Tunnell’s Theorem And Some Simple Applications Of Modular Forms

This thesis is a summary of Tunnell’s work on the congruent number problem.If the positive integer n is the area of a right triangle whose side length is rational number,then n is called the congruent number.The congruent number problem refers to how to determine whether n is the congruent number given a positive integer n.This is a classic Diophantine problem,and the commonly used method to deal with this type of problem is to study the twisted family of its corresponding elliptic curve.In the congruent number problem,the Weierstrass equations corresponding to these elliptic curves is E_n:y~2=x~3-n~2x.To determine whether n is a congruent number,one only needs to know if there is a rational point of infinite order on E_n.The BSD conjecture provides the relationship between the rank of rational point groups on elliptic curves and the critical value of Hasse Weil L-functions.In addition,Coates and Wiles proved that for E_n,the BSD conjecture partial establishment.Then Tunnell used this conclusion to provide a method for determining whether n is a non congruent number.This thesis mainly introduces Tunnell’s conclusion.I first introduce the relationship between the integer weight and the semi integer weight modular form given by Shimura,Then we introduce the relationship between the Fourier coefficients of the semi integer weight modular form and the critical value of the Lfunction of the integer weight modular form under Shimura correspondence given by Waldspurger.Further introduction to Tunnell about the specific structure of the Fourier coefficient of the semi integer weight modular form corresponding to the critical value of the L-function of E_n.Finally,two simple applications of modular form are given.The first application is to give the solution of a simple number theory problem by constructing a modular form and using its Fourier coefficients.The second application is to directly construct the modular form whose Fourier coefficients are special values of L-series.