A Rank-Dependent Secondary Bias in the Sign of the Minimal Discriminant

Abstract

We study singleton non-CM isogeny classes of elliptic curves over $\mathbf Q$ with prime conductor. For each fixed rank $0\le r\le3$, we conjecture a two-term asymptotic for the proportion having negative minimal discriminant. The leading constant is the classical $\sqrt3$ archimedean ratio, while the secondary constants exhibit a strict rank-dependent ordering suggested by the available data.

Statement of the conjecture

For $r\in\{0,1,2,3\}$ and $X\ge1$, let $\mathcal P_r(X)$ be the set of $\mathbf Q$-isogeny classes $[E]$ such that

1. the conductor is a prime $N_E=p\le X$;

2. the isogeny class contains a single $\mathbf Q$-isomorphism class;

3. $E$ is non-CM and $\operatorname{rk}E(\mathbf Q)=r$.

The sign of the minimal discriminant is then unambiguous on the isogeny class. Let $\mathcal P_r^-(X)$ denote the subset for which $\Delta_{\min}(E)<0$.

Conjecture 1. Assume that $\#\mathcal P_r(X)\to\infty$ for each $0\le r\le3$. There exist real constants $d_0,d_1,d_2,d_3$ such that

$$\frac{\#\mathcal P_r^-(X)}{\#\mathcal P_r(X)} =\frac{\sqrt3}{1+\sqrt3}+\frac{d_r}{\log X} +o\!\left(\frac1{\log X}\right),$$

and these constants satisfy

$$d_0>d_1>0>d_2>d_3.$$

The main term is the archimedean mass predicted by the Brumer–McGuinness and Watkins model. The conjecture asserts that conditioning on exact Mordell–Weil rank introduces a stable secondary correction of order $1/\log X$.

Numerical evidence

At $X=3\cdot10^8$, the observed proportions and rescaled deviations are as follows.

$r$	$\#\mathcal P_r^-(X)/\#\mathcal P_r(X)$	$\log X\left(\dfrac{\#\mathcal P_r^-(X)}{\#\mathcal P_r(X)}-\dfrac{\sqrt3}{1+\sqrt3}\right)$
0	$0.663154$	$0.56956$
1	$0.637888$	$0.07639$
2	$0.598767$	$-0.68723$
3	$0.557980$	$-1.48336$

Rank-conditioned discriminant-sign statistics at $X=3\cdot10^8$.

Here $\sqrt3/(1+\sqrt3)=0.633975\ldots$. Between $3\cdot10^7$ and $3\cdot10^8$, the four rescaled quantities remain near $0.59$, $0.07$, $-0.66$, and $-1.47$, respectively. The inspected sample contains $731{,}713$ prime-conductor representatives, with no missing discriminant signs and with algebraic rank agreeing with analytic rank throughout.

Relation to known counting results

Brumer and McGuinness first observed the overall negative-to-positive ratio near $\sqrt3$ in prime-conductor data, and Watkins rederived the corresponding sign-dependent volume heuristic. Shankar–Shankar–Wang prove, for certain large conductor-ordered families defined by local conditions, sign-separated leading asymptotics whose constants satisfy $\alpha_-=\sqrt3\,\alpha_+$. Their theorem does not cover the thin requirement that the conductor be prime, nor does it condition on exact rank or give a secondary term.

By modularity and the Mestre–Oesterlé prime-conductor theorem, a prime-conductor isogeny class contains a curve with prime absolute minimal discriminant. In the singleton case this is the unique curve, so the conjecture can equivalently be viewed as a statement about exact-rank prime-discriminant curves. This reduction does not provide the required distribution.

The unresolved analytic problem

Let $A_{r,+}(X)$ and $A_{r,-}(X)$ be the positive- and negative-discriminant counts. Proving Conjecture 1 requires a rank- and sign-separated two-term asymptotic for these counts, strong enough to identify both the main ratio and the limits

$$d_r=\lim_{X\to\infty}\log X \left(\frac{A_{r,-}(X)}{A_{r,+}(X)+A_{r,-}(X)} -\frac{\sqrt3}{1+\sqrt3}\right).$$

The passage to prime conductor already involves a thin global sieve. Exact-rank conditioning further introduces global quantities such as central $L$-values, Selmer groups, regulators, and Tate–Shafarevich groups. Present lattice-counting and average-Selmer methods do not control these conditions simultaneously, and no existing theorem gives even the required leading exact-rank asymptotics in this family. The numerical pattern is therefore evidence rather than a proof.

References

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• Oisín McGuinness, Summary of Results: Elliptic Curves (1990). https://www.math.columbia.edu/~om/EllipticCurves/Summary.html

• Mark Watkins, Some Heuristics about Elliptic Curves (2008). https://arxiv.org/abs/math/0608766

• Ananth N. Shankar, Arul Shankar, and Xiaoheng Wang, Large families of elliptic curves ordered by conductor (2021). https://doi.org/10.1112/S0010437X21007193

• Jean-François Mestre and Joseph Oesterlé, Courbes de Weil semi-stables de discriminant une puissance m-ième (1989). https://doi.org/10.1515/crll.1989.400.173

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