SCIENTIFIC PROGRAMS AND ACTIVITIES

October 22, 2014

October 7-9, 2011
3rd Montreal-Toronto Workshop in Number Theory
at the Fields Institute
222 College St., Toronto (map)


Organizers:
Valentin Blomer, University of Göttingen
Andrew Granville, Universite de Montreal
The third Montreal-Toronto Workshop in Number Theory is devoted to new developments in analytic number theory, in particular additive combinatorics, sieve methods and automorphic forms.

Abstracts

Valentin Blomer
Counting rational points on a cubic surface


Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a projective algebraic variety. I will present analytic and combinatorial techniques for a proof of Manin's conjecture for a certain singular cubic surface and for the analytic continuation of the corresponding height zeta function.

This is joint work with J. Br\"udern.

Vorrapan Chandee
Bounding S(t) via extremal functions

Assuming the Riemann hypothesis we consider the argument function of $\zeta(s)$ given by $S(t) = \frac{1}{\pi}\arg \zeta\big(\frac{1}{2} + it\big)$. We will prove that \begin{equation*}|S(t)| \leq \left( \frac{1}{4} + o(1)\right)\frac{\log t}{\log \log t},\end{equation*}for large $t$, which is an improvement of the previous work of Goldston and Gonek by a factor of 2. Two different approaches to improve a bound for $S(t)$ will be presented in the talk. The first method is to prove bounds for $S_1(t) = \int_0^{t} S\big(\frac{1}{2} + iu\big) \> du$ via extremal functions discovered in the work Carneiro, Littmann and Vaaler, and then use these bounds to obtain the bound for $S(t)$. The other approach is to bound $S(t)$ by relying on the solution of the Beurling-Selberg extremal problem for the odd function $f(x) =\arctan\left(\tfrac{1}{x}\right) - \tfrac{x}{1 + x^2}$, which falls under the scope of recent work by Carneiro and Littmann.

Chantal David
Elliptic curves with a given number of points over finite fields

Let $E$ be an elliptic curve over $\Q$, and $N$ a positive integer. We consider the problem of counting the number of reductions of $E$ modulo $p$ with exactly $N$ points over $\F_p$. The Hasse bound implies the trivial bound $\sqrt{N}/\log{N}$ for the number of such reductions, but no other bound is known. On average over the set of all elliptic curves over $\Q$, we can obtain bounds which are significantly better, and under some hypothesis for the short interval distribution of primes in arithmetic progressions, we can show an asymptotic formula for the average number of reductions with $N$ points. The average result does not depend solely on the size of the integer $N$ as the naive
heuristic predicts, but also on the arithmetic of the integer $N$, as our result indicate that it is more probably to have that $E(\F_p) = N$ when $N$ has many prime factors. This is in agreement with the Cohen-Lenstra heuristics which predict that random groups $G$ appear with a probability weighted by $1/ \# \mbox{Aut}(G)$.

This is joint work with E. Smith (Michigan Technological University and CRM).

Maria Hamel
Polynomial differences in subsets of the integers

In 1978 Sarkozy and Furstenberg proved independently that any subset of the integers with positive upper density must contain two elements whose difference is a non-zero perfect square. In this talk we will discuss an analogue for polynomials with integer roots. In particular, we provide improved quantitative bounds for quadratic polynomials.

This is joint work with Neil Lyall and Alex Rice.

Ram Murty
The Uncertainty Principle and a Theorem of Tao

Let $G$ be a finite abelian group and $f$ a complex-valued function on $G$. The uncertainty principle states that $|supp(f)||supp(\hat{f})|\geq |G|$ where $\hat{f}$ denotes the Fourier transform of $f$. If $G$ has prime order $p$, Tao recently proved that $|supp(f)| + |supp(\hat{f})| \geq p+1$. The key step in his proof relies on an old result of Chebotarev regarding minors of certain Vandermonde determinants. Using the representation theory of the unitary group, we show how one can deduce these results immediately from Weyl's character formula. This reformulation allows us to generalize Tao's result to cyclic groups of prime power order. This is joint work with my undergraduate student Junho Peter Whang.


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