For the elliptic curve $E$ over $\mathbf{Q}(t)$ found by Kihara, with torsion group $\mathbf{Z}/4\mathbf{Z}$ and rank $\geq 5$, which is the current record for the rank of such curves, by using a suitable injective specialization, we determine exactly the rank and generators of $E(\mathbf{Q}(t))$.

In 2014, T. Komatsu and L. Szalay studied the balancing binomial coefficients. In this paper, we focus on the following Diophantine equation \begin{equation*} \binom{1}{5}+\binom{2}{5}+…+\binom{x-1}{5}=\binom{x+1}{5}+…+\binom{y}{5} \end{equation*} where $y>x>5$ are integer unknowns. We prove that the only integral solution is $(x,y)=(14,15)$. Our method is mainly based on the linear form in elliptic logarithms.

We prove the Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs. We also give a relative vanishing theorem of Reid–Fukuda type for semi-log-canonical pairs.

This is an announcement of the results in the paper~[4] which is one of the series of papers dealing with the existence of spun normal surfaces in 3-manifolds with ideal triangulations. We give short comments for proofs of the results without details.

This short note gives a new proof for the existence of the cofibrations constructed by S. Takayasu~[16], using techniques in the category of unstable modules over the mod two Steenrod algebra.