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

  title={Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)},
  author={Andrew Wiles},
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 theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey… 
Infinite Sums, Diophantine Equations and Fermat’s Last Theorem
Thanks to the results of Andrew Wiles, we know that Fermat’s last theorem is true. As a matter of fact, this result is a corollary of a major result of Wiles: every semi-stable elliptic curve over Q
Fermat's Last Theorem guarantees primes
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such
A 2020 View of Fermat's Last Theorem
The work of Wiles and Taylor–Wiles established the modularity of elliptic curves over the field of rational numbers. (In [7], I had proved that FLT would follow from this modularity.) Their new
From Pythagoras Theorem to Fermat’s Last Theorem and the Relationship between the Equation of Degree n with One Unknown
  • Yufeng Xia
  • Mathematics
    Advances in Pure Mathematics
  • 2020
The most interesting and famous problem that puzzled the mathematicians all around the world is much likely to be the Fermat’s Last Theorem. However, since the Theorem was proposed, people can’t find
Jiang And Wiles Proofs On Fermat Last Theorem(2)
D.Zagier (1984) and K.Inkeri(1990) said[7]: Jiang mathematics is true, but Jiang determinates the irrational numbers to be very difficult for prime exponent p>2. In 1991 Jiang studies the composite
Wiles’ Theorem and the Arithmetic of Elliptic Curves
Thanks to the work of Wiles [Wi], completed by Taylor-Wiles [TW] and extended by Diamond [Di], we now know that all elliptic curves over the rationals (having good or semi-stable reduction at 3 and
The Cartesian method and Fermat's Last Theorem
Fermat’s Last Theorem is proved by using the philosophical and mathematical knowledge of 1637 when the French mathematician Pierre de Fermat claimed to have a truly marvelous proof of his conjecture.
The Way to the Proof of Fermat’s Last Theorem
For this we have to browse through 350 years of mathematics facing the fact that the density of research increases exponentially. One main attraction of Fermat’s claim is that everyone can understand
On a few Diophantine equations, in particular, Fermat's last theorem
This is a survey on Diophantine equations, with the purpose being to give the flavour of some known results on the subject and to describe a few open problems. We will come across Fermat’s last
Fermat's last theorem and Catalan's conjecture in weak exponential arithmetics
It is shown that Fermat's Last Theorem for e is provable (again, under the assumption of ABC in N) in Th(N)+Exp+"coprimality for e" and Catalan's conjecture for e holds in (B,e) and (A,e).


Arithmetic moduli of elliptic curves
This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"
A result on modular forms in characteristic p
d The action of the derivation e = q ~ on the q-expansions of modular forms in characteristic p is one of the fundamental tools in the Serre/Swinnerton-Dyer theory of mod p modular forms. In this
Iwasawa theory for the symmetric square of an elliptic curve.
Up until the present time, most work in Iwasawa theory has dealt with either the cyclotomic theory or descent theory on abelian varieties. We began work on the material in this paper several years
Multiplicities of p-finite mod p Galois representations in Jo(Np)
Let M > 1 be an integer. Let Jo(M) be the Jacobian Pic~ of the modular curve Xo(M)q. Let TM be the .subring of End(Jo(M)) generated by the Hecke operators Tn with n > 1. Suppose that p is a maximal
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.
Ring-Theoretic Properties of Certain Hecke Algebras
The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method
Iwasawa Theory for $p$-adic Representations
Several years ago Mazur and Wiles proved a fundamental conjecture of Iwasawa which gives a precise link between the critical values of the Riemann zeta function (and, more generally, Dirichlet
General Selmer Groups and Critical Values of Hecke L-functions
Let K be an imaginary quadratic eld and let O be the ring of integers of K. Let E be an elliptic curve deened over Q with complex multiplication by O. Let be the Grr ossencharacter attached to the
R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic
On modular representations of $$(\bar Q/Q)$$ arising from modular forms
where G is the Galois group GaI ( I ) /Q) and F is a finite field of characteristic I > 3. Suppose that p is modular of level N, i.e., that it arises from a weight-2 newform of level dividing N and