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 2017), the seminar will be hosted by Kings College, and will be held on Wednesdays starting at 4pm. The organisers are Nadav Yesha, Julien Hauseux and Wansu Kim. Following recent tradition, the talks will be at random locations throughout Kings so you'd better be paying attention to the seminar listings each week. The talks will always be in the Kings building, in rooms K1.28 (first floor), K4U.12 (4th floor upper mezzanine), and K-1.14 (note the minus sign, it's level -1). Here are explicit instructions for how to get to these places!

K1.28 (King’s Building, Level 1)
The first floor of King’s Building is not a "separate floor", but almost like a mezzanine floor between the ground floor and the second floor. The lecture hall K1.28 can be only accessed from the ground floor through a specific set of stairs. (If you start seeing rooms K2.** and the chapel after going up the stairs, then you are on the second floor, not on the first floor. To go to K1.28, one has to go down to the ground floor and find the correct set of stairs.)
From the entrance on Strand, go straight ahead down the corridor (past the reception, the first staircase on your left and lifts on your right). Turn left just after the vending machine, turn right (not immediate left!) up a small flight of stairs. K1.28 is straight ahead through a double-door.

K4U.12 (King’s Building, Level 4, upper mezzanine)
At the reception, take the lifts to the 5th floor of Strand Building. Turn right out of the lifts, and immediately on the right you will find the stairs leading to King’s Building. Halfway down the stairs (before reaching the fourth floor of Strand building), go through the door on your left, leading to King’s Building. After exiting the double door at the other end of the corridor, turn left toward the Department of Geography. Take the second set of stairs on the right leading to the upper mezzanine. Go through the door and turn left, K4U.12 is at the end of the corridor.

K-1.14 (King’s Building, Level minus 1)
From reception, take the first staircase on your left, go down 1 flight of stairs, go left through the fire door and carry on straight ahead. Walk down the ramp and carry on straight down the corridor to the very end and K-1.14 is to the left of the stairs.

The talks start on Wed 26th April and finish on 28th June. The talks will be preceded by tea at 3:30 in S5.21.

26/04/14 Rebecca Bellovin (Imperial)
Room: K1.28
Title: Local $\varepsilon$-isomorphisms in families
Abstract: Given a representation of $Gal_{Q_p}$ with coefficients in a $p$-adically complete local ring $R$, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex. When $R=Z_p$ and the representation is a lattice in a de Rham representation, this trivialization should be related to the $\varepsilon$-factor of the corresponding Weil--Deligne representation. Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations. This is joint work with Otmar Venjakob.

Room: K1.28
Title: Arithmetic of hyperelliptic curves over local fields
Abstract: Let $C:y^2 = f(x)$ be a hyperelliptic curve over a local field $K$ of odd residue characteristic. I will explain how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of $f(x)$. This is joint work with Tim Dokchitser, Celine Maistret and Adam Morgan.

10/05/17 Chris Hughes (Univ of York)
Room: K1.28
Title: A new upper bound on Skewes' number
Abstract: The Prime Number Theorem tells us that the logarithmic integral, $li(x)$, is a good approximation to $\pi(x)$, the number of primes up to x. Numerically it always seems to be an overestimate, so $\pi(x)-li(x)$ is negative. The first point where this ceases to be the case is known as Skewes' number whose true value is as yet unknown. I will report on joint work with Chris Smith and Dave Platt, where we improve the best upper bound on Skewes' number.

Room: K1.28
Title: Pair correlation for quadratic polynomials mod 1.
Abstract: It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.

24/05/17 Stefano Vigni (Università di Genova)
Room: K1.28
Title: A Gross-Zagier formula for a certain anticyclotomic p-adic L-function of a rational elliptic curve.
Abstract: Let E be a (semistable) rational elliptic curve of conductor N, let K be an imaginary quadratic field satisfying a "Heegner hypothesis" relative to N and let p be a prime of split multiplicative reduction for E that splits in K. Following a recipe proposed by Bertolini and Darmon, I will define a p-adic L-function L_p(E/K) in terms of distributions of Heegner points on Shimura curves that are rational over the anticyclotomic Z_p-extension of K. The "special value" of L_p(E/K) is 0, and I will sketch a proof of a Gross-Zagier formula for the first derivative of L_p(E/K) involving a Heegner point over K and a p-adic L-invariant of E à la Mazur-Tate-Teitelbaum. The strategy is based on level raising arguments and Jochnowitz-type congruences. This is joint work (in progress) with Rodolfo Venerucci.

31/5/17 Wushi Goldring (Stockholm Uni)
Room: K1.28
Title: Geometry engendered by G-Zips: Shimura varieties and beyond.
Abstract: Moonen, Pink, Wedhorn and Ziegler initiated a theory of G-Zips, which is modeled on the de Rham cohomology of varieties in characteristic p>0 "with G-structure", where G is a connected reductive F_p-group. Building on their work, when X is a good reduction special fiber of a Hodge-type Shimura variety, it has been shown that there exists a smooth, surjective morphism \zeta from X to a quotient stack G-Zip^{\mu}. When X is of PEL type, the fibers of this morphism recover the Ekedahl-Oort stratification defined earlier in terms of flags by Moonen. It is commonly believed that much of the geometry of X lies beyond the structure of \zeta.

I will report on a project, initiated jointly with J.-S. Koskivirta and developed further in joint work with Koskivirta, B. Stroh and Y. Brunebarbe, which contests this common view in two stages: The first consists in showing that fundamental geometric properties of X are explained purely by means of \zeta (and its generalizations). The second is that, while these geometric properties may appear to be special to Shimura varieties, the G-Zip viewpoint shows that they hold much more generally, for geometry engendered by G-Zips: Any scheme Z equipped with a morphism to GZip^{\mu} satisfying some general scheme-theoretic properties. To illustrate our program concretely, I will describe results and conjectures regarding two basic geometric questions about X, Z: (i) Which automorphic vector bundles on X, Z admit global sections? (ii) Which of these bundles are ample? As a corollary, we also deduce old and new results over the complex numbers. Question (i) was inspired by a conjecture of F. Diamond on Hilbert modular forms mod p.

7/6/17 Jens Marklof (Univ of Bristol)
Room: K4U.12
Title: Higher dimensional Steinhaus problems.
Abstract: The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence α, 2α, . . . , Nα, for any integer N and real number α. This statement was proved in the 1950s independently by various authors. In this talk I will explain a different approach, which is based on the geometry of the space of two-dimensional Euclidean lattices (with Andreas Strombergsson, Amer. Math. Monthly, in press). This approach can in fact be generalised to deal with analogous higher dimensional Steinhaus problems for gaps in the fractional parts of linear forms. Here we are able to shed new light on a question of Erdos, Geelen and Simpson, proving the existence of parameters for which the number of distinct gaps is unbounded (joint work with Alan Haynes).

14/6/17 Beth Romano (Univ of Cambridge)
Room: K4U.12
Title: On the arithmetic of simple singularities of type E.
Abstract: Given a simply laced Dynkin diagram, one can use Vinberg theory of graded Lie algebras to construct a family algebraic curves. In the case when the diagram is of type E7 or E8, Jack Thorne and I have used the relationship between these families of curves and their associated Vinberg representations to gain information about integral points on the curves. In my talk, I’ll focus on the role Lie theory plays in the construction of the curves and in our proofs.

21/6/17 Samit Dasgupta (UC Santa Cruz)
Room: K-1.14
Title: On the characteristic polynomial of Gross's regulator matrix.
Abstract: Let $F$ be a totally real field and $\chi$ a totally odd character of $F$. Gross conjectured that the leading term of the Deligne-Ribet $p$-adic $L$-function associated to $\chi$ at $s=0$ is equal to a $p$-adic regulator of $p$-units in the extension of $F$ cut out by $\chi$. I recently proved this result in joint work with Mahesh Kakde and Kevin Ventullo. The topic of this talk is a refinement of Gross's conjecture. I will propose an analytic formula for the characteristic polynomial of Gross's regulator matrix, rather than just its determinant. The formula is given in terms of the Eisenstein cocycle and in fact applies (conjecturally) to give all the principal minors of Gross's matrix. For the diagonal entries, the conjecture overlaps with the conjectural formula presented in prior work. This is joint work with Michael Spiess.

28/6/17 Morten Risager (Uni of Copenhagen)
Room: K4U.12
Title: Arithmetic statistics of modular symbols.
Abstract: 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. We prove a refined version of this conjecture.
This is joint work with Yiannis Petridis.

As usual, the seminar will be preceded by various study groups, starting at 1230 (note time change) (for more details follow the link).

A list of previous seminar talks is here.

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