Hecke Theta Series And Eta Quotients

Modular form gathers theoretical knowledge of number theory,real analysis and complex analysis,algebraic geometry and so on.It is conceivable that it is one of the directions that mathematicians are interested in.Since Euler,Gauss,Jacobi and others propose the concept of theta function,theta function has been widely popularized.Theta function has been widely used in physics,theoretical chemistry and engineering science,and plays an increasingly important role in number theory and other branches of mathematics.In this thesis,we mainly base on the relation which is given in K?hler's book,that are weight k ? 1 mod 4(i.e.,when k is 5,9,13)and weight k ? 1 mod 6(i.e.,when k is 7,13),Hecke theta series,Eisenstein series,eta quotients relation between these three,to further explore the quadratic field,calculate the relation between the three when the weight k is higher,and explore the lacunary in the special case of weight k ? 1 mod 12.After obtaining the two identities between these three of k = 25 in the two different cases of k ? 1 mod 4 and k ? 1 mod 6,we make a series of calculations and finally get the corresponding identity.Since the left side of this identity is a linear combination of the Hecke theta series,and the Hecke theta series is lacunary,so the right side of this identity is also lacunary.On the basis of Serre's result,we know that the right hand side of the identity we get when k is equal to 25 is the optimal result.Meanwhile,by using Magma software,we verify the lack of terms on the right side of the identity.