The Arithmetic Properties Of Hyperelliptic Curves

The hyperelliptic curve is a kind of algebraic curves with a covering map to P1ofdegree2, and it has some special geometric and arithmetic properties. Some of theseproperties are similar to the case of elliptic curves. But for the cause of its genus, thereis not the Mordell-Weil group structure any longer, so some properties are very diferentfrom the elliptic case. So it is important to study its arithmetic and geometric properties.In my thesis, I state some fundamental properties of hyperelliptic curves, and I em-phasis on the hyperelliptic curves with Weierstrass points. First, I use the methods similarto [Chap.4,[1]] to construct the hyperelliptic surfaces which are similar to elliptic sur-faces. I prove that if the hyperelliptic curve used as the generic fbre over a function feldhas a Weierstrass point, the except fnitely many special fbres will have a Weierstrasspoint coming from this point. Then we can generalize almost all properties of ellipticsurface having no relationship with Mordell-Weil group to the hyperelliptic case.The Szpiro’s conjecture is very important in Diophantine approximate theory, whichis equivalent to the famous ABC conjecture. I use the methods in [Sec.4,[2]] to gener-alize the Szpiro’ conjecture of hyperelliptic case admitting ABC conjecture under someparticular conditions. We use the explicit form of ABC conjecture, so we can see how theconstants in ABC conjecture control our result.