We introduce the primitivity of Fricke families, and give some examples. As its application, we first construct generators of the function field of the modular curve of level N in terms of Fricke functions and Siegel ...
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We introduce the primitivity of Fricke families, and give some examples. As its application, we first construct generators of the function field of the modular curve of level N in terms of Fricke functions and Siegel functions, respectively. Furthermore, we use the special values of a certain function in a totally primitive Fricke family of level N in order to generate ray class fields of imaginary quadratic fields.
The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family ...
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The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family of homotopy functions. The mixed cells are reformulated in terms of a linear inequality system with an additional combinatorial condition. An enumeration tree is constructed among a family of linear inequality systems induced from it such that every mixed cell corresponds to a unique feasible leaf node, and the depth-first search is applied to the enumeration tree for finding all the feasible leaf nodes. How to construct such an enumeration tree is crucial in computational efficiency. This paper proposes a dynamic construction of an enumeration tree, which branches each parent node into its child nodes so that the number of feasible child nodes is expected to be small;hence we can prune many subtrees which do not contain any mixed cell. Numerical results exhibit that the proposed dynamic construction of an enumeration tree works very efficiently for large scale polynomial systems;for example, it generated all mixed cells of the cyclic-15 problem for the first time in less than 16 hours.
Let L be an extended ring class field of an imaginary quadratic field K other than Q(root-1) and Q(root-3). We show that there is a form class group induced from a congruence subgroup which describes the Galois group ...
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Let L be an extended ring class field of an imaginary quadratic field K other than Q(root-1) and Q(root-3). We show that there is a form class group induced from a congruence subgroup which describes the Galois group of L over K in a concrete way. We also construct a primitive generator of L over K as a real algebraic integer which can be applied to certain quadratic Diophantine equations. (C) 2018 Elsevier Inc. All rights reserved.
In this note, we consider the special values of q-analogues of Dirichlet L-functions, namely, the values of the functions L-q(s, chi) = Sigma(infinity)(n=1) (Sigma(d vertical bar n) chi (n/d) d(s-1)) q(n) at positive ...
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In this note, we consider the special values of q-analogues of Dirichlet L-functions, namely, the values of the functions L-q(s, chi) = Sigma(infinity)(n=1) (Sigma(d vertical bar n) chi (n/d) d(s-1)) q(n) at positive integers s, where chi is a primitive Dirichlet character and q = e(2 pi i tau) is a complex number such that vertical bar q vertical bar < 1. We prove that if chi(-1) = (-1)(k) and q is algebraic, then L-q(k, chi) is transcendental. We also prove that if chi(-1) = (-1)(k) and j(tau) is algebraic, then there exists a transcendental number omega(tau) which depends only on tau and is <(Q)over bar>-linearly independent with pi such that (pi/omega(tau))(k)(L(1 - k, chi) + 2L(q)(k, chi)) is algebraic. These results can be viewed as an analogue of the classical result of Hecke on the arithmetic nature of the special values L(k, chi) for chi(-1) = (-1)(k).
This paper establishes the result that numbers of the shape 5(2)p(p = 2 mod 9) and 5(2)p(2)(p = 5 mod 9), where p a prime, are expressible as the sum of two rational cubes by finding modular parametrizations of the co...
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This paper establishes the result that numbers of the shape 5(2)p(p = 2 mod 9) and 5(2)p(2)(p = 5 mod 9), where p a prime, are expressible as the sum of two rational cubes by finding modular parametrizations of the corresponding elliptic curves, picking special complex points on the curve and manipulating them to produce rational points thereon.
In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the k-elongated plane partition function d(k)(n) by various primes. They also conjectured the existence of an infinite congruence fam...
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In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the k-elongated plane partition function d(k)(n) by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function d(7)(n). We prove that such a congruence family exists - indeed, for powers of 8. The proof utilizes only classical methods, i.e. integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for d(k)(n) which require more modern methods to prove.
We provide an exact formula for the complex exponents in the modular product expansion of the modular units in terms of the Kubert-Lang structure theory, and deduce a characterization of the modular units in terms of ...
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We provide an exact formula for the complex exponents in the modular product expansion of the modular units in terms of the Kubert-Lang structure theory, and deduce a characterization of the modular units in terms of the growth of these exponents, answering a question posed by Kohnen.
Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already...
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Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already been considered for algebraic F-2(1)-functions. This article presents Darboux evaluations for the classical case of F-3(2)-functions with the projective monodromy group PSL(2, F-7). The pullback curves are of genus 0 (in the simplest case) or of genus 1. As an application of the genus 0 evaluations, appealing modular evaluations of the same F-3(2)-functions are derived.
Usually, the Weierstrass gap theorem is derived as a straightforward corollary of the Riemann-Roch theorem. Our main objective in this article is to prove the Weierstrass gap theorem by following an alternative approa...
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Usually, the Weierstrass gap theorem is derived as a straightforward corollary of the Riemann-Roch theorem. Our main objective in this article is to prove the Weierstrass gap theorem by following an alternative approach based on "first principles", which does not use the Riemann-Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve X-0(N).
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