# Past Number Theory Seminar

15 May 2014
16:00
Abstract
I'll sketch a construction which associates a canonical p-adic L-function with a 'non-critically refined' cohomological cuspidal automorphic representation of GL(2) over an arbitrary number field F, generalizing and unifying previous results of many authors. These p-adic L-functions have good interpolation and growth properties, and they vary analytically over eigenvarieties. When F=Q this reduces to a construction of Pollack and Stevens. I'll also explain where this fits in the general picture of Iwasawa theory, and I'll point towards the iceberg of which this construction is the tip.
• Number Theory Seminar
14 May 2014
15:00
Abstract

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
measure.

• Number Theory Seminar
8 May 2014
16:00
Abstract
We extend Kisin's construction of integral canonical models of Hodge-type Shimura varieties to p=2, using Dieudonné display theory.
• Number Theory Seminar
1 May 2014
16:00
Abstract
Let $G=SL(2,\R)\ltimes R^2$ and $\Gamma=SL(2,Z)\ltimes Z^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\G$, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of $\sqrt n$ mod 1.
• Number Theory Seminar
13 March 2014
16:00
Mohamed Saidi
Abstract
<p>We investigate certain (hopefully new) arithmetic aspects of abelian varieties defined over function fields of curves over finitely generated fields. One of the key ingredients in our investigation is a new specialisation theorem a la N\'eron for the first Galois cohomology group with values in the Tate module, which generalises N\'eron specialisation theorem for rational points. Also, among other things, we introduce a discrete version of Selmer groups, which are finitely generated abelian groups. We also discuss an application of our investigation to anabelian geometry (joint work with Akio Tamagawa).</p>
• Number Theory Seminar
6 March 2014
16:00
Martin Orr
Abstract
Let Z be a subvariety of the moduli space of abelian varieties, and suppose that Z contains a dense set of points for which the corresponding abelian varieties are isogenous. A corollary of the Zilber-Pink conjecture predicts that Z is a weakly special subvariety. I shall discuss the proof of this conjecture in the case when Z is a curve and obstacles to its proof for higher dimensions. <p>For Logic Seminar: Note change of time and place.</p>
• Number Theory Seminar
20 February 2014
16:00
Paloma Bengoechea
Abstract
Zagier studied in 1999 certain real functions defined in a very simple way as sums of powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular forms which are the coefficients in Fourier expansion of the kernel function for Shimura-Shintani correspondence. He conjectured for these sums a representation in terms of a finite set of polynomials coming from reduction of binary quadratic forms and the infinite set of transformations occuring in a continued fraction algorithm of the real variable. We will prove two different such representations, which imply the exponential convergence of the sums. <p>For Logic Seminar: Note change of time and location!</p>
• Number Theory Seminar
13 February 2014
16:00
Bob Hough
Abstract
A distinct covering system of congruences is a collection $(a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k$ whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.
• Number Theory Seminar