This paper is submitted to Algebraic-Number-Theory Archives for validation by Number Theorists Community. It is an update of the previous versions ANT-0155, ANT-0170, ANT-0205, ANT-0237, ANT-0321, ANT-0333, and ANT-0356, of which the first four were titled `A generalization of Eichler criterium for Fermat's Last Theorem' and the last three were titled `A classical approach on Fermat-Wiles theorem'.
This version contains a complete reorganization of the paper with a first part dealing with cyclotomic fields (independently of FLT) from page 10 to 72 and a second part dealing with FLT from page 73 to end.
This version improves our previous results on cyclotomic fields and contains several significant error corrections.
The proofs rest on classical theory of cyclotomic number fields $\Q(zeta_p)$.
Jacob's ladders, E. C. Titchmarsh's hypothesis (1934) and new $ζ$-equivalents of the Fermat-Wiles theorem or connections between Fermat's rationals and the Gram's sequence
Jacob's ladders and three new equivalents of the Fermat-Wiles theorem and an infinite set of these equivalents that are independent on the Jacob's ladders
Jacob's ladders, new classes of sum variants of almost linear formula and corresponding $ζ$-functionals and $ζ$-equivalents of the Fermat-Wiles theorem
Jacob's ladders, point of contact of the remainder in the prime-number law with the Fermat-Wiles theorem and multiplicative puzzles on some sets of integrals
Jacob's ladders, our asymptotic formulae (1981) and next $ζ$-equivalents of the Fermat-Wiles theorem together with decomposition and synthesis of the Riemann's $ζ$-oscillators
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918), the classical Dirichet's sum of divisors (1849) and their relationship with the Fermat-Wiles theorem
Jacob's ladders, new equivalent of the Fermat-Wiles theorem generated by certain cross-breed of Ingham and Heath-Brown formula (1979) and some chains of equivalents
Jacob's ladders, logarithmic modification of the Hardy-Littlewood integral (1918), Titchmarsh's $Ω$-theorem (1928) and new point of contact with the Fermat-Wiles theorem
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relation to the Titchmarsh's sums (1943) and the Fermat-Wiles theorem
Jacob's ladders, our asymptotic formulae (1981) connected with the equation $Z'(t)=0$ as a base for new $ζ$-equivalents of the Fermat-Wiles theorem and some other results
Jacob's ladders and new equivalents of the Fermat-Wiles theorem connected with some cross-bred of the formulae of Hardy-Littlewood-Ingham (1926) and of Ingham (1926)
Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg's formula (1946) and the Fermat-Wiles theorem
The absolute Galois group of the cyclotomic field $K={\mathbb Q}(ζ_p)$ acts on the étale homology of the Fermat curve $X$ of exponent $p$. We study a Galois cohomology group which is valuable for measuring an obstruction for $K$-rational points on $X$. We analyze a $2$-nilpotent extension of $K$ which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of $K$. For $p=3$, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology, which determin…
We define a "period ring-valued beta function" and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark's conjecture implies that the exponential of the derivatives at $s=0$ of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler's formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative (and partial) proof by using the reci…
We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However we use the derivation method of Eichler as a prerequisite for our purpose, a method which is also local but more effect…
We solve the diophantine equations x^4 + d y^2 = z^p for d=2 and d=3 and any prime p>349 and p>131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat's Last Theorem, and by Ellenberg in the solution of the equation x^4 + y^2 = z^p, and we use Q-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d.
The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring $R_\infty$ classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if $R_\infty$ is Noetherian and certain adjoint $μ$-inv…
In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's Equation.
Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that $\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic Fermat's Last Theorem over $\mathbb{Q}_{n,p}$ for all $n \ge 1$ and all primes $p \ge 5$ that are non-Wieferich, i.e. $2^{p-1} \not \equiv 1 \pmod{p^2}$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over $\mathbb{Q}_{n,p}$.
Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=δ$ where $δ$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid δ$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $δ$ are inert in $K/F$, the Selmer groups are explicitly determined an…
Let $K$ be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation $x^4-y^2=z^p$ over $K$, assuming some deep but standard conjectures of the Langlands programme when $K$ has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above $2$. When modularity of elliptic curves over $K$ is known, for example when $K$ is real quadratic or the $r$-layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, effective asymptoti…
In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have in fact been proved not to have any unknown solutions, these improved bounds are interesting in the context of a new effective abc inequality announced in the paper \cite{MFHMP} based on Mochizuki's \cite{Mo}[IUT-IV, Theorem A]. In this paper we provide a strengthening of the lower bound for FLT2, which is necessary in order to take advantage of the best up…
Information about the absolute Galois group $G_K$ of a number field $K$ is encoded in how it acts on the étale fundamental group $π$ of a curve $X$ defined over $K$. In the case that $K=\mathbb{Q}(ζ_n)$ is the cyclotomic field and $X$ is the Fermat curve of degree $n \geq 3$, Anderson determined the action of $G_K$ on the étale homology with coefficients in $\mathbb{Z}/n \mathbb{Z}$.The étale homology is the first quotient in the lower central series of the étale fundamental group.In this paper, we determine the structure of the graded Lie algebra for $π$. As a consequence, this determines the…
Let $A\to B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond's Theorem 2.1 from his 1997 paper "The Taylor-Wiles construction and multiplicity one". We also prove that if there is a nonzero $A$-flat $B$-module, then $A\to B$ is flat and is a relative complete intersection (i.e. $B/\mathfrak{m}_AB$ is a complete intersection). Then we explain how this result allows to simplify Wiles's proof of Fermat's Last Theorem: we do not need the so-called…
This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v, w_1\in\Z\backslash\{0\}$, $u,v$ coprime and $v\equiv 0 \mod p$ has no solution.} Let $ζ$ be a $p$th primitive root of unity and $K:=\Q(ζ)$. A prime $q$ is said {\it $p$-principal} if the class of any prime ideal $\mathfrak q_K$ of $K$ over $q$ is a $p$-power of a class. Assume that $SFLT2$ fails for $(p,u,v)$. Let $q$ be any odd prime coprime with $puv$, $f$ the …
Recently the second author has associated a finite $\F_q[T]$-module $H$ to the Carlitz module over a finite extension of $\F_q(T)$. This module is an analogue of the ideal class group of a number field. In this paper we study the Galois module structure of this module $H$ for `cyclotomic' extensions of $\F_q(T)$. We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular un…
This article deals with a conjecture generalizing the second case of Fermat's Last Theorem, called $SFLT2$ conjecture: {\it Let $p>3$ be a prime, $K:=\Q(ζ)$ the $p$th cyclotomic field and $\Z_K$ its ring of integers. The diophantine equation $(u+vζ)\Z_K=\mk w_1^p$, with $u,v\in\Z\backslash\{0\}$ coprime, $uv\equiv 0 \bmod p$ and $\mk w_1$ ideal of $\Z_K$, has no solution.} Assuming that $SFLT2$ fails for $(p,u,v)$, let $q$ be an odd prime not dividing $uv$, $n$ the order of $\frac{v}{u}\bmod q$, $ξ$ a primitive $n$th root of unity and $M:=\Q(ξ,ζ)$. The aim of this complement of the article [GQ…
From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, t…
We prove a slight generalization of Iwasawa's `Riemann-Hurwitz' formula for number fields and use it to generalize Ferrero's and Kida's well-known computations of Iwasawa λ-invariants for the cyclotomic Z_2-extensions of imaginary quadratic number fields. In particular, we show that if p is a Fermat prime, then similar computations of Iwasawa λ-invariants hold for certain imaginary quadratic extensions of the unique subfield k of Q(ζ_{p^2}) such that [k:Q] = p. In fact, we actually prove more by explicitly computing cohomology groups of principal ideals. The computation of lambda invariants ob…
This article, complement to the article [Que], deals with some generalizations of Futwängler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $ζ$ a $p$th primitive root of unity, $K:=\Q(ζ)$ and $C\ell_K$ the class group of $K$. A prime $q$ is said $p$-principal if the class $c\ell_K (\mk q_K)\in C\ell_K$ of any prime ideal $\mk q_K$ of $\Z_K$ over $q$ is the $p$th power of a class. Assume that FLT2 fails for $(p,x,y,z)$ where $x, y, z$ are mutually coprime integers, $p$ divides $y$ and $x^p+y^p+z^p=0$. Let $q$ be a prime dividing $\frac{(x^p+y^p)(y^p+z^p…
Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and th…
Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, wher…
