Waldspurger Formula And Its Arithmetic Applications On Certain Modular Curves

Cube sum problem is a classical Diophatine problem in number theory.Its aim is to determine which nonzero integer n can be expressed by the sum of two cubic numbers,that is to say whether there exist two nonzero rational numbers a and b such that n=a3+b3.These equations give a certain cubic twist family of elliptic curves and their Weierstrass equations are En:y2=x3-2433n2.And then the original problem becomes to show the existence of non trivial rational points on each elliptic curve.On the other hand,the Birch-Swinnerton-Dyer conjecture predicts the algebraic rank of elliptic curves should equal the order of their Hasse-Weil L functions at central point.Moreover,there are explicit conjectural formulas connecting these arithmetic information and analytic information.The Birch-Swinnerton-Dyer conjecture is a central problem in arithmetic geometry and quite open for general elliptic curves.When the rank of elliptic curves is 0,Waldspurger formula is a powerful tool for the Birch-Swinnerton-Dyer conjecture.If we want to use it to investigate the explicit Birch-Swinnerton-Dyer conjecture,we need the explicit Waldpurger formula.In this situation,one need to figure out certain local toric integrals in the formula.At first,Gross and Prasad’s work gives how to choose test vectors in some special cases.Afterthat,many people follow their method.In[3],Cai,Shu and Tian get the most general results.In this paper,I mainly recall the explicit Waldspurger formula and how to choose the test vectors in the toric integrals in[3].I also recall its application to the modular curve X0(49)in[7],and give another computing process in choosing the test vector,using some theories from quaternion algebra.At last,I start with an elliptic curve which is associated with the cube sum problem and give some arithmetic applications of the explicit Waldspurger formula.