Main Conjectures in the geometric case and p-adic coefficients

The well-known analogies between number fields and function fields have led to the transposition of many problemsamong which the Birch and Swinnerton-Dyer (BSD) conjecture, the equivariant Tamagawa number conjecture and theIwasawa Main conjecture. These three conjectures relate the analytic behavior of some power series on one side andarithmetic invariants on the other side of a single coefficient. Classically the coefficient is a Tate motive or an elliptic curveover a number field. The problem is studied at the base level for the first conjecture, while one look at the level of finite(respectively Zp or even bigger) Galois extensions for the second (respectively third) conjecture.While these conjectures have been intensively studied in the number field case by Iwasawa, Mazur, Coates, Wiles, Katoand many others, their analogue in the function field case are less known. Usually considered as the easier case , thecharacteristic p case has nevertheless its own difficulties. For example, we don't have resolution of singularities andfurthermore, an entirely satisfying p-adic cohomology theory with coefficients is still to be found. Another particularity isthe crucial role played by an operator called Frobenius.Following our previous work on the study of p-adic cohomology and its coefficients and on the application of these toolstoward arithmetic problems, we propose in our research plan to tackle the function field analogue of two of the mainquestions concerning the Iwasawa theory of elliptic curves. First the computation of the Euler characteristic of the Selmergroups over Zp-extensions and a BSD-type formula. In characteristic zero, such formula was obtained by Schneider andPerrin-Riou in the number field case. We also expect some results concerning noncommutative extensions. A p-adic toolcalled syntomic cohomology, which has been used in a previous paper with K. Kato, should play a crucial role in theproof.Secondly, we hope to give a proof of the Iwasawa Main Conjecture for abelian variety in the function field case byreducing this conjecture to a conjecture of Katz related to the study of zeroes on the p-adic unit disc of some analyticfunction associated to our coefficient. The proof of Crew and more recently of Burns for the analogue problem in the trivialcoefficient case and our expertise concerning Katz's conjecture make us confident about our chance to succeed.Thirdly, we hope to proceed on our quest of a good p-adic cohomology with coefficients by studying the difficult questionof ramification of p-adic coefficients. Here the recent progress of Saito and Kato for l-adic coefficients (l a prime distinctfrom the characteristic of the base field) and specially their geometric approach of the question should allow analogues forp ( p a prime equal to the characteristic of the base field). Our previous results concerning p-adic coefficients (like theirproperty to be quasi-unipotent or their stability by the 6 operations of Grothendieck) as well as the expected collaborationof T. Saito (world's expert in the l-adic aspect of this question) should be decisive to solve this problem.Finally, we hope to use the results of the two first problems (concerned with Iwasawa theory of elliptic curves in thegeometric case) to explore new cases, not occurring in the classical setting. Namely, we plan to state a very general formof the Main Iwasawa Conjecture for the so-called holonomic differential modules with Frobenius operator, using ageneralization of the syntomic cohomology and of the conjecture of Katz for such coefficients.