[PDF] Top 20 On the asymptotic Fermat's Last Theorem over number fields
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On the asymptotic Fermat's Last Theorem over number fields
... We will be using the following very special case of Serre’s modularity conjecture over number fields. This conjecture concerns the modularity of 2-dimensional mod p Galois representations. While it ... See full document
13
Modular symbols over number fields
... It can be shown, using Diophantine approximation, that in fact singular points lie necessarily in K (see Theorem 3.2.19 below). Thus from now on we will not make any distinction between the terms singular point ... See full document
148
On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields
... curves over have been of great value in mathematical ...in number theory, and Cremona’s index (classification of elliptic curves over ) becomes ...modularity theorem that explains ... See full document
8
Overconvergent modular symbols over number fields
... the first section, we use overconvergent analogues of the classical evaluation maps of Chapter 12.1.2 to attach a canonical ray class distribution to an overconvergent modular eigensymbol Ψ. In the case where Ψ is ... See full document
218
Irreducible binary cubics and the generalised superelliptic equation over number fields
... the number of such primes must be greater than ...set S, so we need to increase the constant C such that the number of prime ideals is strictly greater than ... See full document
24
p Tower Groups over Quadratic Imaginary Number Fields
... Golod-Shafarevich theorem above does not rule out these ...imaginary number fields with finite 3-towers and d = 2, the earliest discovered being the example of Scholz and Taussky mentioned in the ... See full document
10
Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$
... lowering over totally real fields to study Diophantine ...representations over number fields of relatively high degree, and because we are aiming for a “clean” result without any ... See full document
30
On Fermat’s equation over some quadratic imaginary number fields
... Fermat’s Last Theorem inspired mathematicians to attack the Fermat equation over number fields via elliptic curves and ...attempts over totally real quadratic fields had been ... See full document
16
Heights on elliptic curves over number fields, period lattices, and complex elliptic logarithms
... Let c be the least common multiple of all Tamagawa indices as in Section 2.1. As pointed out by an anonymous referee of [Tho10], it may be possible to obtain a larger lower bound by making use of the explicit formulas ... See full document
233
The generalized Fermat equation over totally real number fields
... 3. Fermat wrote that had a marvellous proof of this which unfortu- nately was too large to fit in the ...This theorem is called Fermat’s Last Theorem, despite the fact that it was not proved ... See full document
135
A versatile proof of Fermat‟s last theorem
... called Fermat‟s Last Theorem (FLT) , but is true when irrational and complex numbers are allowed for at least one or two ...the Fermat‟s assertion, many people [1,2,3] attempted ... See full document
6
Why are there so many solutions to the two-dimensional Ising model?
... mathematical problem, What is behind the Dedekind triplet?, or a physical problem, What are the symmetries of the Ising lattice that allow for the various modes of solution? The appearance of elliptic curves in Ising is ... See full document
8
Early Proofs of Fermat‟s Little Theorem and Applications
... The elements of S represent the nonzero numbers modulo p. The key point is that the elements of a S also represent the nonzero numbers modulo p. For any b≢ 0 mod p, we can solve the equation ax ≡ b mod p ... See full document
6
CONFIRMATION OF THE FERMAT LAST THEOREM BY AN ELEMENTARY SHORT PROOF
... of number theory and is not short because it takes 100 ...value theorem, the B. Bolzano (1781-1848)-K. Weierstrass (1815-1897) theorem and the ... See full document
8
II. ASYMPTOTIC SERIES AND MAIN THEOREM
... via asymptotic expansions is often very suitable for cases in which STABLE can't work proficiently, see ...the number of summation terms (n) so that the magnitude of sum of residual terms would be at most ... See full document
5
p Capitulation over Number Fields with p Class Rank Two
... G of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of G . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the ... See full document
14
Darmon points on elliptic curves over number fields of arbitrary signature
... There are several combinations of extensions K/F and curves E satisfying n + s ≤ 1 and leading to new instances of Darmon points for which we have been able to perform explicit computations. Before presenting the ... See full document
36
The symplectic ideal and a double centraliser theorem
... By Theorem 3(iii) applied with k = Q, these polynomials vanish whenever we specialise t to an even integer ≥ ...1. Theorem 3(iii) was proved in another way in [10, ...centraliser theorem for Sp n ... See full document
15
Lattice methods for finding rational points on varieties over number fields
... In each method we construct O K -lattices containing vectors that potentially repre- sent rational points on a variety V . In Chapter 7 we described the conversion of such an O K -lattice to a Z -lattice and how to use ... See full document
109
On p adic estimates of weights in Abelian codes over Galois rings
... In this chapter, we review the fundamental mathematics needed to state and prove our results. We also introduce definitions, notations, and combinatorial devices that allow us to describe and manipulate the objects that ... See full document
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