A Prym criterion for rank preservation under Denef base change

Definitions and mathematical statement

Let $k$ be a field, $t$ transcendental over $k$, and $f\in k[X]$ a cubic chosen so that the smooth projective curve \[ E_t:\quad f(t)Y^2=f(X) \] is an elliptic curve over $k(t)$ with $\operatorname{rank}E_t(k(t))=1$. The rank of an abelian group here means the dimension of its tensor product with ${\mathbb{Q}}$. Zahidi's question asks for conditions on $u\in k(t)$ under which the analogous curve \[ E_u:\quad f(u)Y^2=f(X) \] also has rank one. Only choices giving a nonsingular elliptic curve are meaningful; the original extract does not specify the field characteristic, separability assumptions, or whether constant $u$ is allowed.

The supplied research aid treats the following precise restricted setting. Assume $\operatorname{char}k=0$, $f$ is separable of degree three, and $u\in k(t)$ is nonconstant. Let $E_0$ be the elliptic curve $y^2=f(x)$ with its point at infinity, and let $C_u$ be the smooth projective normalization of \[ s^2=f(u(t)). \] The map $h_u:C_u\to E_0$ induced by $(t,s)\mapsto(u(t),s)$ gives a homomorphism $(h_u)_*:J(C_u)\to E_0$. Put \[ P_u=\left(\ker (h_u)_*\right)^0, \] the identity component of its kernel. All endomorphisms, homomorphisms and isogenies in the following assertions are defined over $k$. The reported rank formula is \[ \operatorname{rank}E_u(k(t)) =\operatorname{rank}\operatorname{End}_k(E_0) +\operatorname{rank}\operatorname{Hom}_k(P_u,E_0). \] In particular, under $\operatorname{End}_k(E_0)={\mathbb{Z}}$, rank preservation is equivalent to \[ \operatorname{Hom}_k(P_u,E_0)=0. \] This is a criterion involving the abelian variety $P_u$, rather than a classification of all rational functions $u$ by explicit coefficients.

For the special substitution $u=t^2$, additionally assume $f(0)\ne0$. The aid reports a $k$-isogeny \[ J(C_{t^2})\ \sim_k\ E_0\times D_f, \qquad D_f:\quad w^2=z f(z). \] Here $D_f$ means the smooth projective genus-one curve with the rational point $(0,0)$ chosen as origin. The quotient coordinates are $z=t^2$ and $w=ts$ for the second factor. Consequently, when $\operatorname{End}_k(E_0)={\mathbb{Z}}$, \[ \operatorname{rank}E_{t^2}(k(t)) =1+\operatorname{rank}\operatorname{Hom}_k(D_f,E_0), \] and rank one holds exactly when $D_f$ and $E_0$ are not $k$-isogenous. A $k$-isogeny gives a rank jump in this setting. Without the endomorphism-ring hypothesis, the initial $1$ must instead be $\operatorname{rank}\operatorname{End}_k(E_0)$, as in the general formula. The $t^2$ example and the Prym criterion are supplied research claims requiring verification, and do not resolve every version of the original question.

Short English statement

Find substitutions preserving rank one for a Denef quadratic twist. In characteristic zero the supplied aid reduces the problem to the absence of maps from an associated Prym variety to the original elliptic curve; for $u=t^2$, this becomes an explicit isogeny test with $w^2=z f(z)$, under the stated endomorphism hypothesis.

Sources

• Zahidi, Question 22, American Institute of Mathematics workshop Extensions of Hilbert's Tenth Problem, https://aimath.org/WWN/hilberts10th/hilberts10th.pdf. The original question calls $f(t)Y^2=f(X)$ the Denef curve.
• Local UnsolvedMath registry, permanent ID 20002359, source code catalogue entry, canonical location aim-logic-notes.json, notes[134], record dated 14 August 2026. Source verification identifies lost superscripts and subscripts; the original literature field is empty.
• Supplied research attempt 1: partial result, problem status at run partially solved, rigor labelled proved, novelty confidence low. The Prym rank formula and explicit second quotient are machine-generated research aid, not peer reviewed; independently verify before citation. No separate authoritative bibliographic reference for these refinements is supplied.