Related papers: Twists of Shimura Curves

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Z_p-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi and…

In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a…

Jordan, Rotger and de Vera-Piquero proved that Shimura curves have no points rational over imaginary quadratic fields under a certain assumption. In this article, we expand their results to the case of number fields of higher degree. We…

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of…

Stark-Heegner points are conjectural substitutes for Heegner points when the imaginary quadratic field of the theory of complex multiplication is replaced by a real quadratic field $K$. They are constructed analytically as local points on…

In this article, we investigate the possible torsion subgroups of twists of abelian varieties with good reduction. As an application, we prove a theorem concerning ramified primes over any quadratic extension where odd-order torsion growth…

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family,…

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic…

Let p be a prime number, F a totally real field such that [F(mu_p): F]=2 and [F:Q] is odd. For delta \in F^times, let [delta] denote its class in F^times/F^{times p}. In this paper, we show Main Theorem. There are infinitely many classes…

We study the Luna slice theorem in the case of quivers with an involution or supermixed quivers as introduced by Zubkov. We construct an analogue to the notion of a local quiver setting. We use this technique to determine dimension vectors…

In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves…

We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our…

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…

We study conditions determining the $L^p$ boundedness of multiple Hilbert transforms associated with polynomials.

Let $p$ be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over $\mathbb{Q}$ endowed with an ordinary $p$-stabilization. Under the Leopoldt and the weak $p$-adic…

In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive…

In this paper we consider the curves $H_{k,t}^{(p)} : y^{p^k}+y=x^{p^{kt}+1}$ over $\mathbb F_p$ and and find an exact formula for the number of $\mathbb F_{p^n}$-rational points on $H_{k,t}^{(p)}$ for all integers $n\ge 1$. We also give…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables…

The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…