Recent Results in the GL2 Iwasawa Theory of Elliptic Curves without Complex Multiplication

Links Between Cyclotomic and GL 2 Iwasawa Theory

We study, in the case of ordinary primes, some connections between the GL2 and cyclotomic Iwasawa theory of an elliptic curve without complex multiplication. 2000 Mathematics Subject Classification: 11G05; 11R23.

Iwasawa Theory of Elliptic Curves with Complex Multiplication

Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer

1. Quick Review of Elliptic Curves 2 2. Elliptic Curves over C 4 3. Elliptic Curves over Local Fields 6 4. Elliptic Curves over Number Fields 12 5. Elliptic Curves with Complex Multiplication 15 6. Descent 22 7. Elliptic Units 27 8. Euler Systems 37 9. Bounding Ideal Class Groups 43 10. The Theorem of Coates and Wiles 47 11. Iwasawa Theory and the “Main Conjecture” 50 12. Computing the Selmer G...

Q-curves and abelian varieties of GL2-type

A Q-curve is an elliptic curve defined over Q that is isogenous to all its Galois conjugates. The term Q-curve was first used by Gross to denote a special class of elliptic curves with complex multiplication having that property, and later generalized by Ribet to denote any elliptic curve isogenous to its conjugates. In this paper we deal only with Q-curves with no complex multiplication, the c...

Uniform Results for Serre’s Theorem for Elliptic Curves

Let E/K be an elliptic curve defined over a number field K and without complex multiplication (CM). For a rational prime , let K(E[ ]) be the th division field of E, which we know is a finite Galois extension of K. By a celebrated result of Serre [18], there exists a positive constant c(E,K), depending on E and K, such that Gal(K(E[ ])/K) GL2(Z/ Z) for all ≥ c(E,K). In [18, 19], Serre asked whe...