Abstract:
Using the Luthar--Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.

Abstract:
For a finite group $G$, let $\tilde{\mathbb{Z}}$ be the semilocalization of $\mathbb{Z}$ at the prime divisors of $|G|$. If $G$ is a Frobenius group with Frobenius kernel $K$, it is shown that each torsion unit in the group ring $\tilde{\mathbb{Z}} G$ which maps to the identity under the natural ring homomorphism $\tilde{\mathbb{Z}} G \rightarrow \tilde{\mathbb{Z}} G/K$ is conjugate to an element of $G$ by a unit in $\tilde{\mathbb{Z}} G$.

Abstract:
Let $G$ be a finite group. Zassenhaus conjectured that every torsion unit of augmentation one of the integral group ring of $G$ is conjugate in the rational group algebra to an element of $G$. The HeLP Method provides a technique to prove this conjecture for an element $u$ of order $n$, by showing that the partial augmentation of the elements of the form $u^d$ with $d\mid n$, satisfying certain linear inequalities, are of a specific form. When this is true, the Zassenhaus Conjecture follows for $u$ by a result of Marciniak, Ritter, Sehgal and Weiss. For the case of special linear groups this have been done for the case when $n$ is a prime power. In this paper we investigate the distributions of partial augmentations satisfying the linear inequalities of the HeLP Method for the case of $G=\text{PSL}(2,p)$ and $n=2t$, with $p$ and $t$ odd primes.

Abstract:
It is shown that for any torsion unit of augmentation one in the integral group ring $\mathbb{Z} G$ of a finite solvable group $G$, there is an element of $G$ of the same order.

Abstract:
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice ([Sil6], [GS], [He]). We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3-torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.

Abstract:
Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups $J_1$, $J_2$ and $J_3$ is the same as that of the normalized unit group of their respective integral group ring.

Abstract:
We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group $\operatorname{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\operatorname{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\operatorname{PGL}(2,9)$. This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative answer for all groups having an order divisible by at most three different primes.

Abstract:
We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zolt\'an Bertalan and we also provide some generalization.

Abstract:
We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characteristic form, a quadratic function on a torsion group. When the associated symmetric bilinear pairing is fixed, we construct an affine embedding of a quotient of the set of characteristic forms into the set of all quadratic functions and determine explicitly its cokernel. We determine a suitable class of torsion groups so that quadratic functions defined on them are classified by the stable class of their lift. This refines results due to A.H. Durfee, V. Nikulin, C.T.C. Wall and E. Looijenga -- J. Wahl. Finally, we show that on this class of torsion groups, two quadratic functions are isomorphic if and only if they have equal associated Gauss sums and there is an isomorphism between the associated symmetric bilinear pairings which preserves the "homogeneity defects". This generalizes a classical result due to V. Nikulin. Our results are elementary in nature and motivated by low-dimensional topology.

Abstract:
Consider the Bianchi groups, namely the SL_2 groups over rings of imaginary quadratic integers. In the literature, there has been so far no example of p-torsion in the integral homology of the full Bianchi groups, for p a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance p = 80737 at the discriminant -1747.