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We prove that for any $t\in 𝐐$, the curve $$5x^3+9y^3+10z^3+12{\left(\frac{t^2+82}{t^2+22}\right)}^3{(x+y+z)}^3=0$$ in ${𝐏}^2$ is a genus $1$ curve violating the Hasse principle. An explicit Weierstrass model for its jacobian $E_t$ is given. The Shafarevich-Tate group of each $E_t$ contains a subgroup isomorphic to $𝐙/3\times 𝐙/3$.

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over $\mathbb{Z}$. We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over ${𝔽}_2$ is smooth is asymptotically $21/64$ as its degree tends to infinity. Much of this paper is an exposition...

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme ${𝔅}_n$ of finite type over $\mathbb{Z}$, with geometrically connected fibers. It is smooth if and only if $n\le 3$. It is reducible if $n\ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of ${𝔅}_n$ is $\frac{2}{27}{n}^3+O\left(n^{8/3}\right)$. The subscheme parameterizing étale algebras is isomorphic to ${GL}_n/S_n$, which is of dimension only $n^2$. For $n\ge 8$, there exist algebras that are not limits of étale algebras. The dimension calculations...

For $p$ an odd prime, we show that the Fekete polynomial $f_p\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right)t^a$ has $\sim {\kappa }_{0}p$ zeros on the unit circle, where $0.500813\>{\kappa }_0\>0.500668$. Here ${\kappa }_0-1/2$ is the probability that the function $1/x+1/(1-x)+{\sum }_{n\in \mathbb{Z}:n\ne 0,1}{\delta }_n/(x-n)$ has a zero in $]0,1[$, where each ${\delta }_n$ is $\pm 1$ with y $1/2$. In fact $f_p\left(t\right)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $|f_p\left(e\right(2i\pi (K+\tau )/p\left)\right)|\<ϵ\sqrt{p}$ for some $\tau \in ]0,1[$ then there is a zero of $f$ close to this arc.