Pub Date : 2021-10-20DOI: 10.1142/9789811238680_0010
{"title":"Product of Two Elliptic Curves","authors":"","doi":"10.1142/9789811238680_0010","DOIUrl":"https://doi.org/10.1142/9789811238680_0010","url":null,"abstract":"","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131586498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
These are notes from a first course on elliptic curves at Leiden university in spring 2015. They are aimed at advanced batchelor/beginning master students. We do not assume any backgound in algebraic geometry. We define varieties via functors points, but only on the category of fields. This makes several things simpler, but is not ideal in all respects for example, defining morphisms of varieties as functors doesn’t give what one wants. The main result of the course is a proof of the Mordell-Weil theorem for elliptic curves over Q with rational 2-torsion, via Selmer groups. Our proof of this is fairly complete, except that at one point we have to assume more algebraic geometry to show that non-constant maps of curves are surjective (but this can just be taken as a black box). Not everything from the lectures has been typeset, in particular some examples and basic definitions are omitted. The handwritten notes on the course website are complete, but then you have to read my handwriting! Comments and corrections are very welcome, please email them to David.
{"title":"Elliptic Curves","authors":"David Holmes, Steve Alberts","doi":"10.1201/b12331-10","DOIUrl":"https://doi.org/10.1201/b12331-10","url":null,"abstract":"These are notes from a first course on elliptic curves at Leiden university in spring 2015. They are aimed at advanced batchelor/beginning master students. We do not assume any backgound in algebraic geometry. We define varieties via functors points, but only on the category of fields. This makes several things simpler, but is not ideal in all respects for example, defining morphisms of varieties as functors doesn’t give what one wants. The main result of the course is a proof of the Mordell-Weil theorem for elliptic curves over Q with rational 2-torsion, via Selmer groups. Our proof of this is fairly complete, except that at one point we have to assume more algebraic geometry to show that non-constant maps of curves are surjective (but this can just be taken as a black box). Not everything from the lectures has been typeset, in particular some examples and basic definitions are omitted. The handwritten notes on the course website are complete, but then you have to read my handwriting! Comments and corrections are very welcome, please email them to David.","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126198753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-20DOI: 10.1142/9789811238680_0002
{"title":"Preliminaries in Algebraic Geometry","authors":"","doi":"10.1142/9789811238680_0002","DOIUrl":"https://doi.org/10.1142/9789811238680_0002","url":null,"abstract":"","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":" 14","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113951501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-29DOI: 10.1142/9789811238680_0014
T. Fonseca
We survey some key developments in the theory of transcendental numbers, paying special attention to Nesterenko's theorem on values of Eisenstein series and emphasizing its underlying geometric aspects. We finish with a brief discussion on periods and related open problems.
{"title":"A Geometric Introduction to Transcendence Questions on Values of Modular Forms","authors":"T. Fonseca","doi":"10.1142/9789811238680_0014","DOIUrl":"https://doi.org/10.1142/9789811238680_0014","url":null,"abstract":"We survey some key developments in the theory of transcendental numbers, paying special attention to Nesterenko's theorem on values of Eisenstein series and emphasizing its underlying geometric aspects. We finish with a brief discussion on periods and related open problems.","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125741213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Facts about Integers Arithmetical Functions Averages of Arithmetical Functions Elementary Results on the Distribution of Primes The Prime Number Theorem Dirichlet Series Primes in Arithmetic Progression.
整数算术函数的事实算术函数的平均素数分布的初等结果素数定理等差数列中的Dirichlet级数素数。
{"title":"Analytic Number Theory for Undergraduates","authors":"H. Chan","doi":"10.1142/7252","DOIUrl":"https://doi.org/10.1142/7252","url":null,"abstract":"Facts about Integers Arithmetical Functions Averages of Arithmetical Functions Elementary Results on the Distribution of Primes The Prime Number Theorem Dirichlet Series Primes in Arithmetic Progression.","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131708639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Main Correspondence Theorem A Fundamental Region The Case > 2 The Case < 2 The Case = 2 Bochner's Generalization of the Main Correspondence Theorem of Hecke and Related Results Identities Equivalent to the Functional Equation and to the Modular Relation.
{"title":"Hecke's Theory of Modular Forms and Dirichlet Series","authors":"B. Berndt","doi":"10.1142/6438","DOIUrl":"https://doi.org/10.1142/6438","url":null,"abstract":"The Main Correspondence Theorem A Fundamental Region The Case > 2 The Case < 2 The Case = 2 Bochner's Generalization of the Main Correspondence Theorem of Hecke and Related Results Identities Equivalent to the Functional Equation and to the Modular Relation.","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127214726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable ("elementary") and complex variable ("analytic") methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed.
{"title":"Analytic Number Theory - An Introductory Course","authors":"P. Bateman, H. Diamond","doi":"10.1142/5605","DOIUrl":"https://doi.org/10.1142/5605","url":null,"abstract":"This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable (\"elementary\") and complex variable (\"analytic\") methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed.","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126354526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Modular and Automorphic Forms & Beyond","authors":"","doi":"10.1142/12325","DOIUrl":"https://doi.org/10.1142/12325","url":null,"abstract":"","PeriodicalId":339674,"journal":{"name":"Monographs in Number Theory","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124610270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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