicts the order and the leading coefficient of p-adic L-functions when trivial zeros appear. We shall then explain how one calculates the first derivative of the standard p-adic L-function of an ordinary Siegel form with level at p.

25/5/16 Gergely Zábrádi (Eötvös Loránd)
Title: Smooth mod p^n representations and direct powers of Galois groups.
Abstract: Let G be a Qp-split reductive group with connected centre and Borel subgroup B=TN. We construct a right exact functor D from the category of smooth modulo p^n representations of B to the category of projective limits of continuous mod p^n representations of a direct power of the absolute Galois group Gal(Qpbar/Qp) of Qp indexed by the set of simple roots. The objects connecting the two sides are (phi,Gamma)-modules over a multivariable (commutative) Laurent series ring which correspond to the Galois side via an equivalence of categories. Parabolic induction from a subgroup P = L_P N_P amounts to the extension of the representation on the Galois side to the copies of Gal(Qpbar/Qp) indexed by the simple roots alpha not contained in the Levi component L_P using the action of the image of the cocharacter dual to alpha and local class field theory. D is exact and yields finite dimensional representations on the category SP of finite length representations with subquotients of principal series as Jordan-Hölder factors. Using the G-equivariant sheaf of Schneider, Vigneras, and the author on the flag variety G/B corresponding to the Galois representation we show that D is fully faithful on the full subcategory of SP with Jordan-Hölder factors isomorphic to irreducible principal series. Breuil has (preliminary) conjectures for the values of D at certain representations of GL_n(Qp) built out from some mod p Hecke isotypic subspaces of global automorphic representations.

1/6/16 Trevor Wooley (Bristol)
Title: Sub-convexity in certain Diophantine problems via the circle method.
Abstract: The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.

6th and 7th June -- London-Paris Number Theory Seminar.

8/6/16 Adam Harper (Cambridge)
Title: Gaussian and non-Gaussian behaviour of character sums
Abstract: Davenport and Erdos, and more recently Lamzouri, have investigated the distribution of short character sums $\sum_{x < n \leq x+H} \chi(n)$ as $x$ varies, for a fixed non-principal character $\chi$ modulo $q$. In particular, Lamzouri conjectured that these sums should have a Gaussian limit distribution (real or complex according as $\chi$ is real or complex) provided $H=H(q)$ satisfies $H \rightarrow\infty$ but $H = o(q/\log q)$. I will describe some work in progress in connection with this conjecture. In particular, I will try to explain that the conjecture cannot quite be correct (one need not have Gaussian behaviour for $H$ as large as $q/\log q$), but on the other hand one should see Gaussian behaviour for even larger $H$ for most characters.

15/6/16 Min Lee (Bristol)
Title: Effective equidistribution of primitive rational points on expanding horospheres
Abstract: The limit distribution of primitive rational points on expanding horospheres on SL(n,Z)\SL(n, R) has been derived in the recent work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. For n=3, in our joint project with Jens Marklof, we prove the effective equidistribution of q-primitive points on expanding horospheres as q tends to infinity.

22/6/16 Maria Valentino (King’s)
Title: On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms
Abstract: In this talk we shall begin with an introduction to Drinfeld cusp forms for arithmetic subgroups using Teitelbaum's interpretation as harmonic cocycles. We shall then address the problem of the diagonalizability of the function field analogous of the Atkin U-operator carrying the Hecke action over harmonic cocycles.

29/6/16 Nina Snaith (Bristol)
Title: Unearthing random matrix theory in the statistics L-functions: the story of Beauty and the Beast
Abstract: There has been very convincing numerical evidence since the 1970s that the positions of zeros of the Riemann zeta function and other L-functions show the same statistical distribution (in the appropriate limit) as eigenvalues of random matrices. Proving this connection, even in restricted cases, is difficult, but if one accepts the connection then random matrix theory can provide unique insight into long-standing questions in number theory. I will give some history of the attempt to prove the connection, as well as propose that the way forward may be to forgo the enticing beauty of the determinantal formulae available in random matrix theory in favour of something a little less elegant (work with Brian Conrey and Amy Mason)

The seminar will be preceded by the Study Groups, and this term there seem to be three: Classification of mod p representations of p-adic groups (1245-1415), an informal study group on Scholze's Lubin-Tate tower paper (1430-1530) and an analytic study group (1430-1530).

A list of previous seminar talks is here.

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