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.
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.
In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.
Let $F_n$ denote the Fermat curve given by $x^n+y^n=z^n$ and let $μ_n$ denote the Galois module of $n$th roots of unity. It is known that the integral homology group $H_1(F_n,\Z)$ is a cyclic $\Z[μ_n\times μ_n]$ module. In this paper, we prove this result using modular symbols and the modular description of Fermat curves; moreover we find a basis for the integral homology group $H_1(F_n,\Z)$. We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.
Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(ζ_p)^+$, and study i…
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.
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…
Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no non-trivial solutions over the quadratic field $\mathbb{Q}(\sqrt{d})$ for $n \ge 4$. Furthermore, we show for $d=17$ that the same holds for prime exponents $n \equiv 3$, $5 \pmod{8}$.
Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat's Last Theorem is true over $K$, criterion related to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$…
Eisenstein series are real analytic functions which play a central role in spectral theory of the hyperbolic Laplacian. Kronecker limit formulas determine their connection to modular forms. The main result of this work is Theorem 7.2 in which a Kronecker limit formula for a family of non-congruence subgroups associated with the Fermat curves is presented. As an application we can determine the scattering constants for the Fermat curves in Theorem 8.1.
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…
We describe a strategy to attack infinitely many Fermat-type equations of signature $(r,r,p)$, where $r \geq 7$ is a fixed prime and $p$ is a prime allowed to vary. We use a variant of the modular method over totally real subfields of $\mathbb{Q}(ζ_r)$. In particular, to a solution $(a,b,c)$ of $x^r + y^r =Cz^p$ we will attach several Frey curves $E=E_{(a,b)}$. We prove modularity of all the Frey curves and the exsitence of a constant constant $M_r$, depending only on $r$, such that for all $p>M_r$ the representations $\barρ_{E,p}$ are absolutely irreducible. Along the way, we also prove modul…
Let $p$ be an odd prime number. Using modular arguments, we give an easy testable condition which allows often to prove Fermat's Last Theorem over the quadratic field ${\bf Q}(\sqrt{5})$ for the exponent $p$. It is related to the Wendt's resultant of the polynomials $X^n-1$ and $(X+1)^n-1$. We deduce Fermat's Last Theorem over this field in case one has $5\leq p<10^7$, and we obtain analogous results on Sophie Germain type criteria.
In this paper we prove that equations of the form $x^{13} + y^{13} = Cz^{p}$ have no non-trivial primitive solutions (a,b,c) such that $13 \nmid c$ if $p > 4992539$ for an infinite family of values for $C$. Our method consists in relating a solution (a,b,c) to the previous equation to a solution (a,b,c_1) of another Diophantine equation with coefficients in $\Q(\sqrt{13})$. We then construct Frey-curves associated with (a,b,c_1) and we prove modularity of them in order to apply the modular approach via Hilbert cusp forms over $\Q(\sqrt{13})$. We also prove a modularity result for elliptic curv…
In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results in a previous paper of the first author. We finally discuss the present limitations of the method by looking at the case d=3.
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…
In this paper, we prove that the only primitive solutions of the equation $a^2+3b^6=c^n$ for $n\geq 3$ are $(a,b,c,n)=(\pm 47,\pm 2,\pm 7,4)$. Our proof is based on the modularity of Galois representations of $\mathbb Q$-curves and the work of Ellenberg for big values of $n$ and a variety of techniques for small $n$.
The Modular Group provides simple proofs of Fermat's representations: X^2+Y^2 for primes congruent to 1 (mod 4) and by X^2+3Y^2 for primes congruent to 1 (mod 3)
We solve the Diophantine equations $x^5 + y^5 = dz^p$ with $d=2, 3$ for a set of prime numbers of density 1/4, 1/2, respectively. The method consists in relating a possible solution to another Diophantine equation and solving the later by using Q-curves and a generalized modular technique as in work of Ellenberg and Dieulefait-Jimenez along with some new techniques for eliminating newforms.
We show that the mod p Galois representations attached to a Q-curve E of degree d over an imaginary quadratic number field K are surjective for all p larger than some constant M_{K,d}, if E has potentially multiplicative reduction at any prime not dividing 6. The proof uses Mazur's formal immersion method, a result of Darmon and Merel on non-split modular curves, and an analytic argument showing that Jacobians of certain twisted modular curves admit quotients with Mordell-Weil rank 0. In combination with a previous theorem of the author and Skinner on modularity of Q-curves, the surjectivity r…
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.
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 $\mathbb{Q}_{n,p}$.
We prove Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\mathbb Q}(\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\mathbb Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbe…
The primary aim of this paper is to study the generalized Fermat equation \[ x^2+y^{2n} = z^{3p} \] in coprime integers $x$, $y$, and $z$, where $n \geq 2$ and $p$ is a fixed prime. Using modularity results over totally real fields and the explicit computation of Hilbert cuspidal eigenforms, we provide a complete resolution of this equation in the case $p=7$, and obtain an asymptotic result for fixed $p$. Additionally, using similar techniques, we solve a second equation, namely $x^{2\ell}+y^{2m} = z^{17}$, for primes $\ell,m \ne 5$.
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 totally real number field and consider a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature $(p,p,2)$ and $(p,p,3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Işik, Kara and Ozman. For example, consider $K$ a totally real field of degree $n$ with $2 \nmid h_K^+$ and $2$ inert. Moreover, suppose there is a prime $q\geq 5$ which totally ram…
We give an account of Mazur's proof that, for an elliptic curve over $\mathbb{Q}$, if it admits a nonconstant mapping from $X(N)$ defined over the complex numbers $\mathbb{C}$, for some $N$, then it also admits a nonconstant mapping from $X_0(M)$ for some (possibly different) $M$ defined over the rational numbers $\mathbb{Q}$. We also briefly discuss two open questions of Khare concerning uniformisations of elliptic curves by noncongruence modular curves. This expository note is based on the author's expository talk in March 2022 during the 2nd Trimester Program on "Modularity and the Generali…
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.
We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We first develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minima…
Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such…
In this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence of newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t o…
The topic of this paper concerns a certain relation between the jacobians of various quotients of the modular curve $X(p)$, which relates the jacobian of the quotient of $X(p)$ by the normaliser of a non-split Cartan subgroup of $GL_2(F_p)$ to the jacobians of more standard modular curves. In this paper, we confirm a conjecture of Merel found in a paper of Darmon-Merel, "Winding quotients and some variants of Fermat's Last Theorem", Crelle, v. 490, p. 81-100, 1997, which describes this relation in terms of explicit correspondences. The method used is to reduce the conjecture to showing a certa…
Let $n$ be a positive integer and consider the Diophantine equation of generalized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns $x,y,z$. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for $n \in \{5, 31\}$ there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for $n \leq 10^7$. Finally, we show that there are also no solutions for $n\equiv -1 \pmod{6}$.
Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case , where we can consider convenient generalized …
We study the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces with respect to morphisms of a particular kind. We obtain upper bounds for the arithmetic self-intersection number of the dualizing sheaf on minimal regular models of the modular curves associated with congruence subgroups $Γ_0(N)$ with square free level, as well as for the modular curves X(N) and the Fermat curves with prime exponent.
We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.
A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part …
We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, …
