國立高雄大學圖資館 |

Language: English

Back

L-Functionsan elementary introduction /

Record Type:	Electronic resources : Monograph/item
Title/Author:	L-Functionsby Davide Lombardo.
Reminder of title:	an elementary introduction /
Author:	Lombardo, Davide.
Published:	Cham :Springer Nature Switzerland :2025.
Description:	xiv, 261 p. :ill. (some col.), digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	L-functions.
Online resource:	https://doi.org/10.1007/978-3-031-85145-2
ISBN:	9783031851452$q(electronic bk.)

L-Functionsan elementary introduction /
Lombardo, Davide.

L-Functionsan elementary introduction /[electronic resource] :by Davide Lombardo. - Cham :Springer Nature Switzerland :2025. - xiv, 261 p. :ill. (some col.), digital ;24 cm. - UNITEXT. La matematica per il 3+2,v. 1712038-5757 ;. - UNITEXT.La matematica per il 3+2 ;v. 136..

- Part I Classical L-functions and applications -- 1. What is an L-function? -- 2. The Prime Number Theorem -- 3. Review of Algebraic Number Theory -- 4. A Primer of Representation Theory -- 5. The L-Function of a Complex Galois Representation -- 6. Dirichlet's Theorem on Arithmetic Progressions -- 7. The Chebotarev Density Theorem -- Part II Prerequisites for Tate's Thesis -- 8. The Haar Measure -- 9. Abstract Fourier Analysis -- 10. Review of Local Fields -- 11. Restricted Direct Products -- Part III Tate's Thesis -- 12. The Local Theory -- 13. The Global Theory -- 14. Hecke L-Functions -- 15. Recovering the Classical Theory -- 16. An Extended Example: the L-Function of a CM Elliptic Curve.

This book provides an accessible introduction to the theory of L-functions, emphasising their central role in number theory and their direct applications to key results. Designed to be elementary, it offers readers a clear pathway into the subject, starting from minimal background. It describes several important classes of L-functions - Riemann and Dedekind zeta functions, Dirichlet L-functions, and Hecke L-functions (for characters with finite image) - by showing how they are all special cases of the construction, due to Artin, of the L-function of a Galois representation. The analytic properties of abelian L-functions are presented in detail, including the full content of Tate's thesis, which establishes analytic continuation and functional equations via harmonic analysis. General Hecke L-functions are also discussed, using the modern perspective of idèles and adèles to connect their analytic theory with the representation-theoretic approach of Artin's L-functions. A distinguishing feature of this book is its accessibility: while largely avoiding arithmetic geometry, it provides introductions to both algebraic number theory and key aspects of representation theory. This approach ensures that the material is accessible to both beginning graduate students and advanced undergraduates. Applications play a central role throughout, highlighting how L-functions underpin significant results in number theory. The book provides complete proofs of the prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, Chebotarev's density theorem, and the analytic class number formula, demonstrating the power of the theory in solving classical problems. It serves as an ideal introduction for advanced undergraduates and beginning graduate students and can also be a useful reference for preparing a course on the subject.

ISBN: 9783031851452$q(electronic bk.)

Standard No.: 10.1007/978-3-031-85145-2doiSubjects--Topical Terms:

240097
L-functions.

LC Class. No.: QA246

Dewey Class. No.: 512.73

L-Functionsan elementary introduction /
LDR:03599nmm a2200337 a 4500 001 678113
003 DE-He213
005 20250427130146.0
006 m d
007 cr nn 008maaau
008 251021s2025 sz s 0 eng d
020 $a 9783031851452$q(electronic bk.)
020 $a 9783031851445$q(paper)
024 7 $a 10.1007/978-3-031-85145-2 $2 doi
035 $a 978-3-031-85145-2
040 $a GP $c GP
041 0 $a eng
050 4 $a QA246
072 7 $a PBH $2 bicssc
072 7 $a MAT022000 $2 bisacsh
072 7 $a PBH $2 thema
082 0 4 $a 512.73 $2 23
090 $a QA246 $b .L842 2025
100 1 $a Lombardo, Davide. $3 929423
245 1 0 $a L-Functions $h [electronic resource] : $b an elementary introduction / $c by Davide Lombardo.
260 $a Cham : $b Springer Nature Switzerland : $b Imprint: Springer, $c 2025.
300 $a xiv, 261 p. : $b ill. (some col.), digital ; $c 24 cm.
490 1 $a UNITEXT. La matematica per il 3+2, $x 2038-5757 ; $v v. 171
505 0 $a - Part I Classical L-functions and applications -- 1. What is an L-function? -- 2. The Prime Number Theorem -- 3. Review of Algebraic Number Theory -- 4. A Primer of Representation Theory -- 5. The L-Function of a Complex Galois Representation -- 6. Dirichlet's Theorem on Arithmetic Progressions -- 7. The Chebotarev Density Theorem -- Part II Prerequisites for Tate's Thesis -- 8. The Haar Measure -- 9. Abstract Fourier Analysis -- 10. Review of Local Fields -- 11. Restricted Direct Products -- Part III Tate's Thesis -- 12. The Local Theory -- 13. The Global Theory -- 14. Hecke L-Functions -- 15. Recovering the Classical Theory -- 16. An Extended Example: the L-Function of a CM Elliptic Curve.
520 $a This book provides an accessible introduction to the theory of L-functions, emphasising their central role in number theory and their direct applications to key results. Designed to be elementary, it offers readers a clear pathway into the subject, starting from minimal background. It describes several important classes of L-functions - Riemann and Dedekind zeta functions, Dirichlet L-functions, and Hecke L-functions (for characters with finite image) - by showing how they are all special cases of the construction, due to Artin, of the L-function of a Galois representation. The analytic properties of abelian L-functions are presented in detail, including the full content of Tate's thesis, which establishes analytic continuation and functional equations via harmonic analysis. General Hecke L-functions are also discussed, using the modern perspective of idèles and adèles to connect their analytic theory with the representation-theoretic approach of Artin's L-functions. A distinguishing feature of this book is its accessibility: while largely avoiding arithmetic geometry, it provides introductions to both algebraic number theory and key aspects of representation theory. This approach ensures that the material is accessible to both beginning graduate students and advanced undergraduates. Applications play a central role throughout, highlighting how L-functions underpin significant results in number theory. The book provides complete proofs of the prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, Chebotarev's density theorem, and the analytic class number formula, demonstrating the power of the theory in solving classical problems. It serves as an ideal introduction for advanced undergraduates and beginning graduate students and can also be a useful reference for preparing a course on the subject.
650 0 $a L-functions. $3 240097
650 1 4 $a Number Theory. $3 274059
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer Nature eBook
830 0 $a UNITEXT. $p La matematica per il 3+2 ; $v v. 136. $3 922584
856 4 0 $u https://doi.org/10.1007/978-3-031-85145-2
950 $a Mathematics and Statistics (SpringerNature-11649)