Related papers: A prime geodesic theorem for higher rank II: singu…
In this article, we consider a compact symmetric space $M$ of higher rank. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\# P(t)$ its cardinality. We obtain the following…
For a closed locally symmetric space M=\Gamma\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the…
Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also for the non-existence of solutions to the unit equation over F. For example, if 2…
We consider geodesic nets (critical points of a length functional on the space of embedded graphs) on doubled polygons (topological 2-spheres endowed with a flat metric away from finitely many cone singularities). We use the theorem of…
Some known fixed point theorems for nonexpansive mappings in metric spaces are extended here to the case of primitive uniform spaces. The reasoning presented in the proofs seems to be a natural way to obtain other general results.
We establish new upper bounds about symmetric bilinear complexity in any extension of finite fields. Note that these bounds are not asymptotical but uniform. Moreover we give examples of Shimura curves that do not descend over their field…
The work of Reid, Chinburg--Hamilton--Long--Reid, Prasad--Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic…
In this note we continue our investigation of geodesics in the space of convex and plurisubharmonic functions. We show optimal regularity for geodesics joining two smooth strictly convex functions. We also investigate the regularity theory…
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…
A mathematically rigorous Hamiltonian formulation for classical and quantum field theories is given. New results include clarifications of the structure of linear fields, and a plausible formulation for nonlinear fields. Many mathematical…
We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama…
The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense…
We investigate the interrelationships of three notions of primary units in the local cyclotomic field of $p$-th roots of~1($p$ being an odd prime number), especially with reference to global units.
We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group…
We show that a locally symmetric space of noncompact type and with finite volume is quasi-isometric to the euclidean cone over a finite simplicial complex. A detailed analysis of metric properties yields a proof of a conjecture of Siegel.
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing…
Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic…