Modular Forms and Special Cycles on Shimura Curves (eBook)

Stephen S. Kudla, Michael Rapoport, Tonghai Yang (Autoren)

eBook Download: PDF

2006
392 Seiten
Princeton University Press (Verlag)
978-1-4008-3716-8 (ISBN)

Infos

Stephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universität, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.

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Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "e;M"e; attached to a Shimura curve "e;M"e; over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soule arithmetic Chow groups of "e;M"e;. The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "e;M"e;. In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Produkt-Details

• Erscheint lt. Verlag: 4.4.2006
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 1 line illus. 3 tables.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• Schlagworte: abelian group • Addition • Adjunction formula • algebraic number theory • Arakelov Theory • arithmetic • automorphism • bijection • Borel subgroup • Calculation • Chow group • coefficient • cohomology • combinatorics • Compact Riemann surface • complex multiplication • complex number • Cup product • deformation theory • Derivative • Dimension • Disjoint union • Divisor • Dual pair • Eigenfunction • Eigenvalues and Eigenvectors • Eisenstein series • Elliptic Curve • Endomorphism • Equation • Explicit formulae (L-function) • Fields Institute • Formal group • Fourier series • Fundamental matrix (linear differential equation) • Galois group • Generating function • Green's Function • group action • induced representation • intersection number • Intersection (set theory) • Irreducible component • isomorphism class • Laurent Series • Level structure • L-Function • line bundle • Local ring • Mathematical Sciences • Mathematics • Metaplectic Group • modular curve • modular form • Modularity (networks) • moduli space • multiple Integral • Number Theory • Numerical Integration • orbifold • Orthogonal complement • P-adic number • Pairing • prime factor • Prime number • pullback • Pullback (category theory) • Pullback (differential geometry) • quadratic form • Quadratic residue • Quantity • quaternion • Quaternion Algebra • Quotient stack • Rational number • real number • residue field • Riemann zeta function • Ring of integers • scientific notation • Shimura variety • Siegel Eisenstein series • Siegel modular form • SL2(R) • Special case • Standard L-function • SUBGROUP • Subset • Summation • tensor product • Test vector • Theorem • Three-dimensional space (mathematics) • Topology • Trace (linear algebra) • Triangular Matrix • two-dimensional space • uniformization • Valuative criterion • Whittaker Function
• ISBN-10: 1-4008-3716-2 / 1400837162
• ISBN-13: 978-1-4008-3716-8 / 9781400837168