The title of this lecture alludes to Ribenboim’s delightful treatise on Fermat’s Last Theorem [Rib1]. Fifteen years after the publication of [Rib1], Andrew Wiles finally succeeded in solving Fermat’s 350-year-old conundrum. That same year, perhaps to console himself of Fermat’s demise, Ribenboim published a second book, this time on Catalan’s conjecture that there are no consecutive perfect pow...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mord...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function LE(s) of E does not vanish at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to E modulo a power r of a prime p. The theory of vis...

Let A be a quotient of J0(N) associated to a newform f such that the special L-value of A (at s = 1) is non-zero. We give a formula for the ratio of the special L-value to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of f with eigen...

To Takayuki Oda on his 60th birthday.

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...

Let E be an elliptic curve over Q attached to a newform f of weight 2 on 00(N ), and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical R/Z-valued “circle pairing” on E(K ) defined by Mazur and Tate [MT1] to a period integral I ( f, K ) defined in terms of f and K . The resulting conjecture can be viewed as an analogue of the classical...