Necessity of the no-CM condition for Heegner bounds

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix an elliptic curve $E/\mathbb Q$ and a modular parametrization $\Phi:X_0(N)\to E$ defined over $\mathbb Q$. Here $X_0(N)$ classifies elliptic curves equipped with a cyclic subgroup of order $N$. For an imaginary quadratic field $k\subset\mathbb C$ with ring of integers $\mathcal O_k$ and an ideal $\mathfrak n\subset\mathcal O_k$ satisfying $\mathcal O_k/\mathfrak n\simeq\mathbb Z/N\mathbb Z$, let $\xi_k$ be the modular point represented by $(\mathbb C/\mathcal O_k,\mathfrak n^{-1}/\mathcal O_k)$. A Heegner point attached to $k$ is $P_k=\Phi(\xi_k)$, allowing the associated ideal choices and conjugates. Write $h_k$ for the ideal class number and $h_k^{\rm odd}$ for its largest odd divisor. Independence means that $\sum_i a_iP_{k_i}=O_E$ with integers $a_i$ implies all $a_i=0$. Complex multiplication (CM) means $\operatorname{End}_{\overline{\mathbb Q}}E$ is larger than $\mathbb Z$.

Statement recorded in the supplied source.

Can the non-CM hypothesis be removed from the following independence theorem? For every fixed $E/\mathbb Q$ and $\Phi$, does there exist a constant $C=C(E,\Phi)>0$ such that, for every finite collection of distinct imaginary quadratic fields $k_1,\ldots,k_t$ admitting the above Heegner construction and corresponding points $P_{k_i}$, \[ h_{k_i}^{\rm odd}>C\quad(1\le i\le t) \quad\Longrightarrow\quad P_{k_1},\ldots,P_{k_t}\text{ are independent}? \] The source establishes this assertion for elliptic curves without CM and asks whether that restriction is necessary. [S2]

Short English statement

Does the same large-odd-class-number independence theorem hold for CM elliptic curves as well?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800090 (AMR-087-0090). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joseph H. Silverman and Michael Rosen, Independence of Heegner Points, IPAM Number Theory and Cryptography: Open Problems workshop (9--13 October 2006), printed pp.8,11,16, PDF pages 11,15,22. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6669.pdf.