Let E be an elliptic curve defined over Q(t) such that E(C(t)), the set of C(t)-rational points o... more Let E be an elliptic curve defined over Q(t) such that E(C(t)), the set of C(t)-rational points of E, is a finitely generated abelian group. The splitting field of E over Q(t) is the smallest subfield K of C for which E(C(t)) = E(K(t)). It is known that K is a Galois extension of Q with the finite Galois group G = Gal (K|Q) such that the G-invariant elements of E(K(t)) are the E(Q(t))-rational points. In this paper, we will determine the splitting field Kn and find an explicit set of independent generators for the Mordell-Weil group En(Kn(t)) of the elliptic K3 surfaces En : y 2 = x 3 + t n + 1/t n for 1 ≤ n ≤ 6.
In this paper, assuming a conjecture of Vojta on the bounded degree algebraic numbers on a number... more In this paper, assuming a conjecture of Vojta on the bounded degree algebraic numbers on a number field k, we determine explicit lower and upper bounds for the cardinal number of the set of polynomials f ∈ k[x] with degree r ≥ 2 whose irreducible factors have multiplicity strictly less than s and the values f (b 1), • • • , f (b M) are s-powerful elements in k * for a certain positive integer M , where b i 's belong to an arbitrary sequence of the pairwise distinct element of k that satisfy certain conditions.
In this paper, we are going to calculate the determinant of a certain type of square matrices, wh... more 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.
We introduce a new generalization of θ-congruent numbers by defining the notion of rational θ-par... more 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∈ (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.
In this paper, we introduce a new generalization of θ-congruent numbers by defining the notion of... more In this paper, we introduce a new generalization of θ-congruent numbers by defining the notion of θ-parallelogram envelops for an angle θ ∈ (0, π) with rational cosine and a positive integer. We study more closely some problems related to θ-parallelogram envelops, using the arithmetic of algebraic curves. Our results generalize the recent work of T. Ochiai, where only the case θ = π/2 was considered, and answer to the open questions contained in his paper and their generalization for any Pythagorean angles. Subjclass 2010: Primary 11G05; Secondary 14H52 keywords: θ-parallelogram envelop, θ-congruent number, elliptic curve..
In \cite{salami1}, we proved a structure theorem on the Mordell-Weil group of abelian varieties o... more In \cite{salami1}, we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as twists of abelian varieties by cyclic covers of quasi projective varieties in terms of Prym varieties associated to covers. In this paper, we are going to provide an explicit way to construct abelian varieties with arbitrary large rank over function fields. This will be done by applying the above mentioned theorem to the twists of Albanese variety of cyclic multiple plane.
In [16], we proved a structure theorem on the Mordell-Weil group of abelian varieties over functi... more In [16], we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as twists of abelian varieties by cyclic covers of quasi projective varieties in terms of Prym varieties associated to covers. In this paper, we are going to provide an explicit way to construct abelian varieties with arbitrary large rank over function fields. This will be done by applying the above mentioned theorem to the twists of Albanese variety of cyclic multiple plane. Subjclass 2010: Primary 11G10; Secondary 14H40 keywords: Mordell-Weil rank, Albanese and Prym varieties, cyclic multiple plane, twists, function fields.
In this paper, we are going to calculate the determinant of a certain type of square matrices, wh... more 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.
A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with ratio... more A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r − s, where θ ∈ (0, π), cos(θ) = s/r, and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara [4] that N is a θ-congruent number if and only if the elliptic curve E N : 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 6= 1, 2, 3, 6 is a θ-congruent number if and only if rank of E N (Q) is greater than zero. In this paper, we answer positively to a question concerning with 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 this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We ... more 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.
Bulletin of the Brazilian Mathematical Society, New Series, 2020
A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with ratio... more A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r 2 − s 2 , where θ ∈ (0, π), cos(θ) = s/r , and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235-241, 1997) that N is a θ-congruent number if and only if the elliptic curve E θ N : 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 = 1, 2, 3, 6 is a θ-congruent number if and only if rank of E θ N (Q) is greater than zero. In this paper, we answer positively to a question concerning with 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 θ N (Q). Then, based on the addition of two distinct points in E θ N (Q), we provide a way to find new rational θ-triangles for the θ-congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E θ N (Q) with corresponding rational θ-triangles.
In [2003], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of... more In [2003], 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 by Beshaj, Shaska, and Shor 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 Yamauchi's formula works only under certain conditions.
Let K = Q(√ m) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose tha... more Let K = Q(√ m) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < |s| < r and gcd(r, s) = 1. A positive integer n is called a (K, θ)-congruent number if there is a triangle, called the (K, θ, n)-triangles, with sides in K having θ as an angle and nα θ as area, where α θ = √ r 2 − s 2. Consider the (K, θ)-congruent number elliptic curve E n,θ : y 2 = x(x + (r + s)n)(x − (r − s)n) defined over K. Denote the squarefree part of positive integer t by sqf(t). In this work, it is proved that if m = sqf(2r(r − s)) and mn = 2, 3, 6, then n is a (K, θ)-congruent number if and only if the Mordell-Weil group E n,θ (K) has positive rank, and all of the (K, θ, n)-triangles are classified in four types.
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ <... more Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For fixed θ, this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2)x. We study two special cases: θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases. 2010 AMS Mathematics subject classification. Primary 11G05. Keywords and phrases. θ-congruent number, elliptic curve, Mordell-Weil rank.
In this article, we describe a method for finding congruent number elliptic curves with high rank... more In this article, we describe a method for finding congruent number elliptic curves with high ranks. The method involves an algorithm based on the Monsky's formula for computing 2-Selmer rank of congruent number elliptic curves, and Mestre-Nagao's sum which is used in ...
We develop the theory of local and global weighted heights a-la Weil for weighted projective spac... more We develop the theory of local and global weighted heights a-la Weil for weighted projective spaces P n w,k via Cartier divisors by extending the definition of weighted heights for weighted projective varieties and their closed subvarieties, and weighted log pairs. We state 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. Furthermore, we introduce generalized weighted greatest common divisors and express them as heights of the weighted projective spaces blownup at a point, relative to an exceptional divisor. We show that a point x ∈ P n w,k is smooth if and only if its generalized logarithmic weighted greatest common divisor log h wgcd x > 0. In the last part we prove that assuming Vojta's conjecture for weighted projective varieties one can bound the log h wgcd for any subvariety of codimension ≥ 2 and a finite set of places S. An analogue result is proved for weighted homogenous polynomials with integer coefficients.
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ ... more Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2 )x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases.
Let E be an elliptic curve defined over Q(t) such that E(C(t)), the set of C(t)-rational points o... more Let E be an elliptic curve defined over Q(t) such that E(C(t)), the set of C(t)-rational points of E, is a finitely generated abelian group. The splitting field of E over Q(t) is the smallest subfield K of C for which E(C(t)) = E(K(t)). It is known that K is a Galois extension of Q with the finite Galois group G = Gal (K|Q) such that the G-invariant elements of E(K(t)) are the E(Q(t))-rational points. In this paper, we will determine the splitting field Kn and find an explicit set of independent generators for the Mordell-Weil group En(Kn(t)) of the elliptic K3 surfaces En : y 2 = x 3 + t n + 1/t n for 1 ≤ n ≤ 6.
In this paper, assuming a conjecture of Vojta on the bounded degree algebraic numbers on a number... more In this paper, assuming a conjecture of Vojta on the bounded degree algebraic numbers on a number field k, we determine explicit lower and upper bounds for the cardinal number of the set of polynomials f ∈ k[x] with degree r ≥ 2 whose irreducible factors have multiplicity strictly less than s and the values f (b 1), • • • , f (b M) are s-powerful elements in k * for a certain positive integer M , where b i 's belong to an arbitrary sequence of the pairwise distinct element of k that satisfy certain conditions.
In this paper, we are going to calculate the determinant of a certain type of square matrices, wh... more 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.
We introduce a new generalization of θ-congruent numbers by defining the notion of rational θ-par... more 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∈ (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.
In this paper, we introduce a new generalization of θ-congruent numbers by defining the notion of... more In this paper, we introduce a new generalization of θ-congruent numbers by defining the notion of θ-parallelogram envelops for an angle θ ∈ (0, π) with rational cosine and a positive integer. We study more closely some problems related to θ-parallelogram envelops, using the arithmetic of algebraic curves. Our results generalize the recent work of T. Ochiai, where only the case θ = π/2 was considered, and answer to the open questions contained in his paper and their generalization for any Pythagorean angles. Subjclass 2010: Primary 11G05; Secondary 14H52 keywords: θ-parallelogram envelop, θ-congruent number, elliptic curve..
In \cite{salami1}, we proved a structure theorem on the Mordell-Weil group of abelian varieties o... more In \cite{salami1}, we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as twists of abelian varieties by cyclic covers of quasi projective varieties in terms of Prym varieties associated to covers. In this paper, we are going to provide an explicit way to construct abelian varieties with arbitrary large rank over function fields. This will be done by applying the above mentioned theorem to the twists of Albanese variety of cyclic multiple plane.
In [16], we proved a structure theorem on the Mordell-Weil group of abelian varieties over functi... more In [16], we proved a structure theorem on the Mordell-Weil group of abelian varieties over function fields that arise as twists of abelian varieties by cyclic covers of quasi projective varieties in terms of Prym varieties associated to covers. In this paper, we are going to provide an explicit way to construct abelian varieties with arbitrary large rank over function fields. This will be done by applying the above mentioned theorem to the twists of Albanese variety of cyclic multiple plane. Subjclass 2010: Primary 11G10; Secondary 14H40 keywords: Mordell-Weil rank, Albanese and Prym varieties, cyclic multiple plane, twists, function fields.
In this paper, we are going to calculate the determinant of a certain type of square matrices, wh... more 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.
A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with ratio... more A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r − s, where θ ∈ (0, π), cos(θ) = s/r, and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara [4] that N is a θ-congruent number if and only if the elliptic curve E N : 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 6= 1, 2, 3, 6 is a θ-congruent number if and only if rank of E N (Q) is greater than zero. In this paper, we answer positively to a question concerning with 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 this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We ... more 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.
Bulletin of the Brazilian Mathematical Society, New Series, 2020
A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with ratio... more A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r 2 − s 2 , where θ ∈ (0, π), cos(θ) = s/r , and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235-241, 1997) that N is a θ-congruent number if and only if the elliptic curve E θ N : 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 = 1, 2, 3, 6 is a θ-congruent number if and only if rank of E θ N (Q) is greater than zero. In this paper, we answer positively to a question concerning with 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 θ N (Q). Then, based on the addition of two distinct points in E θ N (Q), we provide a way to find new rational θ-triangles for the θ-congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara's Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in E θ N (Q) with corresponding rational θ-triangles.
In [2003], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of... more In [2003], 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 by Beshaj, Shaska, and Shor 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 Yamauchi's formula works only under certain conditions.
Let K = Q(√ m) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose tha... more Let K = Q(√ m) be a real quadratic number field, where m > 1 is a squarefree integer. Suppose that 0 < θ < π has rational cosine, say cos(θ) = s/r with 0 < |s| < r and gcd(r, s) = 1. A positive integer n is called a (K, θ)-congruent number if there is a triangle, called the (K, θ, n)-triangles, with sides in K having θ as an angle and nα θ as area, where α θ = √ r 2 − s 2. Consider the (K, θ)-congruent number elliptic curve E n,θ : y 2 = x(x + (r + s)n)(x − (r − s)n) defined over K. Denote the squarefree part of positive integer t by sqf(t). In this work, it is proved that if m = sqf(2r(r − s)) and mn = 2, 3, 6, then n is a (K, θ)-congruent number if and only if the Mordell-Weil group E n,θ (K) has positive rank, and all of the (K, θ, n)-triangles are classified in four types.
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ <... more Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2)n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For fixed θ, this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2)x. We study two special cases: θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases. 2010 AMS Mathematics subject classification. Primary 11G05. Keywords and phrases. θ-congruent number, elliptic curve, Mordell-Weil rank.
In this article, we describe a method for finding congruent number elliptic curves with high rank... more In this article, we describe a method for finding congruent number elliptic curves with high ranks. The method involves an algorithm based on the Monsky's formula for computing 2-Selmer rank of congruent number elliptic curves, and Mestre-Nagao's sum which is used in ...
We develop the theory of local and global weighted heights a-la Weil for weighted projective spac... more We develop the theory of local and global weighted heights a-la Weil for weighted projective spaces P n w,k via Cartier divisors by extending the definition of weighted heights for weighted projective varieties and their closed subvarieties, and weighted log pairs. We state 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. Furthermore, we introduce generalized weighted greatest common divisors and express them as heights of the weighted projective spaces blownup at a point, relative to an exceptional divisor. We show that a point x ∈ P n w,k is smooth if and only if its generalized logarithmic weighted greatest common divisor log h wgcd x > 0. In the last part we prove that assuming Vojta's conjecture for weighted projective varieties one can bound the log h wgcd for any subvariety of codimension ≥ 2 and a finite set of places S. An analogue result is proved for weighted homogenous polynomials with integer coefficients.
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ ... more Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2 )x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases.
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Papers by Sajad Salami