The Navarro–Tiep Galois conjecture for $$p=2$$ p = 2

We investigate the upper $$FC-$$ central series of the unit group of an integral group ring $${\mathbb Z}G$$ of a periodic group G. We prove that $${\mathcal U}={{\mathcal U}}_1({\mathbb Z}G)$$ has $$FC-$$ central height one if and only if the $$FC-$$ hypercenter of $${{\mathcal U}}_1({\mathbb Z}G)$$ is contained in the normalizer of the trivial units. Further, in these conditions, the $$FC-$$ hypercenter of the unit group is non-central if and only if G is a $$Q^{*}-$$ group. Let $$H \vartriangleleft {\mathcal U}, H$$ contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of $${{\mathcal U}}_1({\mathbb Z}G)$$ .     

On the smooth transfer conjecture of Jacquet–Rallis for n=3

We obtain a relative Shalika germ expansion of orbital integrals appeared in the relative trace formulae Jacquet?CRallis when n=3. This is the first example where there are infinitely many nilpotent orbits. As an application we can prove the smooth transfer conjecture of Jacquet?CRallis for n=3.     

The Isaacs–Navarro conjecture for the alternating groups

A recent refinement of the McKay conjecture is verified for the case of the alternating groups. The argument builds upon the verification of the conjecture for the symmetric groups [P. Fong, The Isaacs–Navarro conjecture for symmetric groups, J. Algebra 250 (1) (2003) 154–161].     

The refined Coates–Sinnott conjecture for characteristic p global fields

This article is concerned with proving a refined function field analogue of the Coates–Sinnott conjecture, formulated in the number field context in 1974. Our main theorem calculates the Fitting ideal of a certain even Quillen K-group in terms of special values of L-functions. The techniques employed are directly inspired by recent work of Greither and Popescu in the equivariant Iwasawa theory of arbitrary global fields. They rest on the results of Greither and Popescu on the Galois module structure of certain naturally defined Picard 1-motives associated to an arbitrary Galois extension of function fields.     

The Lang–Trotter conjecture for products of non-CM elliptic curves

The Ramanujan Journal - Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $${mathbb {Q}}$$ and by the subsequent...     

The Fischer–Marsden conjecture and contact geometry

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and \((\kappa ,\mu )\)-contact manifolds. First, we prove that a complete K-contact metric satisfying \(\mathcal {L}^{*}_g(\lambda )=0\) is Einstein and is isometric to a unit sphere \(S^{2n+1}\). Next, we prove that if a non-Sasakian \((\kappa ,\mu )\)-contact metric satisfies \(\mathcal {L}^{*}_g(\lambda )=0\), then \( M^{3} \) is flat, and for \(n > 1\), \(M^{2n+1}\) is locally isometric to the product of a Euclidean space \(E^{n+1}\) and a sphere \(S^n(4)\) of constant curvature \(+\,4\).     

Strict Kneser–Poulsen conjecture for large radii

In this paper we prove the Kneser–Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space ${\mathbb{E}^n}$ is rearranged so that the distance between each pair of points does not decrease, then there exists a positive number r ₀ that depends on the rearrangement of the points, such that if we consider n-dimensional balls of radius r > r ₀ with centers at these points, then the volume of the union (intersection) of the balls before the rearrangement is not less (not greater) than the volume of the union (intersection) after the rearrangement. Moreover, the inequality is strict whenever the new point set is not congruent to the original one. Also under the same conditions we prove a similar result about surface volumes instead of volumes. In order to prove the above mentioned results we use ideas from tensegrity theory to strengthen the theorem of Sudakov (Dokl. Akad. Nauk SSSR 197:43–45, 1971), Alexander (Trans. Am. Math. Soc., 288(2):661–678, 1985) and Capoyleas and Pach (Discrete and computational geometry. American Mathematical Society, Providence, 1991), which says that the mean width of the convex hull of a finite number of points does not decrease after an expansive rearrangement of those points. In this paper we show that the mean width increases strictly, unless the expansive rearrangement was a congruence. We also show that if the configuration of centers of the balls is fixed and the volume of the intersection of the balls is considered as a function of the radius r, then the second highest term in the asymptotic expansion of this function is equal to ${-M_nr^{n-1}}$ , where M _n is the mean width of the convex hall of the centers. This theorem was conjectured by Balázs Csikós in 2009.     

The Chen–Chvátal conjecture for metric spaces induced by distance-hereditary graphs

A classical theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced by connected distance-hereditary graphs—a graph $G$ is called distance-hereditary if the distance between two vertices $u$ and $v$ in any connected induced subgraph $H$ of $G$ is equal to the distance between $u$ and $v$ in $G$.     

The error term in the Sato–Tate conjecture

Let \({f(z) = \sum_{n=1}^\infty a(n)e^{2\pi i nz} \in S_k^{\mathrm{new}}(\Gamma_0(N))}\) be a newform of even weight \({k \geq 2}\) that does not have complex multiplication. Then \({a(n) \in \mathbb{R}}\) for all n; so for any prime p, there exists \({\theta_p \in [0, \pi]}\) such that \({a(p) = 2p^{(k-1)/2} {\rm cos} (\theta_p)}\) . Let \({\pi(x) = \#\{p \leq x\}}\) . For a given subinterval \({[\alpha, \beta]\subset[0, \pi]}\) , the now-proven Sato–Tate conjecture tells us that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} \sim \mu_{ST} ([\alpha, \beta])\pi(x),\quad \mu_{ST} ([\alpha, \beta]) = \int\limits_{\alpha}^\beta \frac{2}{\pi}{\rm sin}^2(\theta) d\theta. $$ Let \({\epsilon > 0}\) . Assuming that the symmetric power L-functions of f are automorphic, we prove that as \({x \to \infty}\) , $$ \#\{p \leq x: \theta_p \in I\} = \mu_{ST} ([\alpha, \beta])\pi(x) + O\left(\frac{x}{(\log x)^{9/8-\epsilon}} \right), $$ where the implied constant is effectively computable and depends only on k,N, and \({\epsilon}\) .     

The Duffin–Schaeffer conjecture with extra divergence II

In 2012 the authors set out a programme to prove the Duffin–Schaeffer conjecture for measures arbitrarily close to Lebesgue measure. In this paper we take a new step in this direction. Given a non-negative function $\psi : \mathbb N \rightarrow \mathbb R $ , let $W(\psi )$ denote the set of real numbers $x$ such that $|nx -a| < \psi (n) $ for infinitely many reduced rationals $a/n \ (n>0) $ . Our main result is that $W(\psi )$ is of full Lebesgue measure if there exists a $c > 0 $ such that $$\begin{aligned} \sum _{n\ge 16} \, \frac{\varphi (n) \psi (n)}{n \exp (c(\log \log n)(\log \log \log n))} \, = \, \infty \, . \end{aligned}$$      

On the Mumford–Tate conjecture for 1-motives

We show that the statement analogous to the Mumford–Tate conjecture for Abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group.     

The γ-borel conjecture

In this paper we prove that it is consistent that every -set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong -set is countable while not every -set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version.     

The Teichmüller TQFT volume conjecture for twist knots

The Teichmüller TQFT, defined by Andersen and Kashaev, gives rise to a quantum invariant of triangulated hyperbolic knot complements; it has an associated volume conjecture, where the hyperbolic volume of the knot appears as a certain asymptotic coefficient.In this note, we announce a proof of this volume conjecture for all twist knots up to 14 crossings; along the way we explicitly compute the partition function of the Teichmüller TQFT for the whole infinite family of twist knots.Among other tools, we use an algorithm of Thurston to construct a convenient ideal triangulation of a twist knot complement, as well as the saddle point method for computing limits of complex integrals with parameters.