Serre-type conjectures and mod p Langlands correspondences

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"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The aim of this project is to investigate some mod p aspects of the Langlands program. For GL_2 over the rational numbers, Serre's conjecture, now a theorem of Khare-Wintenberger and Kisin, predicts the modularity of certain mod p Galois representations. Generalizations of Serre's conjecture have been put forth for other reductive groups by various mathematicians, including the PI. One main goal is to prove strong new results towards the weight parts of such conjectures. Another main goal is to make progress towards the formulation of a local mod p (and ultimately also p-adic) Langlands correspondence for reductive p-adic groups, in particular for the group GL_3 over the p-adic numbers. These correspondences are expected to be closely related to the weights in Serre-types conjectures, but have only been understood so far for GL_2 over the p-adic numbers.The broader context of this project is the area of number theory, one of the oldest branches of mathematics. Number theory concerns the study of whole numbers and the solvability of equations in whole numbers. Its most important practical application has been to cryptography, which is crucial for secure data transmission on the internet. Through the amazing conjectural framework of the Langlands Program, number theory interacts with the seemingly unrelated areas of group theory (study of symmetries), analysis and geometry. This has proved to be very fruitful and led to some of the recent successes such as the proof of Fermat's Last Theorem.