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.

This term (Summer 2014), the seminar will be hosted by Kings College, and will be held on Wednesdays from 4 pm to 5 pm in room S4.23 of the Department of Mathematics, starting on 23rd April. The talks will be preceded by tea at 3:30 in S5.21.

23/04/14 Jack Lamplugh (Cambridge University)
Title: Class numbers in some non-cyclotomic $\mathbb{Z}_p$-extensions.
Abstract: The class numbers of the finite layers of the cyclotomic $\mathbb{Z}_p$-extensions of the rationals have been the subject of much study. By studying the absolute height of cyclotomic units, Horie effectively proved that 100% of prime numbers $q$ never divide the class numbers of the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of the rationals. I will talk about how similar methods can be applied to $\mathbb{Z}_p$-extensions of an imaginary quadratic field $K$ which are unramified outside of a split prime above $p$.

30/04/14 Yiannis Petridis (UCL)
Title: On the hyperbolic lattice-point problem

07/05/14 Avner Ash (Boston College)
Title: Reducible Galois representations and homology of $GL(n,\mathbb{Z})$.
Abstract: In joint work with Darrin Doud, I have connected some reducible 3-dimensional representations of the absolute Galois group of $\mathbb{Q}$ with homology Hecke eigenclasses for congruence subgroups of $GL(3,\mathbb{Z})$. I will explain what we have done and how it fits into the picture of generalized reciprocity laws for $GL(n,\mathbb{Z})$.

14/05/14 David Hansen (Institut de mathématiques de Jussieu)
Title: Eigenvarieties beyond the case of discrete series
Abstract: Given a connected reductive group $G$ over a number field $F$, I'll explain a construction of eigenvarieties parametrizing "Betti-cohomological overconvergent $p$-adic modular forms" on $G$ under a very mild hypothesis (namely that $G$ splits over $F_v$ for each $v|p$). This construction generalizes previous works of a great number of authors, which have largely focused on case where $G(F \otimes R)$ has a discrete series. Beyond the setting of discrete series, the qualitative geometric properties of eigenvarieties change drastically. I'll discuss the known and expected properties of these spaces, and their (largely conjectural) relationship with Galois representations.

21/05/14 4.00 Gergely Zabradi (Eötvös Lorànd University, Budapest)
Title: Algebraic functional equations and completely faithful Selmer groups
Abstract: Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p\geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_{\infty}/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group. This gives an algebraic functional equation of the conjectured $p$-adic $L$-function. As an application we construct completely faithful Selmer groups in case the $p$-adic Lie extension is obtained by adjoining the $p$-power division points of another non-CM elliptic curve $A$. This is joint work with T. Backhausz.

28/05/14 4.00 Pankaj Vishe (University of York)
Title: Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings
Abstract: We use the dynamics on $SL(3,\mathbb{R})/SL(3,\mathbb{Z})$ to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.

04/06/14 4.00 Dzmitry Badziahin (Durham University)

Monday 9/06/14: London-Paris number theory seminar: details are here.

11/06/14 Minhyong Kim (Oxford University)

18/06/14 Andrew Wiles (Oxford)
Title: On class groups of imaginary quadratic fields
NOTE:This talk will be in room S-1.04 and not the usual room (note also the minus sign, not to be confused with S1.04)

25/06/14 Zeev Rudnick (Tel-Aviv University)

The seminar is preceded by the Study Group, which this term will be on Jared Weinstein's recent paper "Semistable models for modular curves of arbitrary level", in room S4.36, and which will run from 1300 to 1530 with a break in the middle.

A list of previous seminar talks is here.

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This page is maintained, poorly, by Kevin Buzzard. If anyone thinks they can do better they're welcome to offer.