For N and k positive integers, let M(N, k)C denote the C-vector space of cuspidal modular forms of level N and weight k. This vector space is equipped with the usual Hecke operators Tn, n ≥ 1. If we need to consider several levels or weights at the same time, we will denote this Tn by T N n , or T N,k n . If p is a prime number dividing N , our Tp is also known under the name Up. One of our mai...

The group of rational points on an elliptic curve is one of the more fascinating number theoretic objects studied in recent times. The description of this group in terms of the special value of the L-function, or a derivative of some order, at the center of the critical strip, as enunciated by Birch and Swinnerton-Dyer is surely one of the most beautiful relationships in all of mathematics; als...

In this short survey paper we present an outline for using the Saito-Kurokawa correspondence to provide evidence for the Bloch-Kato conjecture for modular forms. Specific results will be stated, but the aim is to provide the framework for such results with an aim towards future research. 1. The Bloch-Kato conjecture for modular forms In this section we will review the Bloch-Kato conjecture for ...

In [Tei], Teitelbaum formulates a conjecture relating first derivatives of the Mazur– Swinnerton-Dyer p-adic L-functions attached to a modular forms of even weight k ≥ 2 to certain L-invariants arising from Shimura curve parametrisations. This article formulates an analogue of Teitelbaum’s conjecture in which the cyclotomic Zp extension of Q is replaced by the anticyclotomic Zp-extension of an ...

In [MT1], B. Mazur and J. Tate present a “refined conjecture of Birch and Swinnerton-Dyer type” for a modular elliptic curve E. This conjecture relates congruences for certain integral homology cycles on E(C) (the modular symbols) to the arithmetic of E over Q. In this paper we formulate an analogous conjecture for E over suitable imaginary quadratic fields, in which the role of the modular sym...

We will discuss similarities of L-functions between in the number theory and in the geometry. In particular the Hesse-Weil congruent zeta function of a smooth curve defined over a finite field will be compared to the zeta function associated to a disrete dynamical system. Moreover a geometric analog of the Birch and Swinnerton-Dyre conjecture and of the Iwasawa Main Conjecture will be discussed. 1

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...