Ekin Ozman, Samir Siksek
Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic Fermat conjecture.
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We prove under some assumptions that the Tate conjecture holds for products of Fermat varieties of different degrees.
Fermat-Catalan and Tijdeman-Zagier conjectures for products
本文将 Fermat-Catalan 猜想、Tijdeman-Zagier 猜想与 Fermat 大定理推广到整数乘积情形，并给出 abc 猜想的新显式表述。
On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.
Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}.$ In 1956, Jeśmanowicz conjectured that for any positive integer $n$, the only solution of $(an)^{x}+(bn)^{y}=(cn)^{z}$ in positive integers is $(x,y,z)=(2,2,2)$. Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be a Fermat number. In this paper, we show that Jeśmanowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.
Fermat quotients and the Ankeny-Artin-Chowla conjecture
本文为 Ankeny-Artin-Chowla 关于实二次域基本单位数的结论提供了精简证明，推广了该猜想，并据此将模 p 的二次非剩余的费马商与调和数之和联系起来。
In the present article, we extend previous results of the author and we show that when $K$ is any quadratic imaginary field of class number one, Fermat's equation $a^p+b^p+c^p=0$ does not have integral coprime solutions $a,b,c \in K \setminus \{ 0 \}$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.
Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also for the non-existence of solutions to the unit equation over F. For example, if 2 totally ramifies and 3 splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.
Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.
Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ${\bf Q}$ given by equations $ax^p+by^p+cz^p=0$, with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent $p$ which are counterexamples to the Hasse principle. It is a consequence of the abc-conjecture if $p\geq 5$. Using a cyclotomic approach due to H. Cohen and Chebotarev's density theorem, we obtain a partial result towards this conjecture, by proving it for $p\leq 19$.
Arithmetic dynamics and Generalized Fermat's conjecture
本文在算术动力学框架下提出广义费马猜想，并提供了相关证据及多索引版本的扩展。
In this article we study the three-variable unit equation $x + y + z = 1$ to be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is the $S$-unit group of some global function field. We give upper bounds for the height of solutions and the number of solutions. We also apply these techniques to study the Fermat surface $x^N + y^N + z^N = 1$.
Let $F$ be a number field and $\mathcal{O}_F$ its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the non-existence of solutions to the unit equation $λ+ μ= 1$, $λ, μ\in \mathcal{O}_F^\times$. This is then used to strengthen a criterion for the asymptotic Fermat's Last Theorem due to Freitas and Siksek.
T-Fermat integers
引入T-费马整数概念，推广素数定义；证明合数T-费马整数必为卡迈克尔数，并猜想存在无穷多个合数T-费马整数，为证明ω(p-1)≤2的素数无穷提供可能路径。
We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.
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…
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…
In a recent work the authors prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(ζ_{2^{r+2}})^+$ where $r \ge 0$. A crucial step in their proof is the following conjecture of Kraus. Let $K$ be a number field having odd narrow class number and a unique prime $λ$ above $2$. Then there are no elliptic curves defined over $K$ with conductor $λ$ and a $K$-rational point of order $2$. In this note we give a new elementary proof of Kraus' conjecture that makes use only of basic facts about elliptic curves, Tate curves and Tate modules.
Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show that the solutions of Fermat's curve X^N + y^N = 1 of degree N over K consist of only either 0 or roots of unity for almost positive integers N. More precisely, the density of such N is 1.
We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same.
Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $μ$ is a zero of $X_n$ if and only if the triple $(μ,μ+m,μ+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the irreducibility of these polynomials implies Fermat's last theorem. It is also shown, in a precise asymptotic sense, that for a vast majority of cases, these polynomials are irreducible via Eisenstein's criterion. We conclude by offering a conjecture on powerful numbers.
We consider Thue equations of the form $ax^k+by^k = 1$, and assuming the truth of the $abc$-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $ax^k+by^k+cz^k = 0$ of degree at least six.
Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over $\mathbb Q(i)$. Under the same assumption, we also prove that, for all prime exponents $p \geq 5$, Fermat's equation $a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb Q(\sqrt{-2})$ and $\mathbb Q(\sqrt{-7})$.
Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem holds for many totally real fields. Later this result was extended by Deconinck to generalized Fermat equations of the form $Ax^p +By^p +Cz^p = 0$, where A;B;C are odd integers belonging to a totally real field. Another extension was given by Sengun and Siksek. They showed that the Fermat equation holds asymptotically for imaginary quadratic number fields satisfying usual conjectures about modularity. In this work, combining their techniques we extend their results about the generalized Fermat equatio…
We give necessary and sufficient conditions on a squarefree integer $d$ for there to be non-trivial solutions to $x^{3} + y^{3} = z^{3}$ in $\Q(\sqrt{d})$, conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.
Local solubility of generalised Fermat equations
本文确定了绝对值有界的整数三元组使得广义费马方程处处局部可解的渐近公式，计算了主常数并验证了相关猜想。
We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.
We consider some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of "unlikely intersections." We prove a special case of the Zilber-Pink conjecture for curves.
Fermat's Last Theorem for Selmer sections
本文证明了对于指数ℓ≥7的仿射费马曲线，所有定义在有理数上的塞尔默截面都是尖点的。
Chebyshev's bias for Fermat curves of prime degree
本文证明素数次 Fermat 曲线及其部分商的素数竞赛渐近公式与 DRH 相关部分等价，并据此在 DRH 下计算其二阶矩 L 函数在 s=1 处的零点阶数。
In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the Legendre family. This result provides an interesting explanation to the modularities of counting functions for K3 surfaces from the mirror symmetry point of view. We will also discuss the relations between the arithmetic geometry of smooth fibers of the Fermat pencil (Dwork family) and that of the smooth fibers of the Legendre family, e.g. Shioda-Inose structure…
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 …
Revisiting the Fermat-type equation $x^{13} + y^{13} = 3z^7$
本文结合多种数论工具，完整求解了互素条件下的费马型方程 $x^{13}+y^{13}=3z^7$，证明了其无解性。
ABC Conjecture: $ABC = 2^m p^n q^r$ with Fermat or Mersenne Primes
研究了ABC猜想中关于费马和梅森素数的三元组解的有限性及具体构造。
Let $K$ be a totally real field. By the asymptotic Fermat's Last Theorem over $K$ we mean the statement that there is a constant $B_K$ such that for prime exponents $p>B_K$ the only solutions to the Fermat equation $a^p + b^p + c^p = 0$ with $a$, $b$, $c$ in $K$ are the trivial ones satisfying $abc = 0$. With the help of modularity, level lowering and image of inertia comparisons we give an algorithmically testable criterion which if satisfied by $K$ implies the asymptotic Fermat's Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $…
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 this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases} \end{equation} where $A,B,C,D,n \in\ZZ_{+}$ and $n\geq3$, which may be regarded as a generalization of Fermat's equation $x^n+y^n=z^n$. When $\gcd(A,B,C)=1$, $(1)$ is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for $\gcd(A,B,C)=p^k$ where $p$ is an odd prime. In particular, for $k=1$ we prove that …
On Darmon's program for the generalized Fermat equation, I
本文基于过去二十年模方法进展，扩展Darmon研究广义费马方程的纲领，开发Frey阿贝尔簇相关工具，处理特定签名方程模方法步骤，解决部分方程解问题并推进纲领未处理的困难。
We study rational periodic points of polynomial $f_{d,c}(x)=x^d+c$ over the field of rational numbers, where $d$ is an integer greater than 2. For period 2, we classify all possible periodic points for degrees $d=4,6$. We also demonstrate the nonexistence of rational periodic points of exact period 2 for $d=2k$ such that $3\mid 2k-1$ and $k$ has a prime factor greater than 3. Moreover, assuming the $abc$-conjecture, we prove that $f_{d,c}$ has no rational periodic point of exact period greater than 1 for sufficiently large integer $d$ and $c\neq -1$.
The structure of the group $(\mathbb{Z}/n\mathbb{Z})^\star$ and Fermat's little theorem are the basis for some of best-known primality testing algorithms. Many related concepts arise: Euler's totient function and Carmichael's lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer's totient problem, Giuga's conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group $\mathcal{G}_n:=\{a+bi\in\mathbb{Z}[i]/n\mathbb{Z}[i] : a^2+b^2\equiv 1\ \textrm{$\pmod n$}\}$. In particular we characterize Gau…
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 $\mathbb{Q}_{n,p}$.
On the generalized Fermat equation of signature $(5,p,3)$
本文通过超几何动机和大图像猜想，证明了在特定条件下广义费马方程无非平凡原始解。
Let $K$ be a number field and $p$ a prime number $\geq 5$. Let us denote by $μ_p$ the group of the $p$th roots of unity. We define $p$ to be $K$-regular if $p$ does not divide the class number of the field $K(μ_p)$. Under the assumption that $p$ is $K$-regular and inert in $K$, we establish the second case of Fermat's Last Theorem over $K$ for the exponent $p$. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over $K$ has a finite number of $K$-rational points. Moreover, if $K$ is an imaginary quadratic field, other than ${\bf Q}(…
A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves
本文针对广义费马方程$x^n+y^n=z^5$执行Darmon纲领步骤，构建工具证明满足2进、5进条件的解的指数$n$的最优上界，将问题归约为大像猜想，为维数大于1的Frey阿贝尔簇情形提供测试案例。
Generalized Fermat equation over number fields
在两个标准数论猜想假设下，利用模方法研究数域上广义费马方程的渐近行为，建立基于$S_K'$-单位方程解的渐近判别准则，针对一类虚二次域证明方程无渐近解，且该类无平方因子整数相对密度为5/6。
Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $ϕ(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the study of almost perfect non-linear (APN) functions and exceptional APN functions. Their structure depends fundamentally on the Fermat varieties corresponding to the monomial functions of exceptional degrees $n=2^k+1$ and $n=2^{2k}-2^k+1$ (Gold and Kasami-Welch numbers, respectively). Very important results for these have been obtained by Janwa, McG…
Explicit solutions of the cubic Fermat equation are constructed in ring class fields $Ω_f$, with conductor $f$ prime to $3$, of any imaginary quadratic field $K$ whose discriminant satisfies $d_K \equiv 1$ (mod $3$), in terms of the Dedekind $η$-function. As $K$ and $f$ vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension $\textsf{K}_3$ of the $3$-adic field $\mathbb{Q}_3$. This is used to give a dynamical proof of a class number rela…
We study ``forms of the Fermat equation'' over an arbitrary field $k$, i.e. homogenous equations of degree $m$ in $n$ unknowns that can be transformed into the Fermat equation $X_1^m+...+X_n^m$ by a suitable linear change of variables over an algebraic closure of $k$. Using the method of Galois descent, we classify all such forms. In the case that $k$ is a finite field of characteristic greater than $m$ that contains the $m$-th roots of unity, we compute the Galois representation on $l$-adic cohomology (and so in particular the zeta function) of the hypersurface associated to an arbitrary form…
Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute th…
Solving Fermat-type equations over quadratic fields
本文通过模方法研究二次域上广义费马方程的有效可计算界及特殊解的不存在性。
We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer $n \geq 2$, the equation \[ x^{13} + y^{13} = 3 z^n \] has no non-trivial solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a…
On Darmon's program for the Generalized Fermat equation, II
本文通过扩展广义费马方程 Darmon 程序的方法，结合多 Frey 曲线/簇及大像猜想相关思想，完整解决了 $x^7 + y^7 = 3 z^n$（$n \ge 2$），并推进了 $x^7 + y^7 = z^n$ 的解决依赖条件研究。
Our main source of inspiration was a talk by Hendrik Lenstra on harmonic numbers, which are numbers whose only prime factors are two or three. Gersonides proved 675 years ago that one can be written as a difference of harmonic numbers in only four ways: 2-1, 3-2, 4-3, and 9-8. We investigate which numbers other than one can or cannot be written as a difference of harmonic numbers and we look at their connection to the $abc$-conjecture. We find that there are only eleven numbers less than 100 that cannot be written as a difference of harmonic numbers (we call these $ndh$-numbers). The smallest …
On the generalized Fermat equation $x^{13} + y^{13} = z^n$
本文结合模法、经典下降及多种筛法研究广义费马方程 $x^{13} + y^{13} = z^n$，证明了 $n=5$ 时仅有平凡解，在 GRH 假设下 $n=7$ 时亦仅有平凡解，并对一般 $n$ 给出了部分结论。
In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama--Shimura.
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…
Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve
研究Fermat椭圆曲线扭曲的中心L值代数部分的3-adic赋值，揭示Tate-Shafarevich群3-部分与惰性素因子个数的关系，并证明在分裂素数乘积情形下BSD猜想预测的阶为完全平方。
In this paper we prove new cases of the asymptotic Fermat equation with coefficients. This is done by solving remarkable $S$-units equations and applying a method of Frey-Mazur.
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.
In this paper we construct number fields ramified at 2 and 3 only, with various moderate-sized non-solvable Galois groups. We construct these fields by specializing three point covers, some from the literature and some new here. The specialization points come from solutions to a generalized Fermat equation. We conclude with a specific suggestion for further research and two general conjectures.
The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring and $p$-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is its real place. It al…
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