Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetr... more Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series [Formula: see text] where [Formula: see text] is the [Formula: see text]th Fourier coefficient of a Hecke eigenform [Formula: see text] of weight [Formula: see text] over the full modular group.
Several authors have considered the problem of finding rational points (xi, yi), i = 1, 2, . . . ... more Several authors have considered the problem of finding rational points (xi, yi), i = 1, 2, . . . , n on various curves f(x, y) = 0, including conics, elliptic curves and hyperelliptic curves, such that the x-coordinates xi, i = 1, 2, . . . , n are in arithmetic progression. In this paper we find infinitely many sets of three points, in parametric terms, on the unit circle x+y = 1 such that the x-coordinates of the three points are in arithmetic progression. It is an open problem whether there exist four rational points on the unit circle such that their x-coordinates are in arithmetic progression.
We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular... more We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular numbers t = t(t + 1)/2. We prove that the rank of Et over the function field Q(t) is 1, while the rank is 0 over Q(t). We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.
Integers : electronic journal of combinatorial number theory, 2021
We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorph... more We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve E t : Y 2 = X ( X 2 + ( t 2 4 - 2 ) X + 1 ) . . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E t ( Q ¯ ( t ) ) is 0 and the torsion subgroup of E t ( Q ( t ) ) is isomorphic to Z ∕ 4 Z . Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of E t ( Q ) and conclude with some results regarding integral points in arithmetic progression on Et .
Let D λ,µ be the space of linear differential operators on weighted densities from F λ to Fµ as m... more Let D λ,µ be the space of linear differential operators on weighted densities from F λ to Fµ as module over the orthosymplectic Lie superalgebra osp(3|2), where F λ , λ ∈ C is the space of tensor densities of degree λ on the supercircle S 1|3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.
In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of E... more In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of E m, p : y 2 = x 3 − m 2 x + p 2 , where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of E m, p (Q) is trivial for both the cases {m ≥ 1, m ≡ 0 (mod 3)} and {m ≥ 1, m ≡ 0 (mod 3), with gcd(m, p) = 1}. We also show that given any odd prime p and for any positive integer m with m ≡ 0 (mod 3) and m ≡ 2 (mod 32), the lower bound for the rank of E m, p (Q) is 2. Finally, we find curves of rank 9 in this family.
In this paper we show that there are infnitely many pairs of integer isosce les triangles and int... more In this paper we show that there are infnitely many pairs of integer isosce les triangles and integer parallelograms with a common iintegral) area and common perimeter. \e also show that there are infnitely many Heron trian gles and integer rhombuses with common area and common perimeter. As a corollary, we show there does not exist any Heron triangle and integer square which have a common area and common perimeter.
In 1981, Zagier conjectured that the constant term of the automorphic form $$ y^{12}|\Delta (z)|^... more In 1981, Zagier conjectured that the constant term of the automorphic form $$ y^{12}|\Delta (z)|^2$$ y 12 | Δ ( z ) | 2 , that is, $$ a_0(y):=y^{12} \sum _{n=1}^{\infty } \tau ^2(n) \exp ({-4 \pi ny}), $$ a 0 ( y ) : = y 12 ∑ n = 1 ∞ τ 2 ( n ) exp ( - 4 π n y ) , where $$\tau (n)$$ τ ( n ) is the n th Fourier coefficient of the Ramanujan cusp form $$\Delta (z)$$ Δ ( z ) , has an asymptotic expansion when $$y \rightarrow 0^{+}$$ y → 0 + and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function $$\zeta (s)$$ ζ ( s ) . This conjecture was settled by Hafner and Stopple, and later Chakraborty, Kanemitsu, and the second author have extended this result for any Hecke eigenform over the full modular group. In this paper, we investigate a Lambert series associated to the Fourier coefficients of a cusp form and the Möbius function $$\mu (n)$$ μ ( n ) . We present an exact formula for the Lambert series and interestingly the main term is in terms of the non-trivial zeros of $$\zeta (s)$$ ζ ( s ) , and the error term is expressed as an infinite series involving the generalized hypergeometric series $${}_2F_1(a,b;c;z)$$ 2 F 1 ( a , b ; c ; z ) . As an application of this exact form, we also establish an asymptotic expansion of the Lambert series.
Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetr... more Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series [Formula: see text] where [Formula: see text] is the [Formula: see text]th Fourier coefficient of a Hecke eigenform [Formula: see text] of weight [Formula: see text] over the full modular group.
Several authors have considered the problem of finding rational points (xi, yi), i = 1, 2, . . . ... more Several authors have considered the problem of finding rational points (xi, yi), i = 1, 2, . . . , n on various curves f(x, y) = 0, including conics, elliptic curves and hyperelliptic curves, such that the x-coordinates xi, i = 1, 2, . . . , n are in arithmetic progression. In this paper we find infinitely many sets of three points, in parametric terms, on the unit circle x+y = 1 such that the x-coordinates of the three points are in arithmetic progression. It is an open problem whether there exist four rational points on the unit circle such that their x-coordinates are in arithmetic progression.
We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular... more We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular numbers t = t(t + 1)/2. We prove that the rank of Et over the function field Q(t) is 1, while the rank is 0 over Q(t). We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.
Integers : electronic journal of combinatorial number theory, 2021
We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorph... more We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve E t : Y 2 = X ( X 2 + ( t 2 4 - 2 ) X + 1 ) . . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E t ( Q ¯ ( t ) ) is 0 and the torsion subgroup of E t ( Q ( t ) ) is isomorphic to Z ∕ 4 Z . Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of E t ( Q ) and conclude with some results regarding integral points in arithmetic progression on Et .
Let D λ,µ be the space of linear differential operators on weighted densities from F λ to Fµ as m... more Let D λ,µ be the space of linear differential operators on weighted densities from F λ to Fµ as module over the orthosymplectic Lie superalgebra osp(3|2), where F λ , λ ∈ C is the space of tensor densities of degree λ on the supercircle S 1|3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.
In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of E... more In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of E m, p : y 2 = x 3 − m 2 x + p 2 , where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of E m, p (Q) is trivial for both the cases {m ≥ 1, m ≡ 0 (mod 3)} and {m ≥ 1, m ≡ 0 (mod 3), with gcd(m, p) = 1}. We also show that given any odd prime p and for any positive integer m with m ≡ 0 (mod 3) and m ≡ 2 (mod 32), the lower bound for the rank of E m, p (Q) is 2. Finally, we find curves of rank 9 in this family.
In this paper we show that there are infnitely many pairs of integer isosce les triangles and int... more In this paper we show that there are infnitely many pairs of integer isosce les triangles and integer parallelograms with a common iintegral) area and common perimeter. \e also show that there are infnitely many Heron trian gles and integer rhombuses with common area and common perimeter. As a corollary, we show there does not exist any Heron triangle and integer square which have a common area and common perimeter.
In 1981, Zagier conjectured that the constant term of the automorphic form $$ y^{12}|\Delta (z)|^... more In 1981, Zagier conjectured that the constant term of the automorphic form $$ y^{12}|\Delta (z)|^2$$ y 12 | Δ ( z ) | 2 , that is, $$ a_0(y):=y^{12} \sum _{n=1}^{\infty } \tau ^2(n) \exp ({-4 \pi ny}), $$ a 0 ( y ) : = y 12 ∑ n = 1 ∞ τ 2 ( n ) exp ( - 4 π n y ) , where $$\tau (n)$$ τ ( n ) is the n th Fourier coefficient of the Ramanujan cusp form $$\Delta (z)$$ Δ ( z ) , has an asymptotic expansion when $$y \rightarrow 0^{+}$$ y → 0 + and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function $$\zeta (s)$$ ζ ( s ) . This conjecture was settled by Hafner and Stopple, and later Chakraborty, Kanemitsu, and the second author have extended this result for any Hecke eigenform over the full modular group. In this paper, we investigate a Lambert series associated to the Fourier coefficients of a cusp form and the Möbius function $$\mu (n)$$ μ ( n ) . We present an exact formula for the Lambert series and interestingly the main term is in terms of the non-trivial zeros of $$\zeta (s)$$ ζ ( s ) , and the error term is expressed as an infinite series involving the generalized hypergeometric series $${}_2F_1(a,b;c;z)$$ 2 F 1 ( a , b ; c ; z ) . As an application of this exact form, we also establish an asymptotic expansion of the Lambert series.
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