Define the ECHO sequence {bn} recursively by (b 0 , b 1 , b 2 , b 3 ) = (1, 1, 2, 1) and for n ≥ 4,We relate this sequence {bn} to the coordinates of points on the elliptic curve E : y 2 + y = x 3 − 3x + 4. We use Galois representations attached to E to prove that the density of primes dividing a term in this sequence is equal to 179 336 . Furthermore, we describe an infinite family of elliptic curves whose Galois images match those of E.

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$M \geqslant 2$).

In this paper, we evaluate the Faltings height of an elliptic curve with complex multiplication by an order in an imaginary quadratic field in terms of Euler's Gamma function at rational arguments. BackgroundIn the Seminar Bourbaki article [5], Deligne used the Chowla-Selberg formula [2] to evaluate the stable Faltings height of an elliptic curve with complex multiplication by the ring of integers O K of an imaginary quadratic field K in terms of Euler's Gamma function (s) at rational arguments. He then used this result to calculate the minimum value attained by the stable Faltings height. In this paper, we will establish a similar formula for both the unstable and stable Faltings height of an elliptic curve with complex multiplication by any order in K (not necessarily maximal). We illustrate these results by explicitly evaluating the Faltings height of an elliptic curve over Q with complex multiplication by a non-maximal order (see Sect. 2).We begin by recalling the definition of the (unstable) Faltings height of an elliptic curve, following ([12], Chapter IV, Sect. 6). Let L be a number field with ring of integers O L . Let E/L be an elliptic curve over L, and let E/O L be a Néron model for E/L.

A family F of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups E(F )[tors] of those elliptic curves E /F ∈ F can be made uniformly bounded after removing from F those whose number field degrees lie in a subset of Z + with arbitrarily small upper density. For every number field F , we prove unconditionally that the family E F of elliptic curves over number fields with F -rational j-invariants is typically bounded in torsion. For any integer d ∈ Z + , we also strengthen a result on typically bounding torsion for the family E d of elliptic curves over number fields with degree d j-invariants.

We prove that the family I F 0 \mathcal {I}_{F_0} of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with F 0 F_0 -rational j j -invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family I d 0 \mathcal {I}_{d_0} of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree d 0 d_0 j j -invariant is typically bounded in torsion.