Related papers: Goldfeld conjecture for non-hyperelliptic directio…

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov -…

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…

The Katz-Sarnak philosophy predicts that statistics of zeros of families of L-functions are strikingly universal. However, subtle arithmetical differences between families of the same symmetry type can be detected by calculating lower-order…

We study unit groups of rings of the form $\mathbb{F}_2[x,y]/(y^2 + gy + h)$, for $g, h \in \mathbb{F}_2[x]$ -- in particular, the question of (non)triviality of such unit groups. Up to automorphisms of $\mathbb{F}_2[x,y]$ we classify such…

Let $E$ be an elliptic curve over $\mathbf{Q}$. We conjecture asymptotic estimates for the number of vanishings of $L(E,1,\chi)$ as $\chi$ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on…

A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both…

We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an…

It is a folk conjecture that for alpha > 1/2 there is no alpha-Hoelder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of R^2 into the Heisenberg group that is Hoelder…

We show that the twisted conjugacy problem is solvable for large-type Artin groups whose outer automorphism group is finite, generated by graph automorphisms and the global inversion. This includes XXXL Artin groups whose defining graph is…

In this short note we prove the Borel conjecture for a family of aspherical manifolds that includes higher graph manifolds.

We use the methods of topological Hochschild homology to shed new light on the groups satisfying the Bass trace conjecture. We show that the factorization of the Hattori-Stallings rank map through the Bokstedt-Hsiang-Madsen cyclotomic trace…

We show that the cop number of the Cayley sum graph of a finite group $G$ with respect to a symmetric subset $S$ is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop…

Let C be a hyperelliptic curve of good reduction defined over a discrete valuation field K with algebraically closed residue field k. Assume moreover that char k \ne 2. Given d \in K^*\K^*2, we introduce an explicit description of the…

By focusing on the family $E:y^2=x^3+a$, we present strategies for determining the structure of the torsion subgroup of the Mordell-Weil group of an elliptic curve, $E(K)$, over quadratic field $K$. Generalizations of the Nagell-Lutz…

In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are $0$ and then…

Let S be a closed oriented surface of genus g > 1, and let T denote its Torelli group. First, given a set E of homotopically nontrivial, pairwise disjoint, pairwise nonisotopic simple closed curves on S, we determine precisely when a…

Following Katz-Sarnak, Iwaniec-Luo-Sarnak, and Rubinstein, we use the 1- and 2-level densities to study the distribution of low lying zeros for one-parameter rational families of elliptic curves over Q(t). Modulo standard conjectures, for…

We exhibit several families of elliptic curves with torsion group isomorphic to $ \Z/6\Z$ and generic rank at least $3$. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We…

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology,…

We determine the twisting Sato-Tate group of the genus $3$ hyperelliptic curve $y^2 = x^{8} - 14x^4 + 1$ and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of $y^2 = x^{8}…