Arithmetic algebraic geometry

The research potential of the network offers tremendous possibilities for young researchers to get initiated to this active and exciting field of Arithmetic Algebraic Geometry. The interaction of various strands of mathematics. Every node will employ at least one postdoc during the whole number theory and geometry have led to a complex and far-reaching web of conjectures which describe one of the deepest structures in contemporary funding period. Furthermore, the evident success of the instructional conference "p-adic mathematics. Just as striking is the success in using the techniques of arithmetic geometry to partially resolve some of these conjectures. The methods in arithmetic algebraic geometry" (held at Levico Terme, Italy, from May 21 through June 2, 1995, organised by the "p-adic" HCM network most celebrated achievement is of course the very recent proof, by Wiles and Taylor, of the Taniyama-Shimura-Weil conjecture for the special case mentioned above) encourages us to repeat this kind of training activity, focussing on one or several directions of research of the proposed of semi stable elliptic curves, and thereby of Fermat's Last Theorem. But network. this must not detract from other outstanding successes of recent years, including progress on the Birch-Swinnerton-Dyer conjecture and on the Main Conjecture in Iwasawa theory. Building upon the current successful work of the two (already intertwined) HCM networks on "p-adic methods in arithmetic algebraic geometry" [CHRX-CT93-0403, I/1994- Xll/1996] and on "Arithmetic Geometry and Automorphic Forms" [CHRX-CT94-0440, IX/1994 - VIII/1996], the proposed TMR network will bring together all major European researchers in the field of Arithmetic Algebraic Geometry. Four major research themes will be addressed in the network: 1. p-adic cohomology theory; 2. Rigid geometry and p-adic uniformization; 3. Automorphic forms and the Langlands programme; 4. L-functions of motives and their special values.