２０２５年１２月１３日
２０２５年１２月１３日
（第６０回）
（第６０回）
講演概要： Ideal Class Groups of number fields were developed in attempts to solve Fermat's Last Theorem. Though, the attempt did not succeed but this theory has a prominent place in Number theory. It has found applications in settling several important Diophantine equations (like Catala equation), study of abelian extensions of rationals, and in cryptology (ideal class group cryptosystems- due to complexity of computation of ideal class groups). This talk is about various diffculties and approaches of the study of class groups. In the early twentieth century, class field theory made it possible to visualise ideal class groups as Galois group; namely the Galois group of Hilbert Class Field over the number field) Though this approach has not simplified the computational aspect of ideal class group in any significant way, but it has enriched the overall understanding of the ideal class groups. An important question is when does the Hilbert class field is absolutely abelian? For number fields with absolutely abelian Hilbert class fields, it is easy to decide when their ring of integer is Euclidean. Also for such number fields it is possible to give good estimate on the growth of the class group itself. We should report our recent results on these. This is a joint work with Dr. Mahesh Kumar Ram and Dr. Nimish Kumar Mahapatra.
講演概要： Finding the solutions of an exponential Diophantine equation is a classical problem and even now it is a very active area of research. One of the most important equations of this type is so-called Lebesgue-Ramanujan-Nagell equation
$$x^2+d=\lambda y^n, (x,y,n\in\mathbb{N}, \gcd(x,y)=1, \lambda=1,2,4),$$
where $d$ is a fixed positive integer. In this talk, we will discuss some interesting results concerning the positive integer solutions of certain generalizations of the above equation. More precisely, we will discuss the integer solutions of the equation
$$cx^2+d^{2m+1}=2y^n, \gcd(x, y)=1,$$
where $c>1$ is a square-free integer and $d\geq 1$ is an odd integer.In the last part of the talk, we will discuss some results concerning the positive integer solutions of variants/generalization of the above equations. The last part of this talk is based on a joint work with Kalyan Chakraborty and Kotyada Srinivas.
講演概要： The Jacobi triple product identity is one of the most mysterious equation between infinite products and infinite sums. There are some interpretations of this equation and many mathematicians research its generalization according to each interpretation. In this talk, we consider one of them from the viewpoint of singular weight Jacobi forms and give a characterization of root systems. This characterization is a response to an open problem on generalized Macdonald identities posed by Borcherds.
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