Related papers: CM points on Shimura curves and $p$-adic binary qu…

The goal of this paper is to study the $p$-adic variation of Heegner points and generalized Heegner classes for ordinary families of quaternionic modular forms. We compare classical specializations of big Heegner points (introduced in the…

We prove the Selmer ranks in certain families of $p$-th twists of CM abelian varieties obey the symplectic or unitary distributions. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curves $X^p+Y^p=\delta$ over a…

This is the third of a series of papers relating intersections of special cycles on the integral model of a Shimura surface to Fourier coefficients of Hilbert modular forms. More precisely, we embed the Shimura curve over Q associated to a…

We study the Mahler measures of the polynomial family $Q_k(x,y) = x^3+y^3+1-kxy$ using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers $\leqslant 3$, we employ these…

We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy…

Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K,…

Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian…

Let $C/k$ be a smooth curve over a finite field of characteristic $p>0$. We prove that there are finitely many principally polarized abelian schemes of given dimension $g$ over $C$ up to $p$-power isogeny. For curves over $\overline{k}$, we…

In this paper, we prove some interesting identities, among average representation numbers (associated to definite quaternion algebras) and `degree' of Hecke correspondences on Shimura curves (associated to indefinite quaternion algebras).

Let $E$ be a CM elliptic curve over $\Bbb{Q}$. We refine the work of Cojocaru on the asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is of square-free order. Also, we derive an unconditional…

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two…

We study central simple algebras in various ways, focusing on the role of $p$-central subspaces. The first part of my thesis is dedicated to the study of Clifford algebras. The standard Clifford algebra of a given form is the generic…

We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of…

We develop class field theory of curves over $p$-adic fields which extends the unramified theory of S. Saito. The class groups which approximate abelian \'etale fundamental groups of such curves are introduced in the terms of algebraic…

We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: one related to universal structures, and one related to co-universal structures. First we provide connections between universal…

This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris…

The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue…

Averaging over imaginary quadratic fields, we prove, quantitatively, the equidistribution of CM points associated to 3-torsion classes in the class group. We conjecture that this equidistribution holds for points associated to ideals of any…

The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and,…