arXiv subjects

Sajad Salami

Publications and source records attributed to Sajad Salami.

At least 19 recordsLinked to original sources

arXiv2026

Gaussian Behavior and Geometric Gaps in Decompositions from Recurrences with Zero Coefficients

Zeckendorf's theorem establishes a unique representation for positive integers as sums of non-consecutive Fibonacci numbers. This result has been generalized to Positive Linear Recurrence Sequences (PLRS), where key statistical properties, such as the Gaussian distribution of summands, depend on strictly positive recurrence coefficients. This paper investigates the consequences of relaxing this condition by studying \textit{Zero Linear Recurrence Relations (ZLRRs)}, where the leading coefficient is zero ($c_1=0$). Focusing on the \textit{Lagonacci sequence} ($Z_{n+1}=Z_{n-1}+Z_{n-2}$) as a primary case study, we demonstrate that while the uniqueness of decompositions is lost, fundamental statistical behaviors persist. We prove that the number of summands in the canonical greedy decomposition converges to a \textit{Gaussian distribution} and that the distribution of gaps between indices decays \textit{geometrically}. Furthermore, we utilize the \textit{principle of equivalence of ensembles} to show these properties are robust for a wide class of ZLRRs. Finally, we quantify the non-uniqueness of these systems, proving that the number of legal decompositions grows \textit{exponentially} at a rate $\alpha =2$, significantly exceeding the growth of the underlying sequence.

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arXiv2025

The splitting field and generators of the elliptic surface $Y^2=X^3 +t^{360} +1$

The splitting field of an elliptic surface $\mathcal{E}/\mathbb{Q}(t)$ is the smallest finite extension $\mathcal{K} \subset \mathbb{C}$ such that all $\mathbb{C}(t)$-rational points are defined over $\mathcal{K}(t)$. In this paper, we provide a symbolic algorithmic approach to determine the splitting field and a set of $68$ linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface $Y^2=X^3 +t^{360} +1$. This surface is noted for having the largest known rank 68 for an elliptic curve over $\mathbb{C}(t)$. Our methodology utilizes the known decomposition of the Mordell-Weil Lattice of this surface into Lattices of ten rational elliptic surfaces and one $K3$ surface. We explicitly compute the defining polynomials of the splitting field, which reach degrees of 1728 and 5760, and verify the results via height pairing matrices and specialized symbolic software packages.

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arXiv2025

The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3 +t^{m} +1$ (I)

The splitting field of an elliptic surface $\mathcal E$ defined over ${\mathbb Q}(t)$ is the smallest subfield $\mathcal K$ of $\mathbb C$ such that ${\mathcal E}({\mathbb C}(t))\cong {\mathcal E}({\mathcal K}(t))$. In this paper, we determine the splitting field ${\mathcal K}_m$ and a set of linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface with generic fiber given by ${\mathcal E}_m: y^2=x^3 +t^{m} +1$ over ${\mathbb Q}(t)$ for positive integers $1\leq m\leq 12$.

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arXiv2025

On Properly $\theta$-Congruent Numbers Over Real Number Fields

The notion of $\theta$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $\theta$-congruent if it is the area of a rational triangle with an angle $\theta$ whose cosine is rational. Das and Saikia [2] established criteria for numbers to be $\theta$-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta$-congruent and properly $\theta$-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to $6$, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree~$6$, and examine the exceptional cases $n=1, 2, 3$ and $6$.

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arXiv2025

Orbit counts and twist zeta functions of weighted projective stacks over finite fields

Let ${\mathbb F}_q$ be a finite field and $\mathbf{w}=(w_0,\dots,w_n)$ a vector of positive integer weights. Several finite-field counts attached to the weighted projective space $\mathbb{P}^n_\mathbf{w}$ are easily conflated: the coarse rational-point count and the stacky mass are both weight-independent, whereas the number $A_\mathbf{w}(q)$ of ${\mathbb F}_q^\times$-orbits on nonzero ${\mathbb F}_q$-representatives for the weighted action depends on the weights. We prove the closed formula \[ A_\mathbf{w}(q)=\sum_{\emptyset\ne S\subseteq\{0,\dots,n\}}(q-1)^{|S|-1}\gcd(k_S,q-1), \qquad k_S=\gcd\{w_i:i\in S\}, \] and identify $A_\mathbf{w}(q)$ intrinsically as the number of ${\mathbb F}_q$-isomorphism classes of the weighted projective stack $\mathcal{P}_\mathbf{w}=[({\mathbb A}^{n+1}\setminus\{0\})/\mathbb{G}_m]$ -- equivalently, the number of ${\mathbb F}_q$-twists lying over the coarse points -- the discrepancy from the coarse count being governed by the Kummer groups ${\mathbb F}_q^\times/({\mathbb F}_q^\times)^{k_S}$. We read the behaviour of $A_\mathbf{w}$ under reduction of the weight vector through the ${\mathbb F}_q$-cohomology of the associated $\boldsymbol{\mu}_d$-gerbe, and prove that the twist zeta function $Z_{\mathrm{tw}}(\mathcal{P}_\mathbf{w},t)=\exp\bigl(\sum_{r\ge1}A_\mathbf{w}(q^r)t^r/r\bigr)$ is rational with multiplicity spectrum independent of $q$, admitting a single global functional equation precisely when the weights share a common prime-to-$p$ part -- for reduced $\mathbf{w}$, precisely when the twist theory is trivial. For weighted diagonal hypersurfaces and same-degree pairs in the split regime, we compute the twist zeta function of the substack explicitly, with reciprocal roots given by Gauss sums and, for intersections, by Frobenius eigenvalues of superelliptic curves.

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arXiv2025

Rank of Jacobian Varieties of Curves $y^s=x(ax^r+b)$

Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \in k$ ($ab \neq 0$), and $r \geq 1, s \geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family. Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\mathcal{W}_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\mathcal{X}_n$, a complete intersection variety. The geometric properties of the fibers of $\Xc_n \to \text{Sym}^{n+1}(\mathbb{P}^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it states that the rank of elliptic curves $y^2=x (x^2+B)$ is uniformly bounded subject to the strong version of Lang's conjecture.

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arXiv2025

On the Rank of Jacobian Varieties of the Curves $y^s=ax^r+b$

We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties of general type, we prove that the Mordell-Weil rank of the Jacobian varieties of these curves is uniformly bounded. The proof proceeds by constructing a parameter space for curves in the family with a given number of rational points and analyzing the geometry of its fibers, which are shown to be complete intersection curves of increasing genus.

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arXiv2025

Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant

This paper examines the arithmetic of the loci $\cL_n$, parameterizing genus 2 curves with $(n, n)$-split Jacobians over finite fields $\F_q$. We compute rational points $|\cL_n(\F_q)|$ over $\F_3$, $\F_9$, $\F_{27}$, $\F_{81}$, and $\F_5$, $\F_{25}$, $\F_{125}$, derive zeta functions $Z(\cL_n, t)$ for $n = 2, 3$. Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

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arXiv2023

On the powerful values of polynomials over number fields

Let ${\mathcal B}=\{b_i \}_{i=1}^\infty$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\leq s \leq r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic numbers on a number field $k$, we provide lower and upper bounds for the cardinal number of ${\mathbf G}_{r,s}^{{\mathcal B}_M}$ the set of polynomials $f\in k[x]$ of degree $r\geq 2$ whose irreducible factors have multiplicity strictly less than $s$ and $f(b_1),\cdots, f(b_M)$ are nonzero $s$-powerful elements in $k$, where $M=2r^2+6r +1$ if $r=s$, and $2sr^2+ s r+1$ otherwise. Moreover, considering certain conditions on ${\mathcal B}$, we show the existence of an integer $M_0> M$ such that no polynomial in ${\mathbf G}_{r,s}^{{\mathcal B}_M}$ takes $s$-powerful values at all of $b_1, \cdots, b_n $ for $n\geq M_0$.

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arXiv2023

Local and Global Heights on Weighted Projective Varieties

We investigate local and global weighted heights a-la Weil for weighted projective spaces via Cartier and Weil divisors and extend the definition of weighted heights on weighted projective spaces from arXiv:1902.06563 to weighted varieties and closed subvarieties. We prove that any line bundle on a weighted variety admits a locally bounded weighted $M$-metric. Using this fact, we define local and global weighted heights for weighted varieties in weighted projective spaces and their closed subschemes and show their fundamental properties.

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arXiv2023

Vojta's conjecture on weighted projective varieties

We formulate Vojta's conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. In the process, we introduce generalized weighted general common divisors and express them as heights of weighted projective spaces blown-up relative to an exceptional divisor. Furthermore, we prove that assuming Vojta's conjecture for weighted projective varieties one can bound the $\log {\rm h_{wgcd}}$ for any subvariety of codimension $\geq 2$ and a finite set of places $S$. An analogue result is proved for weighted homogeneous polynomials with integer coefficients.

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arXiv2022

Generators and splitting fields of certain elliptic K3 surfaces

Let $k \subset {\mathbb C}$ be a number field and ${\mathcal E}$ be an elliptic curve defined over $k(t)$, the rational function field of the projective line ${\mathbb P}^1_k$, is isomorphic to the generic fiber of an elliptic surface $\pi:= \Sc_\Ee \rightarrow {\mathbb P}^1_k$. For any subfield ${\mathcal K}\subseteq {\mathbb C}$ of $k$, the set ${\mathcal E}({\mathcal K}(t))$ of ${\mathcal K}(t)$-rational points of ${\mathcal E}$ is known to be a finitely generated abelian group. The splitting field of ${\mathcal E}$ defined over $k(t)$ is the smallest finite extension ${\mathcal K} \subset {\mathbb C}$ of $k$ such that ${\mathcal E} ({\mathbb C} (t)) \iso {\mathcal E} ({\mathcal K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k={\mathbb Q}$ with the generic fiber given by the Weierstrass equation ${\mathcal E}_n: \displaystyle y^2=x^3 + t^n + 1/t^n$, $1\leq n\leq 6$, and determine the splitting field ${\mathcal K}_n$, and find an explicit set of independent generators for ${\mathcal E}_n ({\mathcal K_n}(t))$ for $1\leq n \leq 6$.

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arXiv2021

Twists of Albanese varieties over function fields with large ranks

In this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We achieve this by using a generalization of the notion of Prym variety to higher dimensions and a structure theorem for the Mordell-Weil group of abelian varieties over function fields proven in our previous works. We consider abelian and dihedral covers of the projective space and apply the above results to the twists of their Albanese varieties.

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arXiv2021

Rational $θ$-parallelogram envelopes via $θ$-congruent elliptic curves

We introduce a new generalization of $θ$-congruent numbers by defining the notion of rational $θ$-parallelogram envelope for a positive integer $n$, where $θ\in (0, π)$ is an angle with rational cosine. Then, we study more closely some problems related to the rational $θ$-parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of T.~Ochiai, where only the case $θ=π/2$ was considered. Moreover, we answer the open questions in his paper and their generalizations for any Pythagorean angle.

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arXiv2020

The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

A positive integer $N$ is called a $θ$-congruent number if there is a $\ta$-triangle $(a,b,c)$ with rational sides for which the angle between $a$ and $b$ is equal to $θ$ and its area is $N \sqrt{r^2-s^2}$, where $θ\in (0, π)$, $\cos(θ)=s/r$, and $0 \leq |s|<r$ are coprime integers. It is attributed to Fujiwara \cite{fujw1} that $N$ is a $\ta$-congruent number if and only if the elliptic curve $E_N^\ta: y^2=x (x+(r+s)N)(x-(r-s)N)$ has a point of order greater than $2$ in its group of rational points. Moreover, a natural number $N\neq 1,2,3,6$ is a $\ta$-congruent number if and only if rank of $E_N^\ta(\Q)$ is greater than zero. In this paper, we answer positively to a question concerning the existence of methods to create new rational $θ$-triangle for a $θ$-congruent number $N$ from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle $θ$ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in $E_N^θ({\mathbb Q})$. Then, based on the addition of two distinct points in $E_N^θ({\mathbb Q})$, we provide a way to find new rational $\ta$-triangles for the $θ$-congruent number $N$ using given two distinct ones. Finally, we give an alternative proof for Fujiwara's theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $E_N^θ({\mathbb Q})$ with corresponding rational $θ$-triangles

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arXiv2020

Twists of the Albanese varieties of cyclic multiple planes with large ranks over higher dimension function fields

In [17], we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as the twists of abelian varieties by the cyclic covers of projective varieties in terms of the Prym varieties associated with covers. In this paper, we provide an explicit way to construct the abelian varieties with large ranks over the higher dimension function fields. To do so, we apply the above-mentioned theorem to the twists of Albanese varieties of the cyclic multiple planes.

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arXiv2019

A correction to the paper "On Curves with Split Jacobian"

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the formula given in [5] works only under certain conditions.

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arXiv2019

On special matrices related to Cauchy and Toeplitz matrices

In this paper, we are going to calculate the determinant of a certain type of square matrices, which are related to the well-known Cauchy and Toeplitz matrices. Then, we will use the results to determine the rank of special non-square matrices.

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