We construct an infinite family of elliptic curves of rank at least 4 over Q with a rational point of order 6, which is parametrized by the rational points of an elliptic curve of rank at least 1.

Nilpotence modulo powers of 3, 5 and 7 is proved for Hecke operators on the space of certain modular forms, and is applied to the arithemtic of quadratic forms and Fourier coefficients of modular forms.

In this paper, we construct explicit lifts of quintic Jacobi sums for finite fields via integer solutions of Dickson's system. Namely we give a procedure to compute quintic Jacobi sums for extended field $\mathbf{F}_{p^{s+t}}$ by using quintic Jacobi sums for $\mathbf{F}_{p^s}$ and for $\mathbf{F}_{p^t}$. We also have the multiplication formula from $\mathbf{F}_{p^s}$ to $\mathbf{F}_{p^{ns}}$ as a special case. By the quintuplication formula, we obtain the explicit factorization of the quintic period polynomials for finite fields.

We consider the problem of the compactification of the iso level sets of the Hessenberg matrices which is propounded by Ercolani et al. We determine the structure of the cohomology ring of the compact iso level set and obtain a new expression of the flag variety $G/B$.

We improve an estimate of the constant in Smale's mean value conjecture, by using the Bieberbach theorem for coefficients of univalent functions and an estimate of the hyperbolic density of a certain simply connected domain.

In this paper we present a necessary and sufficient condition for a C^2 Teichmüller mapping to be $\rho$-harmonic. By this result we show that there is no solution to the Schoen conjecture in the class of C^2 Teichmüller mappings. We also obtain two characterizations for $\pi$-harmonic Teichmüller mappings.

In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.