Infinitely many non-jumping specializations for arbitrary nonisotrivial elliptic curves over $\mathbb{Q}(T)$

For a nonisotrivial elliptic curve $E/\mathbb{Q}(T)$ — an algebraic one-parameter family of elliptic curves with non-constant $j$-invariant — of generic Mordell–Weil rank $r$, Silverman's specialization theorem gives $\operatorname{rank}E_t(\mathbb{Q})\ge r$ for all but finitely many $t\in\mathbb{Q}$. Zywina (2025) produced the first unconditional example in which equality holds for infinitely many $t$, and left as the paper's open problem the general conjecture: for every nonisotrivial $E/\mathbb{Q}(T)$, the non-jumping set $\mathcal{N}(E)=\{t\in\mathbb{Q}:\operatorname{rank}E_t(\mathbb{Q})=r\}$ is infinite, with the stronger expectation that $\mathcal{N}(E)$ has positive density with respect to the natural height. All known unconditional results (rank-0 and rank-2 families over $\mathbb{Q}$, ranks 0 through 4 over any number field, and higher-genus analogues) are proved only for specially engineered families whose specializations have few, explicitly analyzable bad primes, using deep theorems on prime patterns. The problem asks for a proof, or a refutation by explicit counterexample, of the general conjecture for arbitrary nonisotrivial families.

Background

An elliptic curve over the rational function field $\mathbb{Q}(T)$ is a one-parameter algebraic family of elliptic curves: a Weierstrass equation $y^{2}=x^{3}+a(T)x^{2}+b(T)x+c(T)$ with coefficients in $\mathbb{Q}[T]$ and discriminant $\Delta(T)$ defines, for each rational $t$ with $\Delta(t)\neq 0$, an elliptic curve $E_t/\mathbb{Q}$ by substituting $T=t$ (an elliptic curve being a smooth genus-one curve with a chosen base point, whose rational points form a finitely generated abelian group by the Mordell–Weil theorem). The family is nonisotrivial when its $j$-invariant — the standard modulus of an elliptic curve — is not constant, i.e. does not lie in $\mathbb{Q}$; geometrically, such a family corresponds to an elliptic surface over $\mathbb{P}^1_{\mathbb{Q}}$ with varying moduli. The group $E(\mathbb{Q}(T))$ is finitely generated (Lang–Néron theorem), and its rank $r$ is called the generic rank of the family. A theorem of Silverman says that the specialization homomorphism $E(\mathbb{Q}(T))\to E_t(\mathbb{Q})$ is injective for all but finitely many $t$, so $\operatorname{rank}E_t(\mathbb{Q})\ge r$ for all but finitely many $t\in\mathbb{Q}$; specializations where the rank is strictly larger are said to have a rank jump. Writing $\mathcal{N}(E)=\{t\in\mathbb{Q}:\Delta(t)\neq 0,\ \operatorname{rank}E_t(\mathbb{Q})=r\}$ for the non-jumping set, the basic question is whether $\mathcal{N}(E)$ is infinite — and conjecturally whether it even has positive density in $\mathbb{Q}$ with respect to the natural height $H(a/b)=\max(|a|,b)$ on $\mathbb{Q}$ (with $a/b$ in lowest terms, $b>0$). The nonisotriviality hypothesis is essential: Cassels and Schinzel exhibited a nonconstant isotrivial family (constant $j$-invariant) of generic rank 0 whose every specialization has global root number $-1$ — the root number being the sign in the functional equation of the Hasse–Weil $L$-function of the curve, linked to the rank by the parity conjecture, which predicts $(-1)^{\operatorname{rank}}$ equals the root number — so the parity conjecture forces a jump for every $t$ and $\mathcal{N}(E)$ is expected to be empty; in the isotrivial (essentially quadratic-twist) case both behaviors are comparatively well understood. For nonisotrivial families the heuristic picture, developed by Helfgott and by Conrad–Conrad–Helfgott in the function-field setting and revisited by Caro–Pasten, combines Silverman's density conjecture (that $\operatorname{rank}E_t(\mathbb{Q})\in\{r,r+1\}$ for a density-one set of $t$) with the prediction that the average root number over the family lies strictly between $-1$ and $1$: both $\mathcal{N}(E)$ and its complement should then have positive density. Before 2025 no nonisotrivial example with infinite $\mathcal{N}(E)$ was known unconditionally. The family of Neumann and Setzer gives rank-0 fibers at every integer $t$ for which $t^2+64$ is prime, such $t$ being infinite in number only under the Bunyakovsky conjecture (the conjecture that an irreducible polynomial with positive leading coefficient and no fixed prime divisor takes infinitely many prime values); Caro and Pasten proved that the Legendre surface $y^2=x(x+1)(x+T)$ has rank-0 fibers at $t=2^q$ for every Mersenne prime $2^q-1$ with $q\ge 5$ (a Mersenne prime being a prime of this form), making their example conditional on the conjectured infinitude of Mersenne primes. Zywina (2025) then gave the first unconditional example, $y^2=x(x^2-x+T)$: for positive integers $m,n$ with $m$, $m+n$, $m+2n$ all primes congruent to $3\pmod 8$ (infinitely many such 3-term arithmetic progressions of primes exist by a theorem of Green, later extended by Green–Tao), the specialization at $t=(m+n)/(2m)$ has rank 0, proved by a 2-descent — the standard technique of computing a Selmer group, a finite effectively computable group that bounds $E_t(\mathbb{Q})/2E_t(\mathbb{Q})$ and hence the rank — which is possible because these fibers have bad reduction at only the four primes $2$, $m$, $m+n$, $m+2n$, all explicitly analyzable. The same strategy of few controllable bad primes plus descent, with prime-pattern theorems (Green, Green–Tao, Tao–Ziegler's theorem on prime values of polynomial patterns, and Kai's number-field analogue) supplying the specializations, underlies all subsequent unconditional results: a rank-2 family over $\mathbb{Q}$, ranks 0 through 4 over arbitrary number fields, and hyperelliptic analogues. What none of these provides is an argument for an arbitrary family $E/\mathbb{Q}(T)$, where the prime divisors of $\Delta(t)$ follow no exploitable pattern. Zywina's Conjecture 1.1 — that $\mathcal{N}(E)$ is infinite for every nonisotrivial $E/\mathbb{Q}(T)$ — and its positive-density strengthening remain open, and no unconditional positive-density statement for $\mathcal{N}(E)$ is known for even a single nonisotrivial family.

Problem Statement

Let $E/\mathbb{Q}(T)$ be an arbitrary nonisotrivial elliptic curve (non-constant $j$-invariant) with generic rank $r=\operatorname{rank}E(\mathbb{Q}(T))$, given by a Weierstrass model over $\mathbb{Q}[T]$ with discriminant $\Delta$, and define the non-jumping set $\mathcal{N}(E)=\{t\in\mathbb{Q}:\Delta(t)\neq 0,\ \operatorname{rank}E_t(\mathbb{Q})=r\}.$ Prove or refute the conjecture (Conjecture 1.1 of Zywina 2025, equivalently the non-jumping half of Conjecture 1.1 of Caro–Pasten 2023): $\mathcal{N}(E)$ is infinite for every nonisotrivial $E/\mathbb{Q}(T)$. The stronger expected form, likewise open, is that $\mathcal{N}(E)$ has positive density with respect to the natural height $H(a/b)=\max(|a|,b)$ on $\mathbb{Q}$ in lowest terms, i.e. $\liminf_{X\to\infty}\frac{\#\{t\in\mathcal{N}(E):H(t)\le X\}}{\#\{t\in\mathbb{Q}:H(t)\le X\}}>0;$ a proof of this stronger statement for all nonisotrivial $E$ resolves the conjecture a fortiori and is the fully expected answer. A refutation consists of an explicit nonisotrivial $E/\mathbb{Q}(T)$ together with an unconditional proof that $\mathcal{N}(E)$ is finite. The statement concerns arbitrary nonisotrivial families: answers restricted to specially constructed families, to a fixed generic rank, to particular geometric types of elliptic surfaces, or conditional on unproved conjectures (parity, Birch–Swinnerton-Dyer, Bunyakovsky-type or Mersenne-type prime infinitude, average-root-number conjectures) do not settle the problem.

References

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