A01=Anthony W. Knapp
Algebra homomorphism
Algebraic extension
Algebraic geometry
Algebraic integer
Algebraic number
Algebraic number theory
Analytic continuation
Analytic function
Associative algebra
Author_Anthony W. Knapp
Automorphism
Big O notation
Binary quadratic form
Birch and Swinnerton-Dyer conjecture
Category=PBG
Change of variables
Characteristic polynomial
Coefficient
Complex number
Conjecture
Coprime integers
Cusp form
Dimension (vector space)
Dirichlet series
Division algebra
Eigenform
Eigenvalues and eigenvectors
Elementary symmetric polynomial
Elliptic curve
Elliptic function
Elliptic integral
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equation
Fourier analysis
Function (mathematics)
Functional equation
General linear group
Group homomorphism
Group isomorphism
Hecke operator
Holomorphic function
Ideal (ring theory)
Integer matrix
Integral domain
Inverse function theorem
Isogeny
J-invariant
Linear fractional transformation
Liouville's theorem (complex analysis)
Mathematical induction
Meromorphic function
Minimal polynomial (field theory)
Modular form
Monic polynomial
Number theory
P-adic number
Polynomial ring
Prime number
Prime number theorem
Principal axis theorem
Projective line
Projective variety
Quadratic equation
Quadratic function
Quadratic reciprocity
Riemann surface
Riemann zeta function
Simultaneous equations
Summation
Theorem
Unique factorization domain
Variable (mathematics)
Weierstrass's elliptic functions
Product details
- ISBN 9780691085593
- Weight: 567g
- Dimensions: 152 x 229mm
- Publication Date: 25 Oct 1992
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves.
The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.
Anthony W. Knapp is Professor of Mathematics at the University of New York, Stony Brook. He is the author of Representation Theory of Semisimple Groups: An Overview Based on Examples and Lie Groups, Lie Algebras, and Cohomology (both published by Princeton University Press).
