Research on the arithmetic of algebraic curves and jacobian varieties

In1994 Wiles and Taylor have settled the proof of Taniyama-Shimura conjecture for(Semistable)elliptic curves over Q.This，with its application to the proof of Fermat‘s Last Theorem，was one of the greatest achievment in this century。In our previous research，we extended the result of Wiles-Taylor proving the modularity of certain abelian varieties over Q，including Q-curves over number fields，and jacobians of QM-curves of GL(2)-type。The aim of the present research has been to provide as many as possible the concrete examples of algebraic curves over Q，for which our modularity criterion for their jacobian can be applied，as well as to investigate various arithmetic properties of such curves。Some of our main results are：·We obtained some families of genus2curves over Q whose jacobian varieties are of GL(2)-type，and checked their modularity numerically and theoretically.·Conversely，for each cusp f(Z)of weight2whose Fourier coefficients generate a quadratic field K，we tried to find an algebraic curve over Q I D4-e D4 shose jacobian variety is isogenous to the Shimura‘s abelian surface A i D2f文件D2 attached to f.We have settled this problem in all known cases for K=Q(I D 8-5 I D 8)，Q(I D 8-1 E D 8).There are 11such f.·We constructed the most general family with7free parameters，of genus2curves over Q which form a double cuver of a family of elliptic curves。Among them we found a generic family of the covering C(J)→E(J)where E(J)is the Tate‘s model of elliptic curve with j(E(J)=j.Then the simple factor of JacC(J)is shown to be a Q-curve over quadratic field Q(Ii)D8j-12i D13个D 1个D 8)。

In 1994 Wiles and Taylor have settled the proof of Taniyama-Shimura conjecture for (semistable) elliptic curves over Q. This, with its application to the proof of Fermat's Last Theorem, was one of the greatest achievment in this century. In our previous research, we extended the result of Wiles-Taylor proving the modularity of certain abelian varieties over Q, including Q-curves over number fields, and jacobians of QM-curves of GL (2) -type. The aim of the present research has been to provide as many as possible the concrete examples of algebraic curves over Q, for which our modularity criterion for their jacobian can be applied, as well as to investigate various arithmetic properties of such curves. Some of our main results are :・ We obtained some families of genus 2 curves over Q whose jacobian varieties are of GL (2) -type, and checked their modularity numerically and theoretically.・ Conversely, for each cusp f (z) of weight 2 whose Fourier coefficients generate a quadratic field K, we tried to find an algebraic curve over QィイD4-ィエD4 shose jacobian variety is isogenous to the Shimura's abelian surface AィイD2fィエD2 attached to f. We have settled this problem in all known cases for K = Q(ィイD8-5ィエD8), Q (ィイD8-1ィエD8). There are 11 such f.・ We constructed the most general family with 7 free parameters, of genus 2 curves over Q which form a double cuver of a family of elliptic curves. Among them we found a generic family of the covering C (j) → E (j) where E (j) is the Tate's model of elliptic curve with j (E (j) ) = j. Then the simple factor of JacC (j) is shown to be a Q-curve over quadratic field Q (ィイD8j-12ィイD13ィエD1ィエD8).