arXiv subjects
Publications and source records attributed to Samir Siksek.
Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.
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 $u_k$ be a Lucas sequence. A standard technique for determining the perfect powers in the sequence $u_k$ combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation $u_k=x^n$ can be translated into a ternary equation of the form $a y^2=b x^{2n}+c$ (with $a$, $b$, $c \in \mathbb{Z}$) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form $u_k=x^n+c$ which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over $\mathbb{Q}$. We illustrate this approach by showing that the quaternary Diophantine equation $x^{2n} \pm 6 x^n+1=8 y^2$ has no solutions in positive integers $x$, $y$, $n$ with $x$, $n>1$.
We prove that all elliptic curves defined over real quadratic fields are modular.
Let K be a totally real Galois number field and let A be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes P such that for p not in P and E in A, the representation on E[p] is irreducible. Our sufficient conditions are often satisfied for Frey elliptic curves associated to solutions of Diophantine equations; in that context, the irreducibility of the mod p representation is a hypothesis needed for applying level-lowering theorems. We illustrate our approach by improving on an existing result for Fermat-type equations of signature (13, 13, p).
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 $K = \mathbb{Q}(\sqrt{d})$ for a subset of $d$ having density $5/6$ among the squarefree positive integers. We can improve this to density 1 if we assume a standard "Eichler-Shimura" conjecture.
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.
We determine the set of primitive integral solutions to the generalised Fermat equation x^2 + y^3 = z^15. As expected, the only solutions are the trivial ones with xyz = 0 and the non-trivial pair (x,y,z) = (+-3, -2, 1).
Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contains two skew lines both defined over K then r(S,K) = 1. One of the difficulties in studying the secant and tangent process on cubic surfaces is that it does not lead to an associative binary operation as in the case of elliptic curves. As a partial remedy we introduce an abelian group H_S(K) associated to a cubic surface S/K, naturally generated by the K-rational points, which retains much information about the secant and tangent process. In particular, r(S, K) is large as soon as the minimal number of generators of H_S(K) is large. In situations where weak approximation holds, H_S has nice local-to-global properties. We use these to construct a family of smooth cubic surfaces over the rationals such that r(S,K) is unbounded in this family. This is the cubic surface analogue of the unboundedness of ranks conjecture for elliptic curves.
Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this factorization with the property that if it is empty then the curve C has no rational points. We shall demonstrate the effectiveness of our new method by solving the generalized Fermat equation with signature (3,4,5), which is unassailable via the previously existing methods.
A well-known open problem is to show that the cubic form $x^3+y^3+2z^3$ represents all integers. An obvious variant of this problem is whether every integer can be {\em primitively} represented by $x^3+y^3+2z^3$. In other words, given an integer $n$, are there coprime integers $x$, $y$, $z$ such that $x^3+y^3+2z^3=n$? In this note we answer this variant question negatively. Indeed, we use cubic reciprocity to show that for every integral solution to $x^3+y^3+2z^3=8^m$,the unknowns $x$, $y$, $z$ are divisible by $2^m$.
Let $C$ be a smooth projective absolutely irreducible curve of genus $g \geq 2$ over a number field $K$ of degree $d$, and denote its Jacobian by $J$. Denote the Mordell--Weil rank of $J(K)$ by $r$. We give an explicit and practical Chabauty-style criterion for showing that a given subset $\cK \subseteq C(K)$ is in fact equal to $C(K)$. This criterion is likely to be successful if $r \leq d(g-1)$. We also show that the only solutions to the equation $x^2+y^3=z^{10}$ in coprime non-zero integers is $(x,y,z)=(\pm 3, -2, \pm 1)$. This is achieved by reducing the problem to the determination of $K$-rational points on several genus $2$ curves where $K=\Q$ or $\Q(\sqrt[3]{2})$, and applying the method of this paper.
We prove a result that finishes the study of primitive arithmetic progressions consisting of squares and fifth powers that was carried out by Hajdu and Tengely in a recent paper: The only arithmetic progression in coprime integers of the form (a^2, b^2, c^2, d^5) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generators for a subgroup of finite index of the Mordell-Weil group of the last curve. Applying Chabauty's method, we prove that the only rational points on this curve are the obvious ones.
We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.
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.