• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1159-1186
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• 2019

In this paper, we study plane curves of degree 6 with points whose multiplicities of the tangents are 5. We determine all the Weierstrass semigroups of ramification points on double covers of the plane curves when the genera of the covering curves are greater than 29 and the ramification points are on the points with multiplicity 5 of the tangent.

• Communications of the Korean Mathematical Society
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• v.7 no.2
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• pp.271-277
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• 1992

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.433-442
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• 2010

$\tilde{G}(P,\;Q)$, a new generalization of the set of gap numbers of a pair of points, was described in [1]. Here we study gap numbers of local subring $R\;=\;\cal{K}\;+\;C$ of algebraic function field over a finite field and we give a formula for the number of elements of $\tilde{G}(P,\;Q)$ depending on pure gaps and R.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.479-485
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• 2008

In this note, we study arithmetic properties for the exponents of modular forms on ${\Gamma}_0(p)$ for primes p. Our aim is to refine the result of [4] by using the geometric property of the modular curve of ${\Gamma}_0(p)$.

• Journal for History of Mathematics
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• v.28 no.2
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• pp.85-102
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• 2015

Elliptic curves are a common theme among various fields of mathematics, such as number theory, algebraic geometry, complex analysis, cryptography, and mathematical physics. In the history of elliptic curves, we can find number theoretic problems on the one hand, and complex function theoretic ones on the other. The elliptic curve theory is a synthesis of those two indeed. As an overview of the history of elliptic curves, we survey the Diophantine equations of 3rd degree and the congruent number problem as some of number theoretic trails of elliptic curves. We discuss elliptic integrals and elliptic functions, from which we get a glimpse of idea where the name 'elliptic curve' came from. We explain how the solution of Diophantine equations of 3rd degree and elliptic functions are related. Finally we outline the BSD conjecture, one of the 7 millennium problems proposed by the Clay Math Institute, as an important problem concerning elliptic curves.