arXiv subjects

Ahmed Matar

Publications and source records attributed to Ahmed Matar.

9 recordsLinked to original sources

arXiv2026

Vanishing of $\mathrm{Sha}(A/K)[\mathfrak{P}^\infty]$ and its consequences for the anticyclotomic Iwasawa theory of $\mathrm{GL}_2$-abelian varieties

In this article we generalise a classical Kolyvagin's result on the vanishing of the $p$-part of the Shafarevich--Tate group of an elliptic curve (defined over $\mathbb{Q}$) over an imaginary quadratic field $K$ to modular abelian varieties of $\mathrm{GL}_2$-type (defined over a totally real number field $F$) and a CM field $K/F$. Combining this result with the abstract Iwasawa-theoretical argument of [MN19], we show that a similar vanishing holds over the layers of a suitably defined anticyclotomic multi-$\mathbb{Z}_p$-extension of $K$.

math.NT↗

arXiv2026

On the $\mathfrak{M}_H(G)$-property for Selmer groups at supersingular reduction

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ which has good supersingular reduction at the odd prime $p$. We study the variation of Iwasawa invariants and the $\mathfrak{M}_H(G)$-property for signed Selmer groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic number field $K$ that lie inside the $\mathbb{Z}_p^2$-extension $\mathbb{K}_\infty$ of $K$ and are not necessarily cyclotomic. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property which involve the growth of $\mu$-invariants of the signed Selmer groups over intermediate shifted $\mathbb{Z}_p$-extensions in $\mathbb{K}_\infty$, and the boundedness of $\lambda$-invariants as one runs over $\mathbb{Z}_p$-extensions of $K$ inside $\mathbb{K}_\infty$. We give examples where the $\mathfrak{M}_H(G)$-property holds, and also examples where we can prove that it does not hold. It is striking that although the case of supersingular reduction is much more difficult than the case of ordinary reduction, we get finer results here; moreover, we are able to derive analogous criteria for the validity of the $\mathfrak{M}_H(G)$-property of the classical Selmer group, as well as the fine Selmer group. Many of the properties that we investigate have not been studied before in this non-torsion setting. Further, we study various implications between the $\mathfrak{M}_H(G)$-properties for Selmer groups, signed Selmer groups and fine Selmer groups. We apply our results to a conjecture of Mazur, and prove implications between the $\mathfrak{M}_H(G)$-property and Conjectures A and B of Coates and Sujatha.

math.NT↗

arXiv2025

FPGA-based Lane Detection System incorporating Temperature and Light Control Units

Intelligent vehicles are one of the most important outcomes gained from the world tendency toward automation. Applications of IVs, whether in urban roads or robot tracks, do prioritize lane path detection. This paper proposes an FPGA-based Lane Detector Vehicle LDV architecture that relies on the Sobel algorithm for edge detection. Operating on 416 x 416 images and 150 MHz, the system can generate a valid output every 1.17 ms. The valid output consists of the number of present lanes, the current lane index, as well as its right and left boundaries. Additionally, the automated light and temperature control units in the proposed system enhance its adaptability to the surrounding environmental conditions.

cs.CV↗

arXiv2021

Kolyvagin's result on the vanishing of $\sha(E/K)[p^\infty]$ and its consequences for anticyclotomic Iwasawa theory

Let $E$ be an elliptic curve defined over ${\bf Q}$ and $K$ an imaginary quadratic field satisfying the Heegner hypothesis. A classical result of Kolyvagin states that, under suitable assumptions, if the basic Heegner point $y_K \in E(K)$ is not divisible by an odd prime $p$, then the groups $E(K)/{\bf Z} y_K$ and $\sha(E/K)$ are finite and their orders are prime to $p$. In this article we develop the following themes: firstly, we discuss improvements of Kolyvagin's result, following Cha (2005) and Lawson and Wuthrich (2016). Secondly, we prove an abstract Iwasawa-theoretical result which allows us to deduce, under several additional assumptions, that similar vanishing holds for all layers in the anticyclotomic ${\bf Z}_p$-extension of $K$. Analogous results hold for CM points on simple quotients of Jacobians of Shimura curves over totally real fields; this will be discussed in a separate article.

math.NT↗

arXiv2020

On the Lambda-cotorsion subgroup of the Selmer group

Let $E$ be an elliptic curve defined over a number field $K$ with supersingular reduction at all primes of $K$ above $p$. If $K_{\infty}/K$ is a $\mathbb{Z}_p$-extension such that $E(K_{\infty})[p^{\infty}]$ is finite and $H^2(G_S(K_{\infty}), E[p^{\infty}])=0$, then we prove that the $Λ$-torsion subgroup of the Pontryagin dual of $\text{Sel}_{p^{\infty}}(E/K_{\infty})$ is pseudo-isomorphic to the Pontryagin dual of the fine Selmer group of $E$ over $K_{\infty}$. This is the Galois-cohomological analog of a flat-cohomological result of Wingberg.

math.NT↗

arXiv2018

Selmer groups and anticyclotomic $\mathbb{Z}_p$-extensions II

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime where $E$ has ordinary reduction and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. We give sufficient conditions on $E$ and $p$ in order to ensure that $\text{Sel}_{p^{\infty}}(E/K_{\infty})$ is a cofree $Λ$-module of rank one. We also show that these conditions imply that $\text{rank}(E(K_n))=p^n$ for all $n \geq 0$ and that the $p$-primary subgroup of the Tate-Shafarevich group of $E/K_n$ is trivial for all $n \geq 0$.

math.NT↗

arXiv2017

On the Fine Selmer Group

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$. In a previous article the author conjectured that the fine $p^{\infty}$-Selmer group $R_{p^{\infty}}(E/K_{\infty})$ is confinitely generated over $\mathbb{Z}_p$. In this note we prove this conjecture assuming some hypotheses on $E$ and $p$.

math.NT↗

arXiv2017

Fine Selmer Groups, Heegner points and Anticyclotomic $\mathbb{Z}_p$-extensions

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper we make a conjecture about the fine Selmer group over $K_{\infty}$. We also make a conjecture about the structure of the module of Heegner points in $E(K_{\mathfrak{p}_{\infty}})/p$ where $K_{\mathfrak{p}_{\infty}}$ is the union of the completions of the fields $K_n$ at a prime of $K_{\infty}$ above $p$. We prove that these conjectures are equivalent. When $E$ has supersingular reduction at $p$ we also show that these conjectures are equivalent to the conjecture in our earlier work. Assuming these conjectures when $E$ has supersingular reduction at $p$, we prove various results about the structure of the Selmer group over $K_{\infty}$.

math.NT↗

arXiv2015

Selmer groups and anticyclotomic $\mathbb{Z}_p$-extensions

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the $Λ$-corank of $\text{Sel}(E/K_{\infty})$ in the case where $E$ has ordinary reduction at $p$. In the case where $E$ has supersingular reduction at $p$ we make a conjecture about the structure of the module of Heegner points mod $p$. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the $Λ$-corank of $\text{Sel}(E/K_{\infty})$ in the case where $E$ has supersingular reduction at $p$.

math.NT↗