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A First Course in Modular Forms [PDF] (Valkura)

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 							A First Course in Modular Forms
  
  Series: Graduate Texts in Mathematics, Vol. 228
  
 
Diamond, Fred, Shurman, Jerry
  
  2005, XV, 436 p. 29 illus.
  
  
  This book introduces the
theory of modular forms with an eye toward the Modularity Theorem:
  
  All rational elliptic curves arise
from modular forms.
  
  The topics covered include
  
  * elliptic curves as complex tori and as
algebraic curves,
  
  * modular curves as Riemann surfaces and as algebraic curves,
  
  * Hecke
operators and Atkin--Lehner theory,
  
  * Hecke eigenforms and their arithmetic properties,
  
  *
the Jacobians of modular curves and the Abelian varieties
  
  associated to Hecke eigenforms,
  
 
* elliptic and modular curves modulo~$p$ and the Eichler--Shimura
  
  Relation,
  
  * the Galois
representations associated to elliptic curves
  
  and to Hecke eigenforms.
  
  As it presents
these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their
applications to number theory.
  
  A First Course in Modular Forms is written for beginning graduate students
and advanced undergraduates. It does not require background in algebraic number theory or algebraic geometry, and it
contains exercises throughout.
  
  Fred Diamond received his Ph.D from Princeton University in 1988 under the
direction of Andrew Wiles and now teaches at Brandeis University. Jerry Shurman received his Ph.D from Princeton
University in 1988 under the direction of Goro Shimura and now teaches at Reed College.
  
  Content Level
» Graduate
  
  Related subjects » Algebra - Number Theory and Discrete Mathematics  						

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