Related papers: Counting and equidistribution over primes in hyper…
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic…
We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To…
We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$.…
We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca.
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$.…
We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for…
It is well known that the orbit of a lattice in hyperbolic $n$-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and…
We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension up to 5. We prove generalisations of Mertens'…
We consider a local average in the hyperbolic lattice point counting problem for the Picard group $\Gamma$ acting on the three-dimensional hyperbolic space. Compared to the pointwise case, we improve the bounds on the remainder in the…
We prove an asymptotic formula for the number of primes of the shape $a^2 +p^4$, thereby refining the well known work of Friedlander and Iwaniec. Along the way, we prove a result on equidistribution of primes up to $x$, in which the moduli…
The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are…
We investigate the distribution of orbits of a non-elementary discrete hyperbolic group acting on the n-dimensional hyperbolic space and its geometric boundary. In particular, we show that if the group $\Gamma$ admits a finite…
We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of perfect Lie groups with a polynomial error term. Even for semisimple quotients, our result provides the first infinite…
In [FI09], Friedlander and Iwaniec studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements gamma in SL(2,Z) such that the norm squared |gamma|^2 = a^2 + b^2 + c^2 + d^2 = p, a prime. Under the…
We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early…
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove…
For a prime number $p$, we study the asymptotic distribution of CM points on the moduli space of elliptic curves over $\mathbb{C}_p$. In stark contrast to the complex case, in the $p$-adic setting there are infinitely many different…
We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume…
We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb C(z)$. For a fixed $K \gg 1$, we show that almost all intervals of length $2\pi/K$ in $(-\pi,\pi]$ contain a multiplier angle with the property that…
Problems of (i) precise (exact) bound for families of hyperelliptic curves over prime finite fields and (ii) equidistribution of angles of Kloosterman sums are discussed.