# 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={Dorian 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…

## 12 Citations

### Lecture on the abc Conjecture and Some of Its Consequences

- Mathematics
- 2015

The abc conjecture was proposed in the 1980s by J. Oesterle and D.W. Masser. This simple statement implies a number of results and conjectures in number theory. We state this conjecture and list a…

### ABC THEOREMS IN THE FUNCTIONAL CASE

- Mathematics
- 2014

In this dissertation we will prove some ABC Theorems, namely for relatively prime by pairs p-adic entire functions in one variable, for p-adic meromorphic functions in several variables without…

### On the Height Conjecture for Algebraic Points on Curves Defined over Number Fields

- Mathematics
- 2005

We study the basic height conjecture for points on curves defined over number fields and show: On any algebraic curve defined over a number field the set of algebraic points contains an unrestricted…

### Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field

- Mathematics
- 2010

We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a…

### Deitmar , Anton and Diamantis , Nikolaos ( 2009 ) Automorphic forms of higher order

- Mathematics
- 2016

In this paper a theory of Hecke operators for higher order modular forms is established. The definition of higher order forms is extended beyond the realm of parabolic invariants. A canonical inner…

### Torsion homology growth and cycle complexity of arithmetic manifolds

- MathematicsDuke Mathematical Journal
- 2016

Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold.
We conjecture that there is a basis for the second homology of M, where each basis
element is represented by a surface of…

### ABC Conjecture–An Ambiguous Formulation

- Mathematics
- 2017

Due to exist forevermore uncorrelated limits of values of real number e≥0, enable ABC conjecture to be able to be both proved and negated. In this article, we find a representative equality…

### On The Torsion Homology of Non-Arithmetic Hyperbolic Tetrahedral Groups

- Mathematics
- 2010

Numerical data concerning the growth of torsion in the first homology of non-arithmetic hyperbolic tetrahedral groups are collected. The data provide support the speculations of Bergeron and…

### Higher order invariants in the case of compact quotients

- Mathematics
- 2010

We present the theory of higher order invariants and higher order automorphic forms in the simplest case, that of a compact quotient. In this case, many things simplify and we are thus able to prove…

### The ABC Conjecture Does Not Hold Water

- Mathematics
- 2016

In this article, the author gave a specific example to negate the ABC conjecture once and for all.

## References

SHOWING 1-10 OF 47 REFERENCES

### Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)

- Mathematics
- 1995

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. This…

### The work of kolyvagin on the arithmetic of elliptic curves

- Mathematics
- 1989

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 modular…

### Bounds for the order of the Tate-Shafarevich group

- Mathematics
- 1995

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 of…

### Effective bounds on the size of the Tate-Shafarevich group

- Mathematics
- 1996

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 denotes…

### On the Passage From Local to Global in Number Theory

- Mathematics
- 1993

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”,…

### Lectures on Elliptic Curves, London Mathematical Society Student Texts 24

- MathematicsThe Mathematical Gazette
- 1995

Lectures on elliptic curves, London Mathematical Society Student Texts 24, by J. W. S. Cassels. Pp 137. £13-95 (paper) £27-95 (hard). 1991. ISBN 0-521-42530-1, -41517-9. (Cambridge University Press)…

### The analytic order of III for modular elliptic curves

- Mathematics
- 1993

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 strong…

### A family of semistable elliptic curves with large Tate-Shafarevitch groups

- Mathematics
- 1983

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-Shafarevitch…

### Properties of Eisenstein series formed with modular symbols

- Mathematics
- 2000

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 consider…