Related papers: $6$-torsion and integral points on quartic threefo…

Consider the smooth quadric Q_6 in P^7. The middle homology group H_6(Q_6,Z) is two-dimensional with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1,p) inside Q_6.

We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic…

We discuss bounds for the number of ordinary triple points on complete intersection Calabi-Yau threefolds in projective spaces and for Calabi-Yau threefolds in weighted projective spaces. In particular, we show that in P5 the intersection…

An optimal 3-point quadrature formula of closed type is derived. Various error inequalities are established. Applications in numerical integration are also given.

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\Z/m\Z \oplus \Z/n\Z$, where $m|n$, be a torsion group such that the modular curve…

We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring…

Ellenberg--Venkatesh proved in 2007 that $h_3(d) \ll_\epsilon |d|^{1/3 + \epsilon}$, where $h_3(d)$ denotes the size of the $3$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. We improve this bound to $h_3(d) \ll_\epsilon |d|^{\kappa…

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the…

Let $G$ be a non-trivial torsion free group and $t$ be an unknown. In this paper we consider three equations (over $G$) of arbitrary length and show that they have a solution (over $G$) provided two relations among their coefficients hold.…

We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing…

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of…

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by…

Merel has shown that the order of torsion subgroup of an elliptic curve over a number field can be bounded in terms of only the degree of the number field. The purpose of this note is to investigate what could be the `right bound'. In this…

We study automorphism groups of smooth quintic threefolds. Especially, we describe all the maximal ones with explicit examples of target quintic threefolds. There are exactly $22$ such groups.

We announce the classification of Sato-Tate groups of abelian threefolds over number fields; there are 410 possible conjugacy classes of closed subgroups of USp(6) that occur. We summarize the key points of the "upper bound" aspect of the…

We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral points…

Let $K$ be a number field and $\mathfrak{p} \mid (2)$ be a prime ideal. We compute the fourth level of the $\mathfrak{p}$-adic completions of $K$ when the ramification index is $4$ and the inertial degree is trivial for the ideal…

For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on…

We classify all geometric torsion points on the Fermat quotients $y^n = x^d + 1$ where $n, d \ge 2$ are coprime. In addition, we classify all geometric torsion points on the generic superelliptic curve $y^n = (x - a_1) \cdots (x - a_d)$,…

For any fixed positive integer $n$, we provide a method to compute all imaginary bicyclic biquadratic number fields with class number $n$, along with their class group structures, using the list of all imaginary quadratic number fields…