Acyclotomy of torsion in the CM case

Chou, Michael; Clark, Pete L.; Milosevic, Marko

doi:10.1007/s11139-020-00271-0

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…In this work we continue our study of two closely related topics: torsion points on CM elliptic curves over number fields and CM points on modular curves. Prior contributions to this program have been made by Kronecker, Weber, Fricke, Hasse, Deuring, Shimura, Olson [Ol74], Silverberg [Si88], [Si92], Parish [Pa89], Aoki [Ao95], [Ao06], Ross [Ro94], Kwon [Kw99], Prasad-Yogananda [PY01], Stevenhagen [St01], Breuer [Br10], Lombardo [Lo17], Lozano-Robledo [LR18], [LR19], Gaudron-Rémond [GR18] and by the present authors and our collaborators [CCS13], [CCRS14], [BCS17], [BCP17], [BP17], [CCM19].…”

Section: Introductionmentioning

confidence: 99%

Torsion points and Galois representations on CM elliptic curves

2020

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Let O be an order in the imaginary quadratic field K. For positive integers M | N , we determine the least degree of an O-CM point on the modular curve X(M, N ) /K(ζ M ) and also on the modular curve X(M, N ) /Q(ζ M ) : that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon [Kw99] that determines the set of positive integers N for which there is an O-CM elliptic curve E admitting a cyclic, Q(j(E))-rational N -isogeny.

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Section: Introductionmentioning

confidence: 99%

Torsion points and Galois representations on CM elliptic curves

2020

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“…In this work, we continue our study of two closely related topics: torsion points on CM elliptic curves over number fields and CM points on modular curves. Prior contributions to this program have been made by Kronecker, Weber, Fricke, Hasse, Deuring, Shimura [29], and Olson [24], Silverberg [30, 31], Parish [25], Aoki [1, 2], Ross [28], Kwon [19], Prasad‐Yogananda [26], Stevenhagen [35], Breuer [7], Lombardo [20], Lozano‐Robledo [21, 22], Gaudron‐Rémond [17] and by the present authors and our collaborators [4–6, 8, 10–13].…”

Section: Introductionmentioning

confidence: 99%

Torsion points and isogenies on CM elliptic curves

Bourdon

Clark

2020

J. London Math. Soc.

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Let O be an order in the imaginary quadratic field K. For positive integers M | N , we determine the least degree of an O-CM point on the modular curve X(M, N) /K(ζ M) and also on the modular curve X(M, N) /Q(ζ M) : that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results, we establish several new theorems on rational cyclic isogenies of complex multiplication elliptic curves. In particular, we extend a result of Kwon (J. Korean Math. Soc. 36 (1999) 945-958) that determines the set of positive integers N for which there is an O-CM elliptic curve E admitting a cyclic, Q(j(E))-rational N-isogeny.

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“…We study elliptic curves with complex multiplication (CM) over number fields, an ongoing project of the present authors and our collaborators [5–8, 10, 13, 15–20].…”

Section: Introductionmentioning

confidence: 99%

The least degree of a CM point on a modular curve

Clark

Genao

Pollack

et al. 2022

Journal of London Math Soc

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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$M \geqslant 2$).

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Acyclotomy of torsion in the CM case

Cited by 4 publications

References 14 publications

Torsion points and Galois representations on CM elliptic curves

Torsion points and Galois representations on CM elliptic curves

Torsion points and isogenies on CM elliptic curves

The least degree of a CM point on a modular curve

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