Conjectures in Arithmetic Algebraic Geometry

1. Front Matter

   Pages i-vii

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2. Introduction

   • Wilfred Hulsbergen

   Pages 1-4

3. The zero-dimensional case: number fields

   • Wilfred W. J. Hulsbergen

   Pages 5-19

4. The one-dimensional case: elliptic curves

   • Wilfred W. J. Hulsbergen

   Pages 21-53

5. The general formalism of L-functions, Deligne cohomology and Poincaré duality theories

   • Wilfred W. J. Hulsbergen

   Pages 55-77

6. Riemann-Roch, K-theory and motivic cohomology

   • Wilfred W. J. Hulsbergen

   Pages 79-100

7. Regulators, Deligne’s conjecture and Beilinson’s first conjecture

   • Wilfred W. J. Hulsbergen

   Pages 101-130

8. Beilinson’s second conjecture

   • Wilfred W. J. Hulsbergen

   Pages 131-136

9. Arithmetic intersections and Beilinson’s third conjecture

   • Wilfred W. J. Hulsbergen

   Pages 137-143

10. Absolute Hodge cohomology, Hodge and Tate conjectures and Abel-Jacobi maps

    • Wilfred W. J. Hulsbergen

    Pages 145-171

11. Mixed realizations, mixed motives and Hodge and Tate conjectures for singular varieties

    • Wilfred W. J. Hulsbergen

    Pages 173-186

12. Examples and Results

    • Wilfred W. J. Hulsbergen

    Pages 187-205

13. The Bloch-Kato conjecture

    • Wilfred W. J. Hulsbergen

    Pages 207-227

14. Back Matter

    Pages 229-246

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In this expository text we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued math­ ematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to introduce L­ functions, the main, motivation being the calculation of class numbers. In partic­ ular, Kummer showed that the class numbers of cyclotomic fields play a decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirichlet had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by properties of L-functions. Twentieth century number theory, class field theory and algebraic geome­ try only strengthen the nineteenth century number theorists's view. We just mention the work of E. H~cke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generalization of Dirichlet's L-functions with a generalization of class field theory to non-abelian Galois extensions of number fields in mind.

• algebra
• algebraic geometry

• KMA, Breda, The Netherlands

  Wilfred W. J. Hulsbergen

Dr. Wilfried Hulsbergen is teaching at the KMA, Breda,Niederlande.

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