Twisted L-Functions and Monodromy (eBook)

Nicholas M. Katz (Autor)

eBook Download: PDF

2009
264 Seiten
Princeton University Press (Verlag)
978-1-4008-2488-5 (ISBN)

Infos

Nicholas M. Katz is Professor of Mathematics at Princeton University. He is the author of four other books in this series: Arithmetic Moduli of Elliptic Curves (with Barry Mazur); Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; and Rigid Local Systems.

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For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves. The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

Produkt-Details

• Erscheint lt. Verlag: 10.1.2009
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• Schlagworte: abelian variety • absolute continuity • Addition • Affine space • Algebraically closed field • Ambient space • average • Betti number • Birch and Swinnerton-Dyer conjecture • Blowing up • Codimension • coefficient • Computation • conjecture • conjugacy class • convolution • Critical Value • differential geometry of surfaces • Dimension • Dimension (vector space) • direct sum • Divisor • Divisor (algebraic geometry) • Eigenvalues and Eigenvectors • Elliptic Curve • Equation • Equidistribution theorem • existential quantification • Factorization • finite field • finite group • Finite set • Flat map • Fourier transform • functional equation • Function field • Goursat's lemma • Ground field • Group representation • hyperplane • hypersurface • Integer • Integer matrix • Irreducible component • irreducible polynomial • irreducible representation • J-invariant • K3 surface • Lebesgue measure • Lefschetz pencil • Level of Measurement • L-Function • Lie algebra • Limit superior and limit inferior • Minimal polynomial (field theory) • modular form • Monodromy • Morphism • Numerical analysis • orthogonal group • percentage • polynomial • Prime number • probability measure • quadratic function • Quantity • Quotient space (topology) • Representation Theory • residue field • Riemann hypothesis • Root of unity • Scalar (physics) • Set (mathematics) • Sheaf (mathematics) • SUBGROUP • Summation • Symmetric group • System of imprimitivity • Theorem • Trivial representation • Zariski topology
• ISBN-10: 1-4008-2488-5 / 1400824885
• ISBN-13: 978-1-4008-2488-5 / 9781400824885