• Honam Mathematical Journal
• /
• v.20 no.1
• /
• pp.57-65
• /
• 1998

In this paper, we consider properties of Dedekind eta-function, modular discrimiant, thata-series and Weierstrass ${\wp}$-function. We prove the integrablities of ${\Delta}({\tau})$ and ${\eta}({\tau})$. Also, we give explicit formulae about ${\Delta}({\tau})$ and ${\tau}(n)$.

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

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1389-1413
• /
• 2013

In this paper, we consider several convolution sums, namely, $\mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $\mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $\mathcal{C}_k(m,n;N)$ ($k=1,2,3,{\cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${\wp}$-function, its derivative and certain linear combination of Eisenstein series is established.