Arithmetic Algebraic Geometry
American Mathematical Society (Verlag)
978-0-8218-2173-2 (ISBN)
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The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.
Elliptic curves, modular forms, and applications, Joe P. Buhler: elliptic curves; points on elliptic curves; elliptic curves over $/mathbf$; modular forms of level 1; L-series - modular forms of higher level; $l$-adic representations; the rank of elliptic curves over $/mathbfQ$; applications of elliptic curves. Open questions in arithmetic algebraic geometry, Alice Silverberg: overview; torsion subgroups; ranks; conjectures of Birch and Swinnerton-Dyer; ABC and related conjectures; some other conjectures. Lectures on Serre's conjectures, Kenneth A. Ribet and William A. Stein: introduction to Serre's conjecture; optimizing the weight; optimizing the level; exercises; appendix by Brian Conrad - The Shimura construction in weight 2; appendix by Kevin Buzzard - A mod $/ell$ multiplicity one result. Deformations of Galois representations, Fernando Q. Gouvea: Galois groups and their representations; deformations of representations; the universal deformation - existence; the universal deformation - properties; explicit deformations; deformations with prescribed properties; modular deformations; $p$-adic families and infinite ferns; a criterion for existence of a universal deformation ring; an overview of a theorem of Flach; an introduction to the $p$-adic geometry of modular curves. Introduction to Iwasawa theory for elliptic curves, Ralph Greenberg: Mordell-Weil groups; Selmer groups; $/Lambda$-modules; Mazur's control theorem. Galois cohomology, John Tate: Galois cohomology. The arithmetic of modular forms, Wen-Ching Winnie Li: introduction to elliptic curves, modular forms, and Calabi-Yau varieties; the arithmetic of modular forms; connections among modular forms, elliptic curves, and representations of Galois groups. Arithmetic of certain Calabi-Yau varieties and mirror symmetry, Noriko Yui: the modularity conjecture for rigid Calabi-Yau threefolds over the field of rational numbers; arithmetic of orbifold Calabi-Yau varieties over number fields $K3$ surfaces, mirror moonshine phenomenon.
| Erscheint lt. Verlag | 10.1.2002 |
|---|---|
| Reihe/Serie | IAS/Park City Mathematics Series ; No. 9 |
| Zusatzinfo | bibliography |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 190 x 260 mm |
| Gewicht | 1218 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 0-8218-2173-3 / 0821821733 |
| ISBN-13 | 978-0-8218-2173-2 / 9780821821732 |
| Zustand | Neuware |
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