The Bloch-Kato conjecture, relating special values of L-functions to algebraic data, is one of the most important open problems in number theory; it includes the Birch-Swinnerton-Dyer conjecture for elliptic curves as a special case. I will describe some recent breakthroughs establishing special cases of this conjecture (and related problems such as the Iwasawa
main conjecture) using the method of Euler systems.
Let N and p at least 5 be primes such that p divides N−1. In his landmark paper on the Eisenstein ideal, Mazur proved the p-part of the BSD conjecture for the p-Eisenstein quotient J(p) of J0(N) over ℚ. Using recent results and techniques of the work of Venkatesh and Sharifi on the Sharifi conjecture, we prove unconditionally a weak form of the BSD conjecture for J(p) over a quadratic field K (which can be real or imaginary). This includes results in positive analytic rank, as the analytic rank of J(p) over K can be greater than or equal to 2 for well-chosen K.
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