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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