Modular forms of small weight and their applications

Abstract

In number theory, as well as many areas in mathematics, modularforms (or in general, automorphic forms) are powerful tools which havemany applications. In this thesis, the author focuses on modular formsof small weight and their applications. The author is particularly inter-ested in weight 3=2 and 2. In fact, cusp forms of weight 3=2 and cuspforms of weight 2 are closely linked together by the well-known Shimuracorrespondences. In general, many properties of half-integral weight cuspforms were exploited from the properties of integral weight cusp formsthrough the Shimura correspondence. Chapter 1 is an introduction wheredenitions and preliminaries are provided. Chapter 2 deals with dierentaltered weight 2 Eisenstein series for the full modular group. Chapter 3deals with a weight 3=2 cusp forms which has an application of provingthe halting of an algorithm which computes a supersingular elliptic curvewith a given endormorphism ring.In Chapter 2, the author reviews a technique of Hecke and give ananalytic continuation of an altered Eisenstein seriesG(z; s) :=Xm;n0 1(mz + n)2jmz + njson the complex s-plane with <(s) > 􀀀1. Then, the author considers aholomorphic seriesG(z; s) :=Xm;n0 1(mz + n)2+sand let s approach 0 along the real line. The author is interested inwhether the holomorphic property for z or the modularity of it is lost.After that, a multiplier system introduced by Petersson is briey in-troduced. The author reviews that the modularity of G(z; s) attachedwith Petersson's MS is obtained and it is expected to have 0 when s ap-proaches 0. A comparison of the modularity and holomorphicity of thethree altered series when s = 0 is made at the end of Chapter 2.In Chapter 3, the author rst gives a background of problems concern-ing the endomorphism ring of an elliptic curve and describe the algorithmby Chevyrev and Galbraith which provides applications in algorithmictheory of elliptic curves over nite elds. Then, the author gives a precisestatement of a conjecture by Chevyrev and Galbraith which ensures thehalting of their algorithm. After that, a detailed proof of the conjecture isgiven. An equivalence among isomorphicity of a pair of maximal orders,agreement of a pair of theta series associated with the pair of maximalorders, and the global equivalency of their associated quadratic formsare given. At the end, the author describes the second conjecture ofChevyrev and Galbraith which are used to nd the running time of theiralgorithm and give some suggestions on further research in this topic.