Mahesh Kakde
University College, London, U.K.
August 25, 2011
On main conjectures in non-commutative Iwasawa theory: Recently the main conjecture of non-commutative Iwasawa theory for totally real fields was proven under the assumption of vanishing of certain $\mu$ invariant. The proof reduces the non-commutative main conjecture to a family commutative main conjecture (which are known due to Wiles) and certain congruences between special values of Artin $L$-functions (which are proven using the Deligne-Ribet $q$-expansion principle). More generally, one can reduce non-commutative main conjectures (for any motive) to commutative main conjectures and certain congruences between special values of $L$-functions of Artin twists of the motive. This is usually referred to as the strategy of Burns-Kato. I will present a formulation of the non-commutative main conjecture and the strategy of Burns-Kato. The construction of non-commutative $p$-adic $L$-function and the proof of non-commutative main conjecture go hand in hand in the Burns-Kato strategy. But now we know enough about $K_1$ of Iwasawa algebras to construct non-commutative $p$-adic $L$-functions by just proving certain congruences (the {\it non-commutative Kummer congruences}) between special values of $L$-functions.
University College, London, U.K.
August 25, 2011
On main conjectures in non-commutative Iwasawa theory: Recently the main conjecture of non-commutative Iwasawa theory for totally real fields was proven under the assumption of vanishing of certain $\mu$ invariant. The proof reduces the non-commutative main conjecture to a family commutative main conjecture (which are known due to Wiles) and certain congruences between special values of Artin $L$-functions (which are proven using the Deligne-Ribet $q$-expansion principle). More generally, one can reduce non-commutative main conjectures (for any motive) to commutative main conjectures and certain congruences between special values of $L$-functions of Artin twists of the motive. This is usually referred to as the strategy of Burns-Kato. I will present a formulation of the non-commutative main conjecture and the strategy of Burns-Kato. The construction of non-commutative $p$-adic $L$-function and the proof of non-commutative main conjecture go hand in hand in the Burns-Kato strategy. But now we know enough about $K_1$ of Iwasawa algebras to construct non-commutative $p$-adic $L$-functions by just proving certain congruences (the {\it non-commutative Kummer congruences}) between special values of $L$-functions.