Michael Stoll
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Michael Stoll has a total of 54 co-authors. Here are the top co-authors by number of shared publications:
- Samir Siksek
Mathematics Institute, University of Warwick, Coventry, United Kingdom
5 shared publications - Tom Fisher
DPMMS, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK
5 shared publications - Bjorn Poonen
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
4 shared publications - John E. Cremona
University of Warwick
4 shared publications - Nils Bruin
Simon Fraser University
4 shared publications - Denis Simon
Université de Caen, Campus II - Boulevard Maréchal Juin, BP 5186–14032, Caen, France
3 shared publications
Latest Publications
- 25 August 2026
- Published by World Scientific Pub Co Pte Ltd in International Journal of Number Theory
- p. 1-17
On some open cases of a conjecture of Conrad, Edixhoven and Stein
Let [Formula: see text] be a prime. In 2003, Conrad et al. conjectured that the rational torsion subgroup of the modular Jacobian [Formula: see text] coincides with the rational cuspidal divisor class group. Using explicit computations in Magma, the open case [Formula: see text] has been proven by [M. Derickx, S. Kamienny, W. Stein and M. Stoll, Torsion points on elliptic curves over number fields of small degree, Algebra Number Theory 17(2) (2023) 267–308, doi:10.2140/ant.2023.17.267]. We extend these results to primes [Formula: see text] and [Formula: see text]. In addition, we provide a list of the groups [Formula: see text] for every prime up to [Formula: see text]. However, our method is general and can be applied to larger primes.
- 25 August 2026
- Published by World Scientific Pub Co Pte Ltd in International Journal of Number Theory
- p. 1-17
On some open cases of a conjecture of Conrad, Edixhoven and Stein
Let [Formula: see text] be a prime. In 2003, Conrad et al. conjectured that the rational torsion subgroup of the modular Jacobian [Formula: see text] coincides with the rational cuspidal divisor class group. Using explicit computations in Magma, the open case [Formula: see text] has been proven by [M. Derickx, S. Kamienny, W. Stein and M. Stoll, Torsion points on elliptic curves over number fields of small degree, Algebra Number Theory 17(2) (2023) 267–308, doi:10.2140/ant.2023.17.267]. We extend these results to primes [Formula: see text] and [Formula: see text]. In addition, we provide a list of the groups [Formula: see text] for every prime up to [Formula: see text]. However, our method is general and can be applied to larger primes.
- Open Access
- 13 July 2026
- Published by Society for Industrial & Applied Mathematics (SIAM)
- p. 484-498
Determining the Rational Points on a Curve of Genus 2 and Mordell–Weil Rank 1
Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis... Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis...
- 13 July 2026
- Published by Society for Industrial & Applied Mathematics (SIAM)
- p. 484-498
Open AccessDetermining the Rational Points on a Curve of Genus 2 and Mordell–Weil Rank 1
Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis... Continuing the legacy of the International Congress of Mathematicians, these proceedings bring together presentations delivered by invited speakers at the 2026 Congress held in Philadelphia. Publis...
- Open Access
- 1 March 2026
- Published by Association for Mathematical Research in Journal of Experimental Mathematics
- Vol. 2 (1) , 194-204
Prime Numbers and Dynamics of the Polynomial x² - 1
Let n ∈ Z_{>=2}. By P(n) we denote the set of all prime divisors of the integers in the sequence n, n^2 - 1, (n^2 - 1)^2 - 1,... . We ask whether the set P(n) determines n uniquely under the assumption that n ≠ m^2 - 1 for m ∈ Z_{>=2}. This problem originates in the structure theory of infinite-dimensional Lie algebras. We show that the sets P(n) generate infinitely many equivalence classes of positive integers under the equivalence relation n_1 ∼ n_2 ⇔ P(n_1) = P(n_2). We also prove that the sets P(n) separate all positive integers up to 229, and we provide some heuristics on why the answer to our question should be positive.
- 1 March 2026
- Published by Association for Mathematical Research in Journal of Experimental Mathematics
- Vol. 2 (1) , 194-204
Open AccessPrime Numbers and Dynamics of the Polynomial x² - 1
Let n ∈ Z_{>=2}. By P(n) we denote the set of all prime divisors of the integers in the sequence n, n^2 - 1, (n^2 - 1)^2 - 1,... . We ask whether the set P(n) determines n uniquely under the assumption that n ≠ m^2 - 1 for m ∈ Z_{>=2}. This problem originates in the structure theory of infinite-dimensional Lie algebras. We show that the sets P(n) generate infinitely many equivalence classes of positive integers under the equivalence relation n_1 ∼ n_2 ⇔ P(n_1) = P(n_2). We also prove that the sets P(n) separate all positive integers up to 229, and we provide some heuristics on why the answer to our question should be positive.
- Open Access
The valuation of the discriminant of a hypersurface
Let $R$ be a discrete valuation ring, with valuation $v \colon R \twoheadrightarrow \mathbb{Z}_{\ge 0} \cup \{\infty\}$ and residue field $k$. Let $H$ be a hypersurface $\operatorname{Proj}(R[x_0,\ldots,x_n]/\langle f \rangle)$. Let $H_k$ be the special fiber, and let $(H_k)_{\mathrm{sing}}$ be its singular subscheme. Let $Δ(f)$ be the discriminant of $f$. We use Zariski's main theorem and degeneration arguments to prove that $v(Δ(f))=1$ if and only if $H$ is regular and $(H_k)_{\mathrm{sing}}$ consists of a nondegenerate double point over $k$. We also give lower bounds on $v(Δ(f))$ when $H_k$ has multiple singularities or a positive-dimensional singularity.Open AccessThe valuation of the discriminant of a hypersurface
Let $R$ be a discrete valuation ring, with valuation $v \colon R \twoheadrightarrow \mathbb{Z}_{\ge 0} \cup \{\infty\}$ and residue field $k$. Let $H$ be a hypersurface $\operatorname{Proj}(R[x_0,\ldots,x_n]/\langle f \rangle)$. Let $H_k$ be the special fiber, and let $(H_k)_{\mathrm{sing}}$ be its singular subscheme. Let $Δ(f)$ be the discriminant of $f$. We use Zariski's main theorem and degeneration arguments to prove that $v(Δ(f))=1$ if and only if $H$ is regular and $(H_k)_{\mathrm{sing}}$ consists of a nondegenerate double point over $k$. We also give lower bounds on $v(Δ(f))$ when $H_k$ has multiple singularities or a positive-dimensional singularity. - Open Access
Determining the rational points on a curve of genus 2 and Mordell-Weil rank 1
We explain how one can efficiently determine the (finite) set of rational points on a curve of genus 2 over $\mathbb Q$ with Jacobian variety $J$, given a point $P \in J(\mathbb Q)$ generating a subgroup of finite index in $J(\mathbb Q)$.Open AccessDetermining the rational points on a curve of genus 2 and Mordell-Weil rank 1
We explain how one can efficiently determine the (finite) set of rational points on a curve of genus 2 over $\mathbb Q$ with Jacobian variety $J$, given a point $P \in J(\mathbb Q)$ generating a subgroup of finite index in $J(\mathbb Q)$. - Open Access
- 15 July 2025
- Published by Centre pour la Communication Scientifique Directe (CCSD)
- Vol. ume 1
Formalizing zeta and L-functions in Lean
The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library, including a proof of Dirichlet's theorem on primes in arithmetic progressions and a formal statement of the Riemann hypothesisFinal version, to appear in Annals of Formalized Mathematics
- 15 July 2025
- Published by Centre pour la Communication Scientifique Directe (CCSD)
- Vol. ume 1
Open AccessFormalizing zeta and L-functions in Lean
The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library, including a proof of Dirichlet's theorem on primes in arithmetic progressions and a formal statement of the Riemann hypothesisFinal version, to appear in Annals of Formalized Mathematics
- Open Access
Coarse length can be unbounded in 3-step nilpotent Lie groups
In "On the asymptotics of the growth of 2-step nilpotent groups" (J. London Math. Soc. (2), 58 (1998)), we remarked that, contrary to 2-step nilpotent simply connected Lie groups, in 3-step nilpotent simply connected Lie groups it is possible that `$\mathbb{R}$-words' in the given generators cannot be replaced by an equally long $\mathbb{R}$-word representing the same group element and having a bounded number of direction changes. In this note, we present an example for this phenomenon.
Open AccessCoarse length can be unbounded in 3-step nilpotent Lie groups
In "On the asymptotics of the growth of 2-step nilpotent groups" (J. London Math. Soc. (2), 58 (1998)), we remarked that, contrary to 2-step nilpotent simply connected Lie groups, in 3-step nilpotent simply connected Lie groups it is possible that `$\mathbb{R}$-words' in the given generators cannot be replaced by an equally long $\mathbb{R}$-word representing the same group element and having a bounded number of direction changes. In this note, we present an example for this phenomenon.
- Open AccessThe Generalized Fermat Equation $x^2 + y^3 = z^{25}$We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form $H(u,v) = w^5$, where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$, $u$ and $v$ are coprime (rational) integers, and $w \in K$.Open AccessThe Generalized Fermat Equation $x^2 + y^3 = z^{25}$We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form $H(u,v) = w^5$, where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$, $u$ and $v$ are coprime (rational) integers, and $w \in K$.
- Open Access
Prime order torsion on elliptic curves over number fields. Part I: Asymptotics
We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$.Open AccessPrime order torsion on elliptic curves over number fields. Part I: Asymptotics
We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$. - Open Access
- 30 January 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$ . We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.- 30 January 2025
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Sigma
- Vol. 13
Open AccessComplete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$ . We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.