Bounded prime-sum criterion for rational points

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.

Let $E/\mathbb Q$ be an elliptic curve, represented by an integral equation $y^2=x^3+Ax+B$ with $A,B\in\mathbb Z$ and $4A^3+27B^2\ne0$. For every prime $p$, define \[ N_p=1+\#\{(x,y)\in\mathbb F_p^2:y^2=x^3+Ax+B\},\qquad f_E(X)=\sum_{\substack{p\text{ prime}\\p\le X}}\log\!\left(\frac{N_p}{p}\right)\quad(X\ge2). \] The extra one accounts for the point at infinity: at good primes $N_p=\#E(\mathbb F_p)$, rather than this projective count plus one. At the finitely many bad primes the same displayed affine-count formula specifies $N_p$. A change to another integral model affects at most finitely many local counts and therefore changes $f_E(X)$ by a constant for all sufficiently large $X$. Bounded means that $|f_E(X)|\le C_E$ for all real $X\ge2$, for some finite constant $C_E$.

Statement recorded in the supplied source.

For every elliptic curve $E/\mathbb Q$, is $f_E(X)$ bounded as $X\to\infty$ if and only if the group $E(\mathbb Q)$ is finite? [S2]

Short English statement

Does boundedness of the logarithmic prime sum of local point counts exactly characterize elliptic curves with only finitely many rational points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8900013 (AMR-088-0013). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Wuthrich, Open problems in number theory, lecture slides (December 2011) slides beginning Let $N_p$ be the number of solutions of $E$ modulo $p$ plus $1$, followed by Birch and Swinnerton-Dyer conjecture. https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf.