DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 김완수 | - |
dc.contributor.author | Daniel, Juan | - |
dc.contributor.author | 대니엘 후안 | - |
dc.date.accessioned | 2024-07-25T19:31:12Z | - |
dc.date.available | 2024-07-25T19:31:12Z | - |
dc.date.issued | 2023 | - |
dc.identifier.uri | http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=1045891&flag=dissertation | en_US |
dc.identifier.uri | http://hdl.handle.net/10203/320662 | - |
dc.description | 학위논문(석사) - 한국과학기술원 : 수리과학과, 2023.8,[iii, 23 p. :] | - |
dc.description.abstract | We review Swinnerton-Dyer's paper [11], which discusses congruence relations on modular forms with integral coefficients. Using Serre-Deligne Theorem, we can associate each modular form $f$ with $\rho_l: \mathrm{Gal}(K_l/\mathbb{Q}) \rightarrow \mathrm{GL}_2(\mathbb{Z}_l)$ with the property $\mathrm{Tr}(\mathrm{Frob}(p))=a_p(f)$. We define $l$ to be an exceptional prime if the image of $\rho_l$ does not contain $\mathrm{SL}_2(\mathbb{Z}_l)$, which for $l \geq 5$ is equivalent to the image of $\rho_l$ under $\bmod$ $l$ reduction, $\overline{\rho_l}$, not containing $\mathrm{SL}_2(\mathbb{F}_l)$. We determine all possible images of $\overline{\rho_l}$ in $\mathrm{GL}_2(\mathbb{F}_l)$ and obtain three types of congruence relations that may arise. Then, we develop the theory of modular forms $\bmod$ $l$. From this setting, we obtain a necessary condition for $l$ to be an exceptional prime of type 1 and 2. This enables us to obtain a complete list of exceptional primes of type 1 and 2 for modular forms with integral coefficients. We also eliminate all but one candidate prime, 59, which may be an exceptional prime of type 3 for the unique cusp form of weight 16. All the contents of this thesis can easily be derived from the original paper of Swinnerton-Dyer's [11], and we do not claim any originality in either mathematical or expositional aspects. | - |
dc.language | eng | - |
dc.publisher | 한국과학기술원 | - |
dc.subject | 모듈러 형식▼a세르-들리뉴 정리▼a$\mathrm{Frob}(p)$▼a예외적인 소수▼a법$l$ 갈루아 표현의 상▼a법$l$ 모듈러 형식 $\bmod$ $l$ | - |
dc.subject | Modular forms▼aSerre-Deligne theorem▼a$\mathrm{Frob}(p)$▼aExceptional primes▼aImage of $\overline{\rho_l}$▼aModular forms $\bmod$ $l$ | - |
dc.title | On $l$-adic representations and congruences of modular forms: following Swinnerton-Dyer | - |
dc.title.alternative | $l$진 갈루아 표현과 모듈러 형식의 합동성: 스위너튼다이어 결과의 서베이 | - |
dc.type | Thesis(Master) | - |
dc.identifier.CNRN | 325007 | - |
dc.description.department | 한국과학기술원 :수리과학과, | - |
dc.contributor.alternativeauthor | Kim, Wansu | - |
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