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postgraduate thesis: Modular forms of small weight and their applications
Title | Modular forms of small weight and their applications |
---|---|
Authors | |
Advisors | |
Issue Date | 2017 |
Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
Citation | Fung, K. [馮競鏘]. (2017). Modular forms of small weight and their applications. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. |
Abstract | In number theory, as well as many areas in mathematics, modular
forms (or in general, automorphic forms) are powerful tools which have
many applications. In this thesis, the author focuses on modular forms
of 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 cusp
forms of weight 2 are closely linked together by the well-known Shimura
correspondences. In general, many properties of half-integral weight cusp
forms were exploited from the properties of integral weight cusp forms
through the Shimura correspondence. Chapter 1 is an introduction where
denitions and preliminaries are provided. Chapter 2 deals with dierent
altered weight 2 Eisenstein series for the full modular group. Chapter 3
deals with a weight 3=2 cusp forms which has an application of proving
the halting of an algorithm which computes a supersingular elliptic curve
with a given endormorphism ring.
In Chapter 2, the author reviews a technique of Hecke and give an
analytic continuation of an altered Eisenstein series
G(z; s) :=
X
m;n
0 1
(mz + n)2jmz + njs
on the complex s-plane with <(s) > 1. Then, the author considers a
holomorphic series
G(z; s) :=
X
m;n
0 1
(mz + n)2+s
and let s approach 0 along the real line. The author is interested in
whether 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) attached
with 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 the
three 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 algorithm
by Chevyrev and Galbraith which provides applications in algorithmic
theory of elliptic curves over nite elds. Then, the author gives a precise
statement of a conjecture by Chevyrev and Galbraith which ensures the
halting of their algorithm. After that, a detailed proof of the conjecture is
given. An equivalence among isomorphicity of a pair of maximal orders,
agreement of a pair of theta series associated with the pair of maximal
orders, and the global equivalency of their associated quadratic forms
are given. At the end, the author describes the second conjecture of
Chevyrev and Galbraith which are used to nd the running time of their
algorithm and give some suggestions on further research in this topic. |
Degree | Master of Philosophy |
Subject | Automorphic forms Forms, Modular |
Dept/Program | Mathematics |
Persistent Identifier | http://hdl.handle.net/10722/249204 |
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | Kane, BR | - |
dc.contributor.advisor | Lau, YK | - |
dc.contributor.author | Fung, King-cheong | - |
dc.contributor.author | 馮競鏘 | - |
dc.date.accessioned | 2017-11-01T09:59:47Z | - |
dc.date.available | 2017-11-01T09:59:47Z | - |
dc.date.issued | 2017 | - |
dc.identifier.citation | Fung, K. [馮競鏘]. (2017). Modular forms of small weight and their applications. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. | - |
dc.identifier.uri | http://hdl.handle.net/10722/249204 | - |
dc.description.abstract | In number theory, as well as many areas in mathematics, modular forms (or in general, automorphic forms) are powerful tools which have many applications. In this thesis, the author focuses on modular forms of 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 cusp forms of weight 2 are closely linked together by the well-known Shimura correspondences. In general, many properties of half-integral weight cusp forms were exploited from the properties of integral weight cusp forms through the Shimura correspondence. Chapter 1 is an introduction where denitions and preliminaries are provided. Chapter 2 deals with dierent altered weight 2 Eisenstein series for the full modular group. Chapter 3 deals with a weight 3=2 cusp forms which has an application of proving the halting of an algorithm which computes a supersingular elliptic curve with a given endormorphism ring. In Chapter 2, the author reviews a technique of Hecke and give an analytic continuation of an altered Eisenstein series G(z; s) := X m;n 0 1 (mz + n)2jmz + njs on the complex s-plane with <(s) > 1. Then, the author considers a holomorphic series G(z; s) := X m;n 0 1 (mz + n)2+s and let s approach 0 along the real line. The author is interested in whether 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) attached with 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 the three 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 algorithm by Chevyrev and Galbraith which provides applications in algorithmic theory of elliptic curves over nite elds. Then, the author gives a precise statement of a conjecture by Chevyrev and Galbraith which ensures the halting of their algorithm. After that, a detailed proof of the conjecture is given. An equivalence among isomorphicity of a pair of maximal orders, agreement of a pair of theta series associated with the pair of maximal orders, and the global equivalency of their associated quadratic forms are given. At the end, the author describes the second conjecture of Chevyrev and Galbraith which are used to nd the running time of their algorithm and give some suggestions on further research in this topic. | - |
dc.language | eng | - |
dc.publisher | The University of Hong Kong (Pokfulam, Hong Kong) | - |
dc.relation.ispartof | HKU Theses Online (HKUTO) | - |
dc.rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works. | - |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.subject.lcsh | Automorphic forms | - |
dc.subject.lcsh | Forms, Modular | - |
dc.title | Modular forms of small weight and their applications | - |
dc.type | PG_Thesis | - |
dc.description.thesisname | Master of Philosophy | - |
dc.description.thesislevel | Master | - |
dc.description.thesisdiscipline | Mathematics | - |
dc.description.nature | published_or_final_version | - |
dc.identifier.doi | 10.5353/th_991043962781103414 | - |
dc.date.hkucongregation | 2017 | - |
dc.identifier.mmsid | 991043962781103414 | - |