For fixed , consider the set of dilated squares modulo . Rudnick and Sarnak conjectured that, for Lebesgue, almost all such the gap-distribution of is consistent with the Poisson model (in the limit as N tends to infinity). In this paper, we prove …
We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic.
Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copi…
Given a compact Kähler manifold X, it is shown that pairs of the form , where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affin…
Given a commutative unital ring R, we show that the finiteness length of a group G is bounded above by the finiteness length of the Borel subgroup of rank one whenever G admits certain R-representations with metabelian image. Combined with results due…
We derive two-sided bounds for the Newton and Poisson kernels of the W-invariant Dunkl Laplacian in the geometric complex case when the multiplicity i.e., for flat complex symmetric spaces. For the invariant Dunkl–Poisson kernel , the estimates are wh…
We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine–Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there e…
We give a formulation of a deformation of Dirac operator along orbits of a group action on a possibly noncompact manifold to get an equivariant index and a K-homology cycle representing the index. We apply this framework to noncompact Hamiltonian torus…
Let be a compact surface with boundary. For a given conformal class c on the functional is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of con…