Sporadic cubic torsion

Derickx, Maarten; Etropolski, Anastassia; Hoeij, Mark van; Morrow, Jackson S.; Zureick-Brown, David

doi:10.2140/ant.2021.15.1837

Cited by 28 publications

(11 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first has canonical model Since X is a Picard curve (which is clear from the y coordinate), we can use ’s command to compute that the rank of is zero. In principle, one can now simply compute and then compute as the preimage of an Abel–Jacobi map ; see [DEvH+, Subsection 5.2] for a longer discussion of such computations.…”

Section: Analysis Of Rational Pointsmentioning

confidence: 99%

See 1 more Smart Citation

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

Rouse

Sutherland

Zureick-Brown

2022

Forum of Mathematics, Sigma

Self Cite

View full text Add to dashboard Cite

We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$ . For each of these $\ell $ , we compute the directed graph of arithmetically maximal $\ell $ -power level modular curves $X_H$ , compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$ , for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$ . Aside from the $\ell $ -adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$ -isomorphism classes of elliptic curves $E/\mathbb {Q}$ , we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$ , or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $ -adic images of Galois for any elliptic curve over $\mathbb {Q}$ . In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$ -type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$ .

show abstract

Section: Analysis Of Rational Pointsmentioning

confidence: 99%

“…We compute (using ’s command) that and . (See [DEvH+, Subsection 4.1] for a longer discussion of computing torsion on modular Jacobians). On the other hand, X has two visible rational points with coordinates , whose difference is a point of order 3 in .…”

Section: Analysis Of Rational Pointsmentioning

confidence: 99%

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

Rouse

Sutherland

Zureick-Brown

2022

Forum of Mathematics, Sigma

Self Cite

View full text Add to dashboard Cite

show abstract

“…Proof Let

N \in \lbrace 31,34,39\rbrace

, and let

J_1(N)_{/\mathbb {Q}}

be the Jacobian abelian variety of the modular curve

X_1(N)_{/\mathbb {Q}}

. By [23, Theorem 3.1] we have

J_1(N)(\mathbb {Q})

is finite. From this it follows — see, for example, [9, Theorem 4.2] — that

\delta (X_1(N)) = \gamma _{\mathbb {Q}}(X_1(N))

.…”

Section: Computationsmentioning

confidence: 99%

The least degree of a CM point on a modular curve

Clark

Genao

Pollack

et al. 2022

Journal of London Math Soc

View full text Add to dashboard Cite

For a modular curve X=X0(N)$X = X_0(N)$, X1(N)$X_1(N)$ or X1(M,N)$X_1(M,N)$ defined over Q$\mathbb {Q}$, we denote by dCM(X)$d_{\operatorname{CM}}(X)$ the least degree of a CM point on X$X$. For each discriminant normalΔ<0$\Delta < 0$, we determine the least degree of a point on X0(N)$X_0(N)$ with CM by the order of discriminant Δ$\Delta$. This places us in a position to study dCM(X)$d_{\operatorname{CM}}(X)$ as an ‘arithmetic function’ and we do so, obtaining various upper bounds, lower bounds and typical bounds. We deduce that all but finitely many curves in each of the families have sporadic CM points. Finally, we supplement these results with a computational study, for example, computing dCM(X0false(Nfalse))$d_{\operatorname{CM}}(X_0(N))$ and dCM(X1false(Nfalse))$d_{\operatorname{CM}}(X_1(N))$ exactly for N⩽106$N \leqslant 10^6$ and determining whether X0(N)$X_0(N)$ (respectively, X1(N)$X_1(N)$, X1(M,N)$X_1(M,N)$) has sporadic CM points for all but 106 values of N$N$ (respectively, 227 values of N$N$, 146 pairs false(M,Nfalse)$(M,N)$ with M⩾2$M \geqslant 2$).

show abstract

“…Mazur [20] determined the possible torsion groups over Q and Kamienny, Kenku and Momose [13,18] determined the possible torsion groups over quadratic fields. More recently, Derickx, Etropolski, van Hoeij, Morrow and Zureick-Brown [11] determined the possible torsion groups over cubic fields and Derickx, Kamienny, Stein and Stoll [12] determined all the primes dividing the order of some torsion group over number fields of degree 4 ≤ d ≤ 7. Merel proved that the set of all possible torsion groups over all number fields of degree d is finite, for any positive integer d [23].…”

Section: Introductionmentioning

confidence: 99%

Cyclic isogenies of elliptic curves over a fixed quadratic field

S.¹,

Najman²,

Oana³

2022

Preprint

View full text Add to dashboard Cite

Building on the work of Mazur on prime degree isogenies, Kenku determined in 1981 all the possible cyclic isogeny degrees of elliptic curves over Q. Although more than 40 years have passed, the possible isogeny degrees have not been determined over a single number field since.In this paper we determine such a classification of all possible cyclic isogeny degrees for the quadratic field Q( √ 213), assuming the Generalised Riemann Hypothesis. Along the way we determine all of the finitely many quadratic points on the modular curves X 0 (125) and X 0 (169).

show abstract

Sporadic cubic torsion

Cited by 28 publications

References 42 publications

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

The least degree of a CM point on a modular curve

Cyclic isogenies of elliptic curves over a fixed quadratic field

Contact Info

Product

Resources

About