Number Theory Seminar at CNU

• Abstract: For the study of (smooth) complex algebraic curves C – i.e., putting it more analytically, of compact Riemann surfaces – one defines the notion of divisors and linear series on C. We can use this to embed C in some complex projective space P which enables us to study C in an extrinsic way, by the geometry of P (e.g., looking at special complex surfaces in P containing C, or at the intersection of quadrics in P containing C). C is equipped with two important invariants: its genus g and its gonality k. Viewing C as a compact Riemann surface g is the number of handles of this compact oriented real surface, and k is a “measure of irrationality” of C, namely the smallest number of sheets for coverings of C over the Riemann sphere ( = the compact Riemann surface of genus 0). In the late 19th century G. Castelnuovo discovered that the genus of C is bounded by numbers occuring if we embed C into P. If this “Castelnuovo bound” for g is achieved C is called an extremal curve in P, and (leaving aside some peculiar cases) we call C a Castelnuovo curve if it is an extremal curve in P for some projective space P. It is well-known that a Castelnuovo curve is contained in a very special complex surface: a Hirzebruch surface (= rational ruled surface). In this talk we aim at the converse, and we will see that a (smooth) curve on a Hirzebruch surface is indeed a Castelnuovo curve if its genus is not too small with respect to its gonality  

• Abstract: Euler introduced zeta values, which have been actively studied. In this talk, we introduce their generalized forms, multiple zeta values, and investigate relationships between them. We also present Zagier's conjecture and Hoffman's conjecture, which remain open questions in the classical setting. As analogues of the classical case, Thakur and Harada introduced multiple zeta values and alternating multiple zeta values in positive characteristic. As our recent result, we have determined the dimension and basis of the span of all alternating multiple zeta values over the rational function field by identifying all linear relations among them. Consequently, we have completely established Zagier-Hoffman's conjectures in positive characteristic, as formulated by Todd and Thakur, which predict the dimension and an explicit basis of the span of multiple zeta values of Thakur of fixed weight. These are joint works with Hojin Kim, Khac Nhuan Le, Tuan Ngo Dac, and Lan Huong Pham.

• Abstract: Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. In this talk, we construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical models due to Shimura. As its applications, by using such class fields, for a positive integer $n$ we find primes of the form $x^2+ny^2$ with additional conditions on $x$ and $y$.

This is a joint work with Ho Yun Jung, Ja Kyung Koo and Dong Hwa Shin.