Number Theory and Allied Topics

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The main thrust of this research is to establish in the setting offunction fields analogues of major conjectures by Rohrlich and,Shimura on the algebraic (in)dependence of coordinates of periods of certain abelian varieties. The current step involves the analysis of the structure of certain types of t-modules, which correspond roughly to abelian varieties. With these tools, one expects further progress on related questions. In particular, the PI and his coworkers will establish precise conditions for the independence of the periods of certain one dimensional t-modules, called Drinfeld modules. Another approach to Nesterenko's recent work on the Ramanujan functions will also be attempted by means of determining the rational solutions to acertain differential equation. This would allow the application of the PI's Lojasiewicz inequality rather than the originalfundamental, but considerably more complicated, independencecriterion due to Philippon. Moreover the PI will provide a quite general arithmetic version of the fundamental Lojasiewiczinequality which is best possible in its important respects.The present proposal centers about the properties of numbers andpolynomials. One central question is whether those numbers whicharise in various contexts in mathematics (analysis or geometry) have unkown linkages. Some of the first research of this type over a hundred years ago resolved problems which had puzzledmathematicians for two millenia. One main objective of the present research is to show that such linkages do not exist in a broad setting analogous to certain central puzzles of today. In that setting, polynomials correspond to integers and power series correspond to real numbers. The core of this work will be carried out by the PI together with M.A. Papanikolas and G.W. Anderson. A coordination of quite diverse elements is crucial to establishing these results on the boundary between the broad fields of analysis, geometry, and number theory. In the classical case, the PI will try to develop an understanding of the behaviour of certain differential equations to simplify our view of certain recent work of Nesterenko. Just as one tries to show that there are no "hidden" linkages between values, one also strives toestablish that polynomial values cannot be "unnaturally" small. The PI has one good version of such a result, and he will extend its applicability even further.