arXiv subjects
Publications and source records attributed to Michael Stoll.
Let $X$ be one of the $28$ Atkin-Lehner quotients of a curve $X_0(N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich-Tate group of $J/\mathbb{Q}$ is trivial. This verifies the strong BSD conjecture for $J$.
We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.
We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $\mathbb Q$ with a $K$-rational point of order 13 and such that $E$ is not defined over $\mathbb Q$.
We study the number of irreducible factors (over $\mathbb{Q}$) of the $n$th iterate of a polynomial of the form $f_r(x) = x^2 + r$ for rational $r$. When the number of such factors is bounded independent of $n$, we call $f_r(x)$ \textit{eventually stable} (over $\mathbb{Q}$). Previous work of Hamblen, Jones, and Madhu shows that $f_r$ is eventually stable unless $r$ has the form $1/c$ for some integer $c \not\in \{0,-1\}$, in which case existing methods break down. We study this family, and prove that several conditions on $c$ of various flavors imply that all iterates of $f_{1/c}$ are irreducible. We give an algorithm that checks the latter property for all $c$ up to a large bound $B$ in time polynomial in $\log B$. We find all $c$-values for which the third iterate of $f_{1/c}$ has at least four irreducible factors, and all $c$-values such that $f_{1/c}$ is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all rational points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we use the more recent variant known as elliptic Chabauty. Finally, we apply all these results to completely determine the number of irreducible factors of any iterate of $f_{1/c}$, for all $c$ with absolute value at most $10^9$.
Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root finding method of Weierstrass, a root finder that tries to approximate all roots of a given polynomial in parallel (in the Jacobi version, i.e., with parallel updates). This method has a good reputation for finding all roots in practice except in obvious cases of symmetry, but very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well known Newton method, is not generally convergent: there are open sets of polynomials $p$ of every degree $d \ge 3$ such that the dynamics of the Weierstrass method applied to $p$ exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to $Z^3 + Z + 180$ have attracting cycles of period $4$. Here, period $4$ is minimal: we show that for cubic polynomials, there are no periodic orbits of length $2$ or $3$ that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree $d\ge 3$ there are orbits that are defined for all iterates but converge to $\infty$; this is a problem that does not occur for Newton's method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra.
We describe how one can use the "Selmer group Chabauty" method developed by the author to show that certain hyperelliptic curves of the form $$ C \colon y^2 = x^N + h(x)^2 \,, $$ where $N = 2g + 1$ is odd, $h \in \mathbb{Z}[x]$ with $\operatorname{deg} h \le g$ and $h(0)$ odd, have only the "obvious" rational points $\infty$ (the unique point at infinity on the smooth projective model of the curve) and $(0, \pm h(0))$. As an application of the method, we prove the following result. Let $c \in \mathbb{Q}$ and write $f_c(x) = x^2 + c$. We denote the iterates of $f_c$ by $f_c^{\circ n}$; i.e., we set $f_c^{\circ 0}(x) = x$ and $f_c^{\circ(n+1)}(x) = f_c(f_c^{\circ n}(x))$. If $f_c^{\circ 2}$ is irreducible, then $f_c^{\circ 6}$ is also irreducible. Assuming the Generalized Riemann Hypothesis (GRH), it also follows that $f_c^{\circ 10}$ is irreducible.
The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in $\mathbb P^4$. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat's diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namely by 777480/8288641. We then discuss a method that allows us to find the Mordell-Weil group of an elliptic curve $E$ defined over the rational function field $\mathbb Q(t)$ when $E$ has full $\mathbb Q(t)$-rational 2-torsion. This builds on recent results of Dujella, Gusić and Tadić. We give several concrete examples to which this method can be applied. One of these results implies that there is only one extension of the diophantine quadruple $\bigl(t-1,t+1,4t,4t(4t^2-1)\bigr)$ over $\mathbb Q(t)$.
We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\operatorname{SL}(2,\mathbb Z)$, either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the $\operatorname{PGL}(2,\mathbb Q)$ conjugacy class of an endomorphism of the projective line. Having a small model of such an endomorphism is useful for various computations.
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 minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p \bmod 24$. Recent results on mod $p$ representations with image in the normalizer of a split Cartan subgroup allow us to reduce the list further in some cases. Our second main result is the complete solution of the equation when $p = 11$, which previously was the smallest unresolved $p$. One relevant new ingredient is the use of the `Selmer group Chabauty' method introduced by the third author in recent work, applied in an Elliptic Curve Chabauty context, to determine relevant points on $X_0(11)$ defined over certain number fields of degree 12. This result is conditional on GRH, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p = 13$.
We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $\mathbb Q$, where $c, d \in \mathbb Z[x]$ are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that $x^N - k x^2 + 1$ is irreducible for all $N \ge 5$ and $k \in \{3, 4, \ldots, 24\} \setminus \{9, 16\}$. We also give a complete description of the factorization of polynomials of the form $x^N + k x^{N-1} \pm (l x + 1)$ with $k, l \in \mathbb Z$, $k \neq l$.
We provide a complete classification of possible graphs of rational preperiodic points of quadratic rational functions defined over the rationals with a rational periodic critical point of period 3, under two assumptions: that these functions have no periodic points of period at least 5 and the conjectured enumeration of points on a certain genus 6 affine plane curve. We show that there are exactly six such possible graphs, and that rational functions satisfying the conditions above have at most eleven rational preperiodic points.
We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as étale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus 1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a given primary Burniat surface $S$ defined over $\mathbb Q$. This involves an explicit description of the relevant twists of the étale covering of $S$ coming from the second construction mentioned above and methods for finding the set of rational points on a given twist.
We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field $k$ of characteristic $\neq 2$. In particular, we provide explicit equations defining the Kummer variety $\mathcal K$ as a subvariety of $\mathbb P^7$, together with explicit polynomials giving the duplication map on $\mathcal K$. A careful study of the degenerations of this map then forms the basis for the development of an explicit theory of heights on such Jacobians when $k$ is a number field. We use this input to obtain a good bound on the difference between naive and canonical height, which is a necessary ingredient for the explicit determination of the Mordell-Weil group. We illustrate our results with two examples.
Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve $C$, given only the $p$-Selmer group $S$ of its Jacobian (or some other abelian variety $C$ maps to) and the image of the $p$-Selmer set of $C$ in $S$. The method is more likely to succeed when the genus is large, which is when it is usually rather difficult to obtain generators of a finite-index subgroup of the Mordell-Weil group, which one would need to apply Chabauty's method in the usual way. We give some applications, for example to generalized Fermat equations of the form $x^5 + y^5 = z^p$.
Let $K$ be a number field and let $C/K$ be a curve of genus 2 with Jacobian variety $J$. In this paper, we study the canonical height $\hat{h} \colon J(K) \to \mathbb R$. More specifically, we consider the following two problems, which are important in applications: (1) for a given $P \in J(K)$, compute $\hat{h}(P)$ efficiently; (2) for a given bound $B > 0$, find all $P \in J(K)$ with $\hat{h}(P) \le B$. We develop an algorithm running in polynomial time (and fast in practice) to deal with the first problem. Regarding the second problem, we show how one can tweak the naive height $h$ that is usually used to obtain significantly improved bounds for the difference $h - \hat{h}$, which allows a much faster enumeration of the desired set of points. Our approach is to use the standard decomposition of $h(P) - \hat{h}(P)$ as a sum of local `height correction functions'. We study these functions carefully, which leads to efficient ways of computing them and to essentially optimal bounds. To get our polynomial-time algorithm, we have to avoid the factorization step needed to find the finite set of places where the correction might be nonzero. The main innovation at this point is to replace factorization into primes by factorization into coprimes. Most of our results are valid for more general fields with a set of absolute values satisfying the product formula. An analogous approach to (1) above for elliptic curves was used in our recent paper arXiv:1509.08748.
We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.
We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an explicit bound is 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic `annulus' on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k). We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over Q whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus g tends to infinity.
We improve a result due to Masser and Zannier, who showed that the set $$ \{λ\in {\mathbb C} \setminus \{0,1\} : (2,\sqrt{2(2-λ)}), (3,\sqrt{6(3-λ)}) \in (E_λ)_{\text{tors}}\} $$ is finite, where $E_λ\colon y^2 = x(x-1)(x-λ)$ is the Legendre family of elliptic curves. More generally, denote by $T(α, β)$, for $α, β\in {\mathbb C} \setminus \{0,1\}$, $α\neq β$, the set of $λ\in {\mathbb C} \setminus \{0,1\}$ such that all points with $x$-coordinate $α$ or $β$ are torsion on $E_λ$. By further results of Masser and Zannier, all these sets are finite. We present a fairly elementary argument showing that the set $T(2,3)$ in question is actually empty. More generally, we obtain an explicit description of the set of parameters $λ$ such that the points with $x$-coordinate $α$ and $β$ are simultaneously torsion, in the case that $α$ and $β$ are algebraic numbers that not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that ${\mathbb Q}(α, β)$ has transcendence degree 1. In this case we show that $\#T(α, β) \le 1$ and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating $α$ and $β$. This leads to a more precise description of $T(α, β)$ also in the case when both $α$ and $β$ are algebraic. We performed extensive computations that support several conjectures, for example that there should be only finitely many pairs $(α, β)$ such that $\#T(α, β) \ge 3$.