Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $. We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p, $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$-module by elements with norm in S. The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function.

We study the joint distribution of values of a pair consisting of a quadratic form ${\mathbf q}$ and a linear form ${\mathbf l}$ over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of $\mathrm{SL}_3(\mathbb{R})$. Math. Ann. 286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$, then under the assumptions that for every $(\alpha , \beta ) \in {\mathbb {R}}^2 \setminus \{ (0,0) \}$, the form $\alpha {\mathbf q} + \beta {\mathbf l}^2$ is irrational and that the signature of the restriction of ${\mathbf q}$ to the kernel of ${\mathbf l}$ is $(p, n-1-p)$, where ${3\le p\le n-2}$, the number of vectors $v \in {\mathbb {Z}}^n$ for which $\|v\| < T$, $a < {\mathbf q}(v) < b$ and $c< {\mathbf l}(v) < d$ is asymptotically $ C({\mathbf q}, {\mathbf l})(d-c)(b-a)T^{n-3}$ as $T \to \infty $, where $C({\mathbf q}, {\mathbf l})$ only depends on ${\mathbf q}$ and ${\mathbf l}$. The density of the set of joint values of $({\mathbf q}, {\mathbf l})$ under the same assumptions is shown by Gorodnik [Oppenheim conjecture for pairs consisting of a linear form and a quadratic form. Trans. Amer. Math. Soc. 356(11) (2004), 4447–4463].

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.

Let $p$ be a prime, and let ${{\zeta }_{p}}$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p\,-\,1)$-dimensional and are geometrical representations of the integral $\mathbb{Z}[{{\zeta }_{p}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p\,-\,1$ where $149\,\le \,p\,\le \,3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p\,-\,1)(q\,-\,1)$-dimensional lattices from the integral $\mathbb{Z}[{{\zeta }_{pq}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}{{\left\langle 1\,-\,{{\zeta }_{q}} \right\rangle }^{j}}$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

We prove a new upper bound for the smallest zero $x$ of a quadratic form over a number field with the additional restriction that $x$ does not lie in a finite number of $m$ prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on $m$.