Open problems in the Iwasawa theory of quadratic twist families of elliptic curves: density of $(\mu,\lambda)$-invariants, $2$-primary Tate-Shafarevich finiteness, supersingular $p=2$ foundations, real quadratic twists, and fine Selmer triviality

Two recent studies of ranks of elliptic curves in quadratic twist families - Hatley-Ray (arXiv:2412.07308) via 2-adic Iwasawa theory, and Kundu-Müller (arXiv:2507.21339) via half-integral weight modular forms and Selmer-group algebra - state explicitly which inputs are missing to turn their conditional results toward Goldfeld's rank distribution conjecture into unconditional and more general ones. Five source-stated gaps remain open: (1) a rigorous density theorem for elliptic curves with vanishing Iwasawa $\mu$-invariant and bounded $\lambda$-invariant, at $p=2$ in particular (the only such theorem currently known is at $p=5$); (2) finiteness of the $2$-primary Tate-Shafarevich group over the twist family, needed to convert Selmer-corank statements into rank statements; (3) the foundational supersingular $p=2$ Iwasawa theory, namely the sharp/flat Selmer groups over the cyclotomic $\mathbb{Z}_2$-extension - where cotorsionness fails - and the associated 2-adic $L$-functions; (4) a Waldspurger-type formula for real quadratic twists, extending the analytic method beyond imaginary quadratic twists; (5) verifiable density criteria for triviality of the $p$-primary fine Selmer group. All five remain open; the closest adjacent progress is Adachi (arXiv:2607.05305, 2026), which develops the analytic side of the supersingular $p=2$ theory conditional on an unproven $\mu=0$ hypothesis. Together, the five gaps point toward a common objective: an unconditional, general Iwasawa-theoretic route to rank distribution in quadratic twist families.

Background

An elliptic curve $E/\mathbb{Q}$ is a nonsingular plane cubic given by an equation $y^2=x^3+Ax+B$ with $A,B\in\mathbb{Q}$; its $\mathbb{Q}$-rational points form a finitely generated abelian group $E(\mathbb{Q})$ (Mordell-Weil theorem) whose free rank $r_{\mathrm{alg}}(E)$ is the central arithmetic invariant. For a squarefree integer $d$, the quadratic twist $E^{(d)}$ is the curve $dy^2=x^3+Ax+B$, isomorphic to $E$ over $\mathbb{Q}(\sqrt{d})$ but generally not over $\mathbb{Q}$. Goldfeld's conjecture predicts that when the twists of a fixed $E$ are ordered by $|d|$, half have rank $0$, half have rank $1$, and rank $\ge 2$ occurs for density zero [Goldfeld 1979]. The best unconditional results toward the average-rank version of this circle are due to Bhargava-Shankar, who bounded the average rank of all elliptic curves over $\mathbb{Q}$ using their computation of average sizes of Selmer groups.

Proving rank statements almost always passes through the Selmer group $\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})$: the group of Galois cohomology classes with coefficients in the $p$-power torsion $E[p^\infty]$ satisfying local Kummer conditions at every prime. Its $\mathbb{Z}_p$-corank bounds the rank, and the discrepancy with the rank is measured by the Tate-Shafarevich group $\mathrm{X}(E/\mathbb{Q})$ (the kernel of the localization map from global to product of local Galois cohomology, which measures failure of the local-global principle for principal homogeneous spaces). The Birch and Swinnerton-Dyer conjecture (BSD) predicts that $r_{\mathrm{alg}}(E)$ equals the order of vanishing of the Hasse-Weil $L$-function $L(E,s)$ at $s=1$ and that $\mathrm{X}(E/\mathbb{Q})$ is finite; both remain open in general.

Iwasawa theory studies these objects along the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}_\infty/\mathbb{Q}$, the infinite Galois tower with group isomorphic to the $p$-adic integers $\mathbb{Z}_p$, contained in the field of all $p$-power roots of unity. The $p$-primary Selmer group over $\mathbb{Q}_\infty$ is a module over the Iwasawa algebra $\Lambda=\mathbb{Z}_p[\![T]\!]$; when it is cotorsion over $\Lambda$, its structure is encoded by two invariants $\mu_p$ and $\lambda_p$ (the power of $p$ and the degree of the distinguished polynomial in a characteristic power series, via $p$-adic Weierstrass preparation), and one has the chain $r_{\mathrm{alg}}(E)\le \mathrm{corank}_{\mathbb{Z}_p}\,\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})\le \lambda_p(E/\mathbb{Q})$, the first inequality being an equality exactly when $\mathrm{X}(E/\mathbb{Q})[p^\infty]$ is finite. Greenberg conjectured that $\mu_p(E/\mathbb{Q})=0$ for curves with good ordinary reduction at $p$ and irreducible mod-$p$ representation [Greenberg 1999]. A prime of good reduction is called ordinary if the Frobenius trace is not divisible by $p$ (equivalently, is a $p$-adic unit) and supersingular otherwise. Kida-type formulas - analogues for Selmer groups and $p$-adic $L$-functions of Kida's classical formula for class groups in $\mathbb{Z}_p$-extensions - describe how $\lambda$ changes in $p$-extensions and under quadratic twists; Hatley-Ray build on the 2-adic version of Matsuno.

Hatley-Ray use Matsuno's formula to construct, for a curve $E/\mathbb{Q}$ of good ordinary reduction at $2$ with $\lambda_2(E/\mathbb{Q})=0$, many squarefree $d$ with $\lambda_2(E^{(d)})=\lambda_2(E)$ or $\lambda_2(E^{(d)})=\lambda_2(E)+2$, and deduce: if $E$ satisfies good ordinary reduction at $2$ with squarefree conductor, root number (sign of the functional equation of $L(E,s)$) $-1$, no rational $2$-torsion, $\mu_2=0$ and $\lambda_2\le 2$, then the number of squarefree $d<X$ with Selmer corank $1$ is $\gg X/(\log X)^{11/12}$, matching Goldfeld's prediction up to a power of $\log X$. Rank conclusions from these corank statements require finiteness of $\mathrm{X}(E^{(d)})[2^\infty]$. All four hypotheses are expected to hold for a positive density of curves ordered by height, but Hatley-Ray state that establishing the fourth rigorously 'remains an open problem, even for a positive density of curves'; the only proven analogue is at $p=5$ [Ray 2024]. For curves with supersingular reduction at $2$, the Selmer group over $\mathbb{Q}_\infty$ is not even $\Lambda$-cotorsion, and the signed ('sharp' $\sharp$ and 'flat' $\flat$) Selmer groups and $2$-adic $L$-functions that decompose the theory in that setting - constructed analytically by Sprung, generalizing Kobayashi's odd-prime plus/minus Selmer groups (a pair of Selmer groups splitting the local condition at a supersingular prime into its two branches) - are described by Hatley-Ray as underdeveloped at $p=2$.

Kundu-Müller approach the same circle at primes $p\ge 5$ of good ordinary reduction, from two directions. Analytically, they use the Shimura correspondence, which attaches to the weight-$2$ modular form of $E$ a half-integral weight modular form (weight $3/2$; 'half-integral weight' means weight $k+1/2$, whose Fourier coefficients carry data about quadratic twists), together with the classical formula of Waldspurger relating central values $L(E^{(-n)},1)$ of imaginary quadratic twists to squares of Fourier coefficients of the half-integral weight form; they prove constancy of analytic Iwasawa invariants across imaginary quadratic twist families, conditionally translating to algebraic invariants via the Iwasawa main conjecture (the conjectural equality of the characteristic ideal of the Selmer group with the $p$-adic $L$-function, a theorem in many ordinary cases). Algebraically, they use the BDP Selmer group (a Selmer-type group over a quadratic field introduced in work of Bertolini, Darmon, and Prasanna, combining anticyclotomic and cyclotomic local conditions; see the source for the precise definition) and the fine Selmer group $\mathrm{Sel}^0_{p^\infty}$ - the subgroup of the Selmer group in which the local condition at primes above $p$ is strengthened to the unramified quotient, capturing the $p$-primary Tate-Shafarevich group together with local torsion. They prove that under explicit hypotheses - including triviality of the $p$-primary fine Selmer group of $E/K$ - the Selmer groups of $E$ and its quadratic twist $E_K$ are isomorphic as $\Lambda$-modules along the cyclotomic tower, and they compute the density of primes $\ell$ for which the checkable hypotheses hold; but the resulting set is 'thin' among all quadratic fields, and triviality of the fine Selmer group itself is hard to verify. Their outlook states that the analytic approach is restricted to imaginary quadratic twist families because 'no analogue for real quadratic twists is known' for the Waldspurger input, and defers supersingular primes to future work.

The five gaps stated by the two sources share one objective: to convert the conditional Iwasawa-theoretic statements about rank and Selmer distribution in quadratic twist families into unconditional theorems, and to extend the method beyond the ordinary, imaginary-twist, $p\ge 5$ setting in which it currently runs.

Problem Statement

Resolve the open problems stated by Hatley-Ray (arXiv:2412.07308) and Kundu-Müller (arXiv:2507.21339) that block an unconditional Iwasawa-theoretic route to the rank distribution of quadratic twist families of elliptic curves over $\mathbb{Q}$ (Goldfeld's conjecture). The two papers state these gaps as separate open problems, each in service of the common objective of making the existing conditional results unconditional and extending them beyond the ordinary, imaginary-twist, $p\ge 5$ scope in which they currently run.

1. $(\mu,\lambda)$-density theorem at $p=2$. Prove rigorously, for a positive density of elliptic curves $E/\mathbb{Q}$ ordered by height, that $\mu_2(E/\mathbb{Q})=0$ and $\lambda_2(E/\mathbb{Q})\le 2$ - condition (4) of Hatley-Ray's Theorem A. No heuristic input and no Shafarevich-finiteness assumption may be used. The only proven analogue is the $p=5$ density theorem of Ray (arXiv:2404.09009).

2. $2$-primary Tate-Shafarevich finiteness. Establish finiteness of $\mathrm{X}(E^{(d)}/\mathbb{Q})[2^\infty]$ for the twists produced by Hatley-Ray's Theorem A - the positive-proportion family of squarefree $d<X$ counted by their Selmer-corank bound $\gg X/(\log X)^{11/12}$ - or uniformly over the quadratic twist family of a given curve, so that those Selmer-corank statements can be converted into Mordell-Weil rank statements.

3. Supersingular $p=2$ foundations. Develop the Iwasawa theory of elliptic curves with good supersingular reduction at $2$ over the cyclotomic $\mathbb{Z}_2$-extension: the algebraic theory of the sharp ($\sharp$) and flat ($\flat$) Selmer groups - structure, control theorems (comparisons between the Selmer groups over the finite layers of the tower and over the whole tower), and growth properties, in a setting where the ordinary Selmer group fails cotorsionness - together with the associated sharp/flat 2-adic $L$-functions.

4. Real quadratic twists. Determine whether a real-quadratic analogue of the Waldspurger input used in the Kundu-Müller analytic approach exists and, if so, supply the input needed to carry through the argument of their Theorem 3.1 - comparison of the evaluations of the $p$-adic $L$-functions at the trivial character - for real quadratic twist families, yielding the corresponding constancy of analytic Iwasawa invariants there. The form of such an analogue is left open. The comparison input for their argument must be supplied; the existence of a classical sign-conditioned positive-discriminant Waldspurger or Kohnen-Zagier formula alone does not supply it.

5. Fine Selmer triviality criteria. Give verifiable density criteria for triviality of the $p$-primary fine Selmer group in quadratic twist families: conditions under which the hypotheses of Kundu-Müller's algebraic theorem can be checked for a positive density of twists (beyond the thin sets currently covered), or which otherwise identify a large explicit subfamily with trivial fine Selmer group.

The five components are separate source-stated open problems, each contributing to the common objective: (1) and (2) remove the two conditional inputs in the 2-adic rank-distribution results of Hatley-Ray; (3) extends the same program to curves excluded by the good-ordinary-at-2 hypothesis; (4) and (5) widen, respectively, the analytic and algebraic tools to settings they do not currently reach. Each of the five is an independently standing open problem - the common objective above is what unifies them, not a joint condition of resolution.

References

1. Jeffrey Hatley and Anwesh Ray, Iwasawa theory and ranks of elliptic curves in quadratic twist families, arXiv:2412.07308 (2024), https://arxiv.org/abs/2412.07308
2. Debanjana Kundu and Katharina Müller, Iwasawa Theory of Elliptic Curves in Quadratic Twist Families, arXiv:2507.21339 (2025), https://arxiv.org/abs/2507.21339
3. Taiga Adachi, Iwasawa invariants of sharp/flat 2-adic L-functions for quadratic twists of elliptic curves, arXiv:2607.05305 (2026), https://arxiv.org/abs/2607.05305
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