Learn More
The central point in the Bloch-Kato conjectures is to establish formulas for the order of the Selmer groups attached to Galois representations in terms of the special values of their L-functions. In order to give upper bound, the main way is to construct Euler systems following Kolyvagin. Besides, lower bounds have been obtained by using congruences between(More)
In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(phi) of a two-dimensional modular Galois representation phi. We start with the p-adic Galois representation phi0 of a modular elliptic curve E and present a formula(More)
Irregular primes—37 being the first such prime—have played a great role in number theory. This article discusses Ken Ribet’s construction— for all irregular primes p—of specific abelian, unramified, degree p extensions of the number fields Q(e2πi/p). These extensions with explicit information about their Galois groups (they are Galois over Q) were predicted(More)
The following are extended notes of a lecture given by the author at the international colloquium on L-functions and Automorphic Representation held at TIFR in january 2012. This lecture reported on some joint work of Chris Skinner and the author on the link between central L-values and Selmer groups of elliptic curves. The detailed proofs of our results(More)
In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the threedimensional adjoint representation ad(f) of a twodimensional modular Galois representation f. We start with the p-adic Galois representation f0 of a modular elliptic curve E and present a formula expressing(More)
Here are the notes I am taking for Eric Urban’s ongoing course on non-vanishing results of special values of L-functions offered at Columbia University in Spring 2015 (MATH G6675: Topics in Number Theory). As the course progresses, these notes will be revised. I recommend that you visit my website from time to time for the most updated version. Due to my(More)
In this paper, we set up a strategy to prove one divisibility toward the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpolating the critical values of the symmetric square of the modular forms in(More)
  • 1