arXiv subjects
Publications and source records attributed to Samir Siksek.
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.
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 $\mathbb{Q}_{n,p}$.
Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(ζ_{2^r})^+$.
We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.
We develop a general framework for finding all perfect powers in sequences derived by shifting non-degenerate quadratic Lucas-Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in logarithms and results based upon the modularity of elliptic curves defined over totally real fields, we are able to answer a question of Bugeaud, Luca, Mignotte and the third author by explicitly finding all perfect powers of the shape $F_k \pm 2 $ where $F_k$ is the $k$-th term in the Fibonacci sequence.
In this paper we determine the quadratic points on the modular curves X_0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81. As well as determining the non-cuspidal quadratic points, we give the j-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic \Q-curves.
If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
Let $X/\mathbb{Q}$ be a curve of genus $g \ge 2$ with Jacobian $J$ and let $\ell$ be a prime of good reduction. Using Selmer varieties, Kim defines a decreasing sequence \[ X(\mathbb{Q}_\ell) \supseteq X(\mathbb{Q}_\ell)_1 \supseteq X(\mathbb{Q}_\ell)_2 \supseteq \cdots \] all containing the rational points of $X$. Thanks to the work of Coleman, the `Chabauty set' $X(\mathbb{Q}_\ell)_1$ is known to be finite provided the Mordell--Weil rank of $J$ is smaller than $g$. In this case one has a practical strategy that often succeeds in computing the set of rational points of $X$. Balakrishnan and Dogra have recently shown that the `quadratic Chabauty set' $X(\mathbb{Q}_\ell)_2$ is finite provided the Mordell--Weil rank is less than $g + ρ-1$, where $ρ$ is the Néron-Severi rank of $J/\mathbb{Q}$. In view of this it is interesting to give families of curves where $ρ\ge 2$ and where therefore quadratic Chabauty is more likely to succeed than classical Chabauty. In this note we show that this is indeed the case for all modular curves of genus at least 3.
Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most $50$ consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most $50$ consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.
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 $k \ge 2$ be even, and let $r$ be a non-zero integer. We show that for almost all $d \ge 2$ (in the sense of natural density), the equation $$ x^k+(x+r)^k+\cdots+(x+(d-1)r)^k=y^n, \qquad x,~y,~n \in \mathbb{Z}, \qquad n \ge 2, $$ has no solutions.
Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.
Let $K$ be a totally real field, and let $S$ be a finite set of non-archimedean places of $K$. It follows from the work of Merel, Momose and David that there is a constant $B_{K,S}$ so that if $E$ is an elliptic curve defined over $K$, semistable outside $S$, then for all $p>B_{K,S}$, the representation $\barρ_{E,p}$ is irreducible. We combine this with modularity and level lowering to show the existence of an effectively computable constant $C_{K,S}$, and an effectively computable set of elliptic curves over $K$ with CM $E_1,\dotsc,E_n$ such that the following holds. If $E$ is an elliptic curve over $K$ semistable outside $S$, and $p>C_{K,S}$ is prime, then either $\barρ_{E,p}$ is surjective, or $\barρ_{E,p} \sim \barρ_{E_i,p}$ for some $i=1,\dots,n$.
Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\barρ_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\barρ_{A,\ell}$ contains a transvection then $\barρ_{A,\ell}$ is either reducible or surjective. With the help of this we study surjectivity of $\barρ_{A,\ell}$ for semistable principally polarized abelian threefolds, and give an example of a genus $3$ hyperelliptic curve $C/\mathbb{Q}$ such that $\barρ_{J,\ell}$ is surjective for all primes $\ell \ge 3$, where $J$ is the Jacobian of $C$.
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 its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne 7$.
A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on the basis of extensive calculations performed by the famous computationalist Zacharias Dase. We complete the proof of this conjecture, building on previous work of Linnik, Watson, McCurley, Ramaré, Boklan, Elkies, and many others.
Given $k \geq 2$, we show that there are at most finitely many rational numbers $x$ and $y \neq 0$ and integers $\ell \geq 2$ (with $(k,\ell) \neq (2,2)$) for which $$ x (x+1) \cdots (x+k-1) = y^\ell. $$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell)$ satisfy either $y=0$ or $\log \ell < 3^k$.
Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \ge 2$) was considered by Zhongfeng Zhang who solved it for $k=2$, $3$, $4$ using Frey-Hellegouarch curves and their Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for $k=5$ is $x=z=0$, and that there are no solutions for $k=6$. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.