Catalogue of AI-generated conjectures from LMFDB

Abstract

We formulate a conjectural power-law distribution for local Tamagawa numbers at marked split multiplicative primes of semistable elliptic curves over $\mathbb Q$, ordered by conductor and conditioned by Mordell–Weil rank and the sign of the minimal discriminant. We record the numerical evidence motivating the conjecture and explain the gap between the proposed conductor-ordered statement and known fixed-prime, height-ordered local density results.

Statement of the conjecture

Fix an integer $r\geq 0$ and a sign $\sigma\in\{-1,+1\}$. For $X>0$, let

$$\begin{aligned} \mathcal S_{r,\sigma}(X)=\bigl\{(E,q):{}& N_E\leq X,\ E/\mathbb Q\text{ semistable and non-CM},\\ &\mathop{\mathrm{rank}}E(\mathbb Q)=r,\quad [E]_{\mathbb Q\text{-isog}}=\{E\},\\ &\mathop{\mathrm{sgn}}(\Delta_{\min}(E))=\sigma,\quad E\text{ has split multiplicative reduction at }q\bigr\}. \end{aligned}$$

Here elliptic curves are counted up to $\mathbb Q$-isomorphism, while the prime $q$ is marked. Thus a curve with several split multiplicative primes contributes once for each such prime.

For a prime $\ell$, define, whenever the limit exists,

$$\delta_{r,\sigma}(\ell) =\lim_{X\to\infty} \frac{\#\{(E,q)\in\mathcal S_{r,\sigma}(X):c_q(E)=\ell\}} {\#\mathcal S_{r,\sigma}(X)}.$$

Conjecture 1. For every $r\geq0$, every $\sigma\in\{-1,+1\}$ and every prime $\ell$, the density $\delta_{r,\sigma}(\ell)$ exists. Moreover,

$$\lim_{\substack{\ell\to\infty\\ \ell\ \mathrm{prime}}} \ell^2\delta_{r,\sigma}(\ell)=r+1.$$

The order of limits is part of the statement: one first lets the conductor bound $X$ tend to infinity for fixed $\ell$, and only afterwards lets $\ell$ tend to infinity through the primes.

Numerical evidence

In the complete conductor range $N_E\leq150000$, the marked split-multiplicative samples contained $62395$, $116608$ and $41390$ pairs in ranks $0$, $1$ and $2$, respectively. The following table gives the observed values of $\ell^2\Pr(c_q(E)=\ell)$ after pooling both signs of the minimal discriminant.

rank	$\ell=5$	$\ell=7$	$\ell=11$	$\ell=13$
$0$	$1.162$	$1.253$	$1.187$	$1.181$
$1$	$1.678$	$1.986$	$2.018$	$1.935$
$2$	$1.883$	$2.388$	$2.850$	$2.683$

Finite-conductor estimates for the scaled local Tamagawa frequency.

Six conductor windows gave comparable values. Splitting by the sign of $\Delta_{\min}$ preserved the same rank dependence. At $\ell=11$, the negative- and positive-sign estimates were respectively $1.237$ and $1.032$ in rank $0$, $2.028$ and $1.992$ in rank $1$, and $2.835$ and $2.879$ in rank $2$.

A wider check in the uniform range $N_E\leq500000$ involved $751417$ marked pairs. The predicted ordering by rank persisted across five cumulative conductor cutoffs. For $\ell$ beyond roughly $17$–$23$, the finite-cutoff estimates decrease; this is consistent with the fact that a fixed conductor cutoff suppresses very large valuations of the minimal discriminant.

Heuristic motivation

At a split multiplicative prime of Kodaira type $I_n$, Tate’s algorithm gives

$$c_q(E)=n=v_q(\Delta_{\min}(E)).$$

Thus Conjecture 1 is a statement about the tail of a marked local discriminant valuation. Conductor ordering records the presence of the bad prime $q$ but does not directly charge for the size of $v_q(\Delta_{\min})$, making a polynomial tail plausible. The data suggest that the exponent is stable while conditioning on rank changes the leading coefficient. Marking $q$ removes variation in the number of bad places, and restricting to singleton rational isogeny classes avoids transfers of Tamagawa factors within an isogeny class.

Rank-conditioned Selmer heuristics offer a possible explanation for the factor $r+1$: conditioning on a larger Mordell–Weil rank may size-bias local arithmetic data. At present this remains only a heuristic analogy.

Relation with known results

Known local density formulae at a fixed residue prime are in a different regime. For fixed $p$, the density of minimal Weierstrass models of type $I_m$ decays exponentially in $m$. Global results on products of Tamagawa numbers are likewise generally height-ordered. Existing conductor-ordered counting theorems allow only restricted families and finitely many prescribed local conditions; they do not impose exact Mordell–Weil rank, a marked varying prime, or a large-prime uniformity statement of the form above.

Open difficulties

A proof would require asymptotics for

$$A_{r,\sigma}(X)=\#\mathcal S_{r,\sigma}(X),\qquad A_{r,\sigma,\ell}(X)= \#\{(E,q)\in\mathcal S_{r,\sigma}(X):v_q(\Delta_{\min}(E))=\ell\},$$

first for each fixed $\ell$ and then uniformly enough in $\ell$ to obtain $\delta_{r,\sigma}(\ell)\sim(r+1)/\ell^2$. No available theorem gives such simultaneous control under exact-rank conditioning. The universal quantifier in $r$ is also far beyond the numerical range: the available data are statistically meaningful only for $r=0,1,2$.

References

• J. E. Cremona and M. Sadek, Local and global densities for Weierstrass models of elliptic curves, Math. Res. Lett. 28 (2021), 413–461.

• M. Griffin, K. Ono and W.-L. Tsai, Tamagawa products of elliptic curves, J. Number Theory 247 (2023), 182–201.

• B. Poonen and E. Rains, Random maximal isotropic subspaces and Selmer groups, J. Amer. Math. Soc. 25 (2012), 245–269.

Abstract

We propose a rank-conditioned distribution for ordinary and signed cyclotomic Iwasawa $\lambda$-invariants of elliptic curves over $\mathbb Q$. After removing the zero forced by Mordell–Weil rank, the remaining half-difference is conjectured to be geometrically distributed, with asymptotic independence across primes and between the two signed invariants at a supersingular prime.

The conditioned family

Let $S$ be a finite set of primes $p\geq11$, let $r\geq1$, and prescribe for each $p\in S$ either ordinary or supersingular reduction, denoted by a type function $\tau$. Let $F_{S,r,\tau}(X)$ be the set of $\mathbb Q$-isogeny classes $[E]$ of non-CM elliptic curves $E/\mathbb Q$ satisfying $N_E\leq X$ and $\mathop{\mathrm{rank}}E(\mathbb Q)=r$, together with the following conditions at every $p\in S$:

1. $E$ has good reduction at $p$ of the prescribed type;

2. the residual representation on $E[p]$ is surjective;

3. $p\nmid\mathop{\mathrm{Tam}}(E)$;

4. the relevant analytic cyclotomic $\mu$-invariant vanishes;

5. in the ordinary case, $p\nmid\#E(\mathbb F_p)$.

Put

$$I_\tau=\{(p,0):\tau(p)=\mathrm{ord}\} \cup\{(p,+),(p,-):\tau(p)=\mathrm{ss}\}.$$

For ordinary $p$, define

$$K_{p,0}(E)=\frac{\lambda_p(E)-r}{2},$$

and for supersingular $p$, using the Pollack signed normalization, define

$$K_{p,\pm}(E)=\frac{\lambda_p^\pm(E)-r}{2}.$$

Conjecture 1. Assume that $\#F_{S,r,\tau}(X)\to\infty$. Then, for every vector $\mathbf k=(k_{p,s})_{(p,s)\in I_\tau}\in\mathbb Z_{\geq0}^{I_\tau}$,

$$\lim_{X\to\infty} \frac{\#\{[E]\in F_{S,r,\tau}(X):K_{p,s}(E)=k_{p,s} \text{ for all }(p,s)\in I_\tau\}} {\#F_{S,r,\tau}(X)} \mathrel{=} \prod_{(p,s)\in I_\tau}(1-p^{-1})p^{-k_{p,s}}.$$

In particular, all the coordinates are asymptotically independent, including the two signs at a single supersingular prime.

Numerical evidence

The available Iwasawa data cover conductor at most $149996$. After passing to $\mathbb Q$-isogeny classes, the rank-one ordinary sample at $p=11$ contained $204573$ classes. The observed counts for $K_{11,0}=0,1,2$ and $K_{11,0}\geq3$ were

$$186137,\qquad 16792,\qquad 1427,\qquad 217,$$

close to the geometric probabilities $10/11$, $10/121$, $10/1331$ and the remaining tail.

For $p=13$ supersingular and rank $1$, a sample of $23707$ classes gave

$$\begin{array}{c|rrrr} (K_{13,+},K_{13,-})&(0,0)&(1,0)&(0,1)&(1,1)\\ \hline \text{count}&20240&1516&1566&127. \end{array}$$

These figures are close to the prediction from two independent geometric variables. Same-prime signed covariances were small, and the largest absolute observed covariance was approximately $0.00211$.

For the pair of primes $11$ and $13$ in rank $1$, the observed probabilities that all relevant coordinates were minimal were

$$0.839467,\quad0.778247,\quad0.763577,\quad0.701206$$

for the ordinary/ordinary, ordinary/supersingular, supersingular/ordinary and supersingular/supersingular strata. The corresponding product predictions were

$$0.839161,\quad0.774610,\quad0.762873,\quad0.704191.$$

The same comparison was performed for all $55$ pairs among the primes from $11$ through $47$ for which reliable data were available.

Random-coefficient heuristic

After extracting the rank-forced factor $T^r$ from a self-dual $p$-adic $L$-function with $\mu=0$, functional-equation symmetry forces the remaining Weierstrass degree to have the same parity as $r$. Write the symmetry-compatible coefficients as

$$a_0,a_1,a_2,\ldots$$

in degrees $r,r+2,r+4,\ldots$. If their reductions modulo $p$ behaved as independent uniform elements of $\mathbb F_p$, then

$$K=\min\{j:a_j\neq0\}$$

would satisfy

$$\Pr(K=k)=(1-p^{-1})p^{-k}.$$

Independence of the coefficient systems would yield the product law in Conjecture 1. The local exclusions in the definition of the family remove deterministic contributions, but they do not prove this randomness.

Relation with known results

Ordinary and signed Iwasawa theory provide the parity constraints and the two supersingular invariants. Existing arithmetic-statistical results give bounds or positive-density statements for prescribed Iwasawa invariants, and recent work gives criteria for the minimal rank-one event $\mu=0$, $\lambda=1$. None of these results proves the full geometric tail, fixed-rank conductor-ordered limits, cross-prime independence, or independence of the two signed invariants at one supersingular prime.

The latter is particularly delicate: the two signed $p$-adic $L$-functions have different Coleman maps, but share the underlying Mordell–Weil lattice and may share factors arising from Tate–Shafarevich or fine Selmer groups.

Open difficulties

Even the subcase consisting of one ordinary prime, rank $1$, and $K=0$ would require proving a conductor-ordered density exactly equal to $1-p^{-1}$. The full statement asks for simultaneous equidistribution of all symmetry-compatible coefficients, conditional on rank, residual surjectivity, vanishing $\mu$-invariant and the stated local restrictions. No theorem presently controls these correlations, either across different primes or between the two signed theories at one prime.

References

• R. Greenberg, Iwasawa theory for elliptic curves, in Arithmetic Theory of Elliptic Curves, Lecture Notes in Math. 1716, Springer, 1999, 51–144.

• S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), 1–36.

• R. Pollack, On the $p$-adic $L$-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), 523–558.

• D. Kundu and A. Ray, Statistics for Iwasawa invariants of elliptic curves, Parts I–III, 2021–2024.

• F. Nuccio Mortarino Majno di Capriglio and R. Sujatha, Residual supersingular Iwasawa theory and signed Iwasawa invariants, J. Number Theory 209 (2020), 224–249.

Abstract

We conjecture that monogenicity and unit-signature rank are asymptotically independent in the discriminant-ordered family of totally real non-Galois cubic fields. We present the numerical ratios motivating the conjecture and discuss why existing results on monogenic fields and on narrow class groups do not presently imply it.

Statement of the conjecture

For $X>0$, let

$$F(X)=\left\{K:\ [K:\mathbb Q]=3,\ r_2(K)=0,\ \mathop{\mathrm{Gal}}(\widetilde K/\mathbb Q)\simeq S_3,\ |\mathop{\mathrm{Disc}}K|\leq X\right\},$$

where fields are counted up to $\mathbb Q$-isomorphism. Let

$$M(X)=\{K\in F(X):\mathcal O_K=\mathbb Z[\alpha]\text{ for some }\alpha\in\mathcal O_K\}$$

be the subfamily of monogenic fields.

For a totally real cubic field $K$, define its unit-signature rank by

$$\operatorname{sgnrk}(K)= \dim_{\mathbb F_2}\mathop{\mathrm{im}}\bigl(\operatorname{sgn}:\mathcal O_K^\times\longrightarrow\{\pm1\}^3\bigr).$$

For $s\in\{1,2,3\}$, put

$$F_s(X)=\{K\in F(X):\operatorname{sgnrk}(K)=s\}, \qquad M_s(X)=M(X)\cap F_s(X).$$

Conjecture 1. For each $s\in\{1,2,3\}$,

$$\lim_{X\to\infty} \frac{\#M_s(X)\,\#F(X)}{\#M(X)\,\#F_s(X)}=1.$$

Equivalently, the limiting distribution of the unit-signature rank is unchanged after conditioning on monogenicity.

Numerical evidence

The proportions of signature ranks $1,2,3$ among monogenic fields at several discriminant cutoffs were as follows.

$X$	rank $1$	rank $2$	rank $3$
$100000$	$0.51\%$	$56.87\%$	$42.62\%$
$500000$	$0.76\%$	$60.03\%$	$39.22\%$
$1000000$	$1.05\%$	$60.53\%$	$38.42\%$
$2000000$	$1.29\%$	$61.31\%$	$37.41\%$

Unit-signature ranks among monogenic fields.

These proportions drift toward the Dummit–Voight full-family prediction

$$(0.019097,\ 0.618304,\ 0.362599).$$

At the largest apparently contiguous cutoff, $X=3375000$, the data contain

$$\#F(X)=193179,\qquad \#M(X)=75689.$$

For $s=1,2,3$, the pairs $(\#F_s,\#M_s)$ are

$$(1739,1044),\qquad(110348,46842),\qquad(81092,27803),$$

giving the finite ratios

$$1.532244,\qquad1.083423,\qquad0.875066.$$

Thus convergence to $1$ is not yet visible, especially in the rare signature-rank-one stratum.

Arithmetic motivation

Unit signatures are controlled by the archimedean part of the $2$-Selmer signature map, whereas monogenicity is the global integral condition that the binary cubic index form represent $\pm1$. These mechanisms are different enough that asymptotic independence is plausible. At the same time, monogenicity is known to alter certain $2$-class-group statistics, so the conjecture is not a formal consequence of existing heuristics.

The fieldwise exact sequence relating signatures, narrow class groups and ordinary class groups gives

$$\frac{h_K^+}{h_K}=2^{3-\operatorname{sgnrk}(K)}.$$

This identity explains the relation with narrow class groups but does not determine the distribution of the signature rank inside the monogenic subfamily.

Relation with known results

Dummit and Voight conjecture the unconditioned distribution of unit-signature ranks in totally real $S_3$-cubic fields. Results on monogenized cubic fields compute moments of ordinary and narrow $2$-torsion under a different ordering, in which a chosen generator is part of the object. They do not count each monogenic field once by field discriminant. Moreover, while a positive proportion of cubic fields is known to be nonmonogenic, no asymptotic formula is known for the number of monogenic cubic fields of bounded discriminant.

Open difficulties

A proof of Conjecture 1 requires the correlation estimate

$$\#M_s(X)\#F(X)-\#M(X)\#F_s(X) =o\bigl(\#M(X)\#F_s(X)\bigr).$$

A stronger sufficient statement would be the existence of common constants $p_s$ such that

$$\#F_s(X)\sim p_s\#F(X),\qquad \#M_s(X)\sim p_s\#M(X).$$

Neither asymptotic is presently available. The difficulty is to control simultaneously the global index-form equation defining monogenicity and the archimedean $2$-Selmer condition defining the signature rank.

References

• D. S. Dummit and J. Voight, The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units, Proc. Lond. Math. Soc. 117 (2018), 682–726.

• M. Bhargava, J. Hanke and A. Shankar, The mean number of $2$-torsion elements in the class groups of $n$-monogenized cubic fields, Forum Math. Sigma 10 (2022), e37.

• L. Alpöge, M. Bhargava and A. Shnidman, A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so, 2021.

Abstract

We study pairs of distinct totally real $S_3$-cubic fields with the same discriminant. For every prime $\ell\geq5$, we conjecture that divisibility of the two class numbers by $\ell$ is asymptotically independent under the natural ordered-pair weighting. The exclusion of $\ell=3$ is essential and reflects the $3$-primary class-field-theoretic mechanism producing common-discriminant multiplets.

Common-discriminant multiplets

For a positive discriminant $D$, let $\mathcal M_D$ denote the set of $\mathbb Q$-isomorphism classes of totally real $S_3$-cubic fields of discriminant $D$. Define the set of ordered sibling pairs

$$\overrightarrow{\mathcal P}(X) =\{(K,K'):K\neq K',\ K,K'\in\mathcal M_D \text{ for some }D\leq X\}.$$

For a prime $\ell\geq5$, put

$$\begin{aligned} u_\ell(X) &=\frac{\#\{(K,K')\in\overrightarrow{\mathcal P}(X):\ell\mid h_K\}} {\#\overrightarrow{\mathcal P}(X)},\\ b_\ell(X) &=\frac{\#\{(K,K')\in\overrightarrow{\mathcal P}(X): \ell\mid h_K\text{ and }\ell\mid h_{K'}\}} {\#\overrightarrow{\mathcal P}(X)}. \end{aligned}$$

Conjecture 1. For every prime $\ell\geq5$,

$$\#\overrightarrow{\mathcal P}(X)\longrightarrow\infty, \qquad \liminf_{X\to\infty}u_\ell(X)>0,$$

and

$$\lim_{X\to\infty}\frac{b_\ell(X)}{u_\ell(X)^2}=1.$$

Numerical evidence

In a primary sample of $414436$ ordered pairs, the results for the first few primes were as follows.

$\ell$	endpoint events	joint ordered events	$u_\ell$	$b_\ell/u_\ell^2$
$5$	$3921$	$34$	$0.00946105$	$0.9165$
$7$	$1335$	$6$	$0.003221$	$1.3952$
$11$	$234$	$0$	$0.000565$	—

Class-number divisibility among ordered sibling pairs.

For $\ell=5$, independence predicts about $37.10$ joint ordered observations, compared with $34$ observed. For $\ell=7$, the six ordered observations correspond to only three unordered double events, so the deviation has little statistical significance. The data for $\ell\geq11$ are too sparse for a meaningful test.

The contrast at $\ell=3$ is striking: every tested range had perfect synchronization between siblings. This confirms that the restriction $\ell\geq5$ is arithmetic rather than cosmetic.

Heuristic explanation

Sibling fields share their discriminant, quadratic resolvent and all discriminant-level ramification data. Their existence and multiplicity are controlled by $3$-primary ring-class information in the common quadratic resolvent. For primes $\ell\geq5$, which are coprime to $|S_3|$, Cohen–Martinet heuristics suggest that the $\ell$-primary class groups should behave as fresh random data for each sibling. Conjecture 1 asserts that this remains true even under the strong conditioning that the complete discriminants agree.

A proved part of the statement

The divergence of the number of sibling pairs follows from known lower bounds for real quadratic fields with large $3$-rank. If a real quadratic field of discriminant $d$ has $3$-rank $r$, then its unramified cyclic cubic extensions give

$$\frac{3^r-1}{2}$$

distinct totally real cubic fields of discriminant $d$. Quantitative results producing infinitely many quadratic fields with $r\geq4$ therefore imply

$$\#\overrightarrow{\mathcal P}(X)\gg \frac{X^{1/30}}{\log X}.$$

This proves the first assertion of Conjecture 1, but gives no control of good-prime divisibility in the cubic class numbers.

Open difficulties

Let

$$m_D=\#\mathcal M_D, \qquad k_D=\#\{K\in\mathcal M_D:\ell\mid h_K\}.$$

Then

$$\begin{aligned} N(X)&=\sum_{D\leq X}m_D(m_D-1),\\ A(X)&=\sum_{D\leq X}k_D(m_D-1),\\ B(X)&=\sum_{D\leq X}k_D(k_D-1). \end{aligned}$$

The desired conclusion is equivalent to

$$\liminf_{X\to\infty}\frac{A(X)}{N(X)}>0, \qquad \frac{B(X)N(X)}{A(X)^2}\longrightarrow1.$$

This is a second-factorial-moment problem with the strongly size-biased weight $m_D(m_D-1)$. Conditional independence at each fixed $D$ would not suffice if the marginal probability varied with $D$; uniformity in the quadratic resolvent, conductor and multiplet size is also required.

References

• D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831–847.

• D. C. Mayer, Classifying multiplets of totally real cubic fields, Int. J. Number Theory 18 (2022), 813–852.

• H. Cohen and J. Martinet, Class groups of number fields: numerical heuristics, Math. Comp. 48 (1987), 123–137.

• W. Wang and M. M. Wood, Moments and interpretations of the Cohen–Lenstra–Martinet heuristics, Comment. Math. Helv. 96 (2021), 339–387.

Abstract

For ordered pairs of distinct totally real $S_3$-cubic fields with the same discriminant, we conjecture that the two unit-signature ranks are asymptotically independent. The conjecture separates the $2$-primary archimedean invariant from the $3$-primary ring-class mechanism producing common-discriminant multiplets.

Statement of the conjecture

For a positive discriminant $D$, let $\mathcal M_D$ be the set of $\mathbb Q$-isomorphism classes of totally real $S_3$-cubic fields of discriminant $D$, and put

$$\overrightarrow{\mathcal P}(X) =\{(K,K'):K\neq K',\ K,K'\in\mathcal M_D \text{ for some }D\leq X\}.$$

For a totally real cubic field $K$, define

$$s(K)=\dim_{\mathbb F_2}\mathop{\mathrm{im}}\bigl(\operatorname{sgn}:\mathcal O_K^\times \longrightarrow\{\pm1\}^3\bigr)\in\{1,2,3\}.$$

For $s,t\in\{1,2,3\}$, let

$$\begin{aligned} u_s(X)&= \Pr_{(K,K')\in\overrightarrow{\mathcal P}(X)}\bigl[s(K)=s\bigr],\\ b_{s,t}(X)&= \Pr_{(K,K')\in\overrightarrow{\mathcal P}(X)} \bigl[s(K)=s,\ s(K')=t\bigr]. \end{aligned}$$

Conjecture 1. Every marginal $u_s(X)$ has positive lower limit, and for all $s,t\in\{1,2,3\}$,

$$\lim_{X\to\infty} \frac{b_{s,t}(X)}{u_s(X)u_t(X)}=1.$$

Numerical evidence

The common-discriminant sample exhibits no large persistent correlation after accounting for the one-field marginal distribution. The most stable cells are those involving signature ranks $2$ and $3$; the rank-$1$ cells are much rarer and therefore converge more slowly. Across cumulative cutoffs and in the wider stress sample, the observed joint-frequency matrix was close to the outer product of its marginals, with the largest visible discrepancies concentrated in the sparse rank-$1$ rows and columns.

The common-discriminant weighting is important. A discriminant supporting $m_D$ fields contributes $m_D(m_D-1)$ ordered pairs, so large multiplets receive substantial weight. Numerical agreement therefore tests more than independence in the ordinary one-field family.

Heuristic motivation

The multiplicity of totally real cubic fields with one discriminant is controlled by $3$-primary class or ring-class data in the common quadratic resolvent. By contrast, the unit-signature rank is an archimedean invariant tied to the $2$-Selmer signature map. This separation of primes suggests that conditioning on a large $3$-primary multiplet should not change the limiting $2$-primary signature distribution.

The conjecture is nevertheless nontrivial. Siblings share the complete discriminant and the same quadratic resolvent, and the pair measure is strongly biased toward resolvents with unusually large $3$-torsion. It is therefore necessary to control both arithmetic correlations within a multiplet and heterogeneity between different discriminants.

Relation with known heuristics

Dummit–Voight heuristics predict the one-field distribution of unit-signature ranks in totally real $S_3$-cubic fields. Common-discriminant multiplicity formulas describe the $3$-primary mechanism producing the sibling family. Neither framework gives a joint law for two distinct cubic subfields of the same ring-class construction. Existing average results for $2$-torsion under local or shape restrictions do not cover the size-biased pair measure appearing here.

Open difficulties

Write

$$m_D=\#\mathcal M_D, \qquad k_{D,s}=\#\{K\in\mathcal M_D:s(K)=s\}.$$

Then the conjecture asks for asymptotics of

$$\sum_{D\leq X}k_{D,s}k_{D,t} \quad\text{and}\quad \sum_{D\leq X}k_{D,s}(m_D-1)$$

relative to

$$\sum_{D\leq X}m_D(m_D-1).$$

The diagonal case $s=t$ requires the appropriate factorial correction because $K\neq K'$. A proof would need uniform control of the signature distribution within variable ring-class multiplets and of the tails of the $m_D(m_D-1)$ weighting. No present theorem supplies such a decorated multiplet count.

References

• D. S. Dummit and J. Voight, The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units, Proc. Lond. Math. Soc. 117 (2018), 682–726.

• D. C. Mayer, Classifying multiplets of totally real cubic fields, Int. J. Number Theory 18 (2022), 813–852.

• M. Bhargava and I. Varma, The mean number of $3$-torsion elements in the class groups and ideal groups of quadratic orders, Proc. Lond. Math. Soc. 112 (2016), 235–266.

Abstract

For a fixed imaginary quadratic field, we conjecture that finite-order twists of cyclic base change have density zero among non-CM rational Bianchi newforms of parallel weight two and trivial central character, ordered by the norm of the level. We record the observed decay in the deepest available fixed-field datasets and isolate the counting problem required for a proof.

Rational Bianchi newforms

Fix an imaginary quadratic field $K$. Let $\mathcal R_K(X)$ be the set of Galois orbits of cuspidal Bianchi newforms $F$ over $K$ satisfying

$$k_F=2,\qquad \omega_F=1,\qquad \mathbb Q_F=\mathbb Q, \qquad F\text{ non-CM},\qquad \mathrm N\mathfrak N_F\leq X,$$

where each Galois orbit is counted once. Let $\mathcal L_K(X)$ be the subset consisting of forms of the shape

$$F\simeq \mathop{\mathrm{BC}}_{K/\mathbb Q}(f)\otimes\psi,$$

where $f$ is a classical newform and $\psi$ is a finite-order Hecke character of $K$.

Conjecture 1. For every imaginary quadratic field $K$,

$$\#\mathcal R_K(X)\longrightarrow\infty \quad\Longrightarrow\quad \lim_{X\to\infty}\frac{\#\mathcal L_K(X)}{\#\mathcal R_K(X)}=0.$$

Numerical evidence

The available data contain $232143$ qualifying non-CM rational orbits over $501$ represented fields; $17053$ are marked as direct or twisted base change. Because the level coverage is highly nonuniform, the meaningful statistic is fixed-field decay rather than the pooled proportion.

For the five deepest datasets, the cumulative lifted proportion changed as follows:

$\mathop{\mathrm{Disc}}(K)$	proportion at $X=1000$	terminal proportion
$-3$	$0.169231$	$0.031664$
$-4$	$0.130890$	$0.029200$
$-7$	$0.126582$	$0.027332$
$-8$	$0.142045$	$0.034867$
$-11$	$0.163077$	$0.034765$

Decay of the lifted proportion in the deepest fixed-field datasets.

The final disjoint high-level bins in these five fields contain approximately $1.7\%$–$2.6\%$ lifted forms. All nine fields with substantial data beyond level norm $1000$ showed a lower terminal cumulative ratio than at $X=1000$.

Why density zero is plausible

Base change and its finite-order twists arise from a lower-dimensional lifting construction. The full Bianchi cohomology should contain genuinely three-dimensional automorphic phenomena that become dominant as the level grows. The observed concentration of lifted forms at very small level and their subsequent decay is consistent with this expectation.

The rationality restriction makes the conjecture substantially harder than a dimension comparison. Complex dimensions weight a Galois orbit by the degree of its Hecke field, whereas $\mathcal R_K(X)$ counts only degree-one orbits.

A lower bound for the lifted family

The growth assumption in Conjecture 1 is in fact easy to satisfy. Fix a non-CM elliptic curve over $\mathbb Q$ and let $\Pi$ be its cyclic base change to $K$. Twisting $\Pi$ by quadratic Hecke characters $\chi/K$ preserves parallel weight $2$, rational Hecke field, non-CM status and trivial central character. Away from a fixed finite set,

$$\mathfrak n(\Pi\otimes\chi)=\mathfrak n(\Pi)\mathfrak f(\chi)^2.$$

Counting quadratic characters by conductor therefore yields

$$\#\mathcal L_K(X)\gg_K X^{1/2}.$$

Consequently $\#\mathcal R_K(X)\to\infty$ for every $K$. This lower bound does not determine the ratio: the full rational family may grow faster, at the same rate, or irregularly.

Relation with known results

Dimension formulae for lifted and non-genuine Bianchi subspaces are available at certain squarefree, Galois-stable levels. They concern full complex dimensions and do not count rational Galois orbits. Published finite computations of rational Bianchi forms provide evidence but no varying-level asymptotic. The possibility that twisting a classical form with nonrational coefficient field produces a rational Bianchi form also prevents one from counting the numerator using only rational classical sources.

Open difficulties

A proof requires

$$\#\mathcal L_K(X)=o\bigl(\#\mathcal R_K(X)\bigr).$$

A plausible upper bound for the numerator is

$$\#\mathcal L_K(X)\ll_{K,\varepsilon}X^{1/2+\varepsilon},$$

but this must include ramified finite-order twists and sources whose rationality changes after twisting. The essential missing input is a lower bound for the denominator that grows faster than $X^{1/2}$, or a direct abundance theorem for genuine rational Bianchi forms. Cohomological dimension estimates do not provide such a bound because they do not isolate Hecke-field degree one.

References

• R. P. Langlands, Base Change for $\mathrm{GL}(2)$, Annals of Mathematics Studies 96, Princeton University Press, 1980.

• M. H. Şengün and P. Tsaknias, Dimension formulae for spaces of lifted Bianchi modular forms, Math. Comp. 83 (2014), 1469–1490.

• A. D. Rahm and P. Tsaknias, Genuine Bianchi modular forms of higher level, at varying weight and discriminant, J. Théor. Nombres Bordeaux 31 (2019), 617–646.

• J. E. Cremona, L. Dembélé, A. Pacetti, C. Schembri and J. Voight, On rational Bianchi newforms and abelian surfaces with quaternionic multiplication, Math. Comp. 91 (2022), 1491–1515.

• D. J. Wright, Distribution of discriminants of abelian extensions, Proc. Lond. Math. Soc. 58 (1989), 17–50.

Abstract

We conjecture that the two global root numbers occur with equal limiting frequency among genuine non-CM rational Bianchi newforms of parallel weight two and trivial central character over a fixed imaginary quadratic field, ordered by level norm. We describe the finite-level bias visible in current data and explain the obstruction created by constant-sign quadratic-twist families.

The family and the conjecture

Retain the notation $\mathcal R_K(X)$ and $\mathcal L_K(X)$ for rational non-CM Bianchi newforms and their twisted-base-change subfamily. Define the genuine family by

$$\mathcal G_K(X)=\mathcal R_K(X)\setminus\mathcal L_K(X).$$

For $F\in\mathcal G_K(X)$, let $w(F)\in\{\pm1\}$ be the sign in the functional equation of its completed standard $L$-function.

Conjecture 1. For every imaginary quadratic field $K$, if $\#\mathcal G_K(X)\to\infty$, then for each $\varepsilon\in\{\pm1\}$,

$$\lim_{X\to\infty} \frac{\#\{F\in\mathcal G_K(X):w(F)=\varepsilon\}} {\#\mathcal G_K(X)}=\frac12.$$

Equivalently, if

$$A_K(X)=\#\mathcal G_K(X), \qquad S_K(X)=\sum_{F\in\mathcal G_K(X)}w(F),$$

then Conjecture 1 is the assertion

$$S_K(X)=o\bigl(A_K(X)\bigr).$$

Numerical evidence

The available data contain $215090$ genuine non-CM rational orbits, of which $98391$ have sign $+1$ and $116699$ have sign $-1$. The pooled proportion is not the conjectured statistic because the level coverage differs strongly from one field to another.

For the deepest fixed-field datasets, the terminal positive proportions are:

$\mathop{\mathrm{Disc}}(K)$	$-3$	$-4$	$-7$	$-8$	$-11$	$-19$	$-23$	$-31$
$\Pr(w=+1)$	.4470	.4459	.4514	.4573	.4729	.5005	.4712	.4725

Positive root-number proportions at the largest available fixed-field cutoffs.

The deepest fields show an initial bias toward sign $-1$ that weakens as the norm cutoff grows. For example, the field of discriminant $-3$ moves from positive proportion $0.4280$ at norm $10000$ to $0.4470$ at norm $149968$. Both signs occur in every meaningfully populated fixed-field sample.

Heuristic motivation

The global root number is a product of local signs. Once CM forms and all base-change mechanisms are removed, no visible global symmetry forces one sign. This suggests cancellation in a sufficiently large fixed-field level family, in analogy with classical sign-equidistribution results.

The rationality and genuineness restrictions are global and very thin. A trace formula on the full newspace gives a weighted sum

$$\sum_F [\mathbb Q_F:\mathbb Q]w(F),$$

not the unweighted sum over rational genuine orbits required here. There is no known projector imposing simultaneously $\mathbb Q_F=\mathbb Q$, non-CM status and exclusion of every twisted-base-change orbit.

The quadratic-twist obstruction

There are rational genuine level-one forms over $K=\mathbb Q(\sqrt{-643})$. Let $\pi$ be one such form. For a quadratic Hecke character $\chi/K$, twisting preserves parallel weight $2$, rational Hecke eigenvalues, trivial central character, non-CM status and genuineness. Since $\pi$ is unramified at every finite place, the local twisting formula and the product formula give

$$w(\pi\otimes\chi)=w(\pi)$$

for every quadratic $\chi$. Non-CM status rules out nontrivial quadratic self-twists, so this produces infinitely many distinct genuine rational forms with the same root number.

This does not disprove Conjecture 1: the constant-sign twist family may have density zero inside $\mathcal G_K(X)$. It does show that equidistribution cannot be proved by a naive sign-reversing quadratic-twist pairing.

Relation with known results

Classical trace-formula results prove root-number equidistribution in full newspaces, and recent automorphic results treat self-dual families in other aspects. These theorems do not isolate the rational genuine Bianchi locus. Existing structural work describes the genuine/non-genuine decomposition and provides examples of rational genuine forms, but no signed counting theorem in the level aspect is known.

Open difficulties

For each fixed $K$, one needs both a sufficiently regular lower bound for $A_K(X)$ and the signed estimate

$$\sum_{F\in\mathcal G_K(X)}w(F)=o\bigl(A_K(X)\bigr).$$

It is unknown whether the explicit constant-sign quadratic-twist families are negligible, have positive density, or are balanced by other twist classes. A proof would therefore require a counting theory for genuine rational Bianchi twist classes that is substantially finer than current dimension formulae.

References

• A. D. Rahm and P. Tsaknias, Genuine Bianchi modular forms of higher level, at varying weight and discriminant, J. Théor. Nombres Bordeaux 31 (2019), 617–646.

• T. Berger, L. Dembélé, A. Pacetti and M. H. Şengün, Theta lifts of Bianchi modular forms and applications to paramodularity, J. Lond. Math. Soc. 92 (2015), 353–370.

• K. Martin, Refined dimensions of cusp forms, and equidistribution and bias of signs, J. Number Theory 188 (2018), 1–17.

• J. E. Cremona, L. Dembélé, A. Pacetti, C. Schembri and J. Voight, On rational Bianchi newforms and abelian surfaces with quaternionic multiplication, Math. Comp. 91 (2022), 1491–1515.

• T. Dokchitser and V. Dokchitser, Elliptic curves with all quadratic twists of positive rank, Acta Arith. 137 (2009), 193–197.