A Panorama in Number Theory or The View from Baker's Garden: Modular Forms, Elliptic Curves and the ABC -Conjecture

@inproceedings{Goldfeld2002API,
  title={A Panorama in Number Theory or The View from Baker's Garden: Modular Forms, Elliptic Curves and the ABC -Conjecture},
  author={D. Goldfeld},
  year={2002}
}
The ABC–conjecture was first formulated by David Masser and Joseph Osterlé (see [Ost]) in 1985. Curiously, although this conjecture could have been formulated in the last century, its discovery was based on modern research in the theory of function fields and elliptic curves, which suggests that it is a statement about ramification in arithmetic algebraic geometry. The ABC–conjecture seems connected with many diverse and well known problems in number theory and always seems to lie on the… Expand
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References

SHOWING 1-10 OF 63 REFERENCES
Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)
When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. ThisExpand
Wieferich's criterion and the abc-conjecture
Abstract We show that the abc-conjecture of Masser and Oesterle implies that there are infinitely many primes for which 2p−1 n= 1 (mod p2). More precisely, we show that there are at least O(log X)Expand
The work of kolyvagin on the arithmetic of elliptic curves
The main reference is the paper “The work of Kolyvagin on the arithmetic of elliptic curves”(1989) by Karl Rubin. Let E be an elliptic curve defined over Q with conductor N , and fix a modularExpand
Bounds for the order of the Tate-Shafarevich group
In this paper we show, over Q, with classical conjectures like Birch and Swinnerton-Dyer, the equivalence for modular elliptic curves between the following two conjectures: relating the order 1 ID ofExpand
Effective bounds on the size of the Tate-Shafarevich group
Throughout this paper we will work over Q, although our methods generalize quite naturally to any number field. Consider an elliptic curve E defined over Q. We use the following notation: XE denotesExpand
The abc Conjecture
Let’s start with a theorem about polynomials. You probably think that one knows everything about polynomials. Most mathematicians would think that, including myself. It came as a surprise when R. C.Expand
On the Passage From Local to Global in Number Theory
Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local” versus “global”,Expand
The analytic order of III for modular elliptic curves
In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class; in [4] we considered the strongExpand
A family of semistable elliptic curves with large Tate-Shafarevitch groups
We present a family of elliptic curves defined over the rationals Q such that each curve admits only good or multiplicative reduction and for every integer n there is a curve whose Tate-ShafarevitchExpand
Properties of Eisenstein series formed with modular symbols
In this work a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied. For an arbitrary Fuchsian group of the first kind we fix a holomorphic cusp form and considerExpand
...
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3
4
5
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