A version-1 arXiv preprint reports an explicit elliptic K3 surface over Q(t) whose generic fibration has exact Mordell-Weil rank 17. In ordinary terms, the rank records independent sections, and the construction exhibits 17 of them. The same work identifies a selected fiber with rank at least 28 and states that infinitely many elliptic curves over Q have rank at least 19.
The generic result concerns the surface before a particular rational value of t is selected. The later fiber results concern individual elliptic curves, while the base-change results examine new parameter spaces built from the same construction.
Building and checking the surface
At the core is an explicit Weierstrass model, written with integer polynomials S(t) and T(t). Alongside it, the construction exhibits 17 sections of the fibration and checks their independence in the Mordell-Weil group.
That check is made through a height-pairing Gram matrix for the selected sections. Its reported determinant is 948, a nonzero value, and the paper concludes that the 17 sections are independent.
The 17 independent sections establish the rank from below. The paper combines them with the stated upper bound that an elliptic K3 surface over Q(t) has rank less than 18. Together, those two parts yield an exact generic Mordell-Weil rank of 17.
The abstract separately reports Neron-Severi rank 19 for the announced surface, alongside the fibration's Mordell-Weil rank 17. The two figures describe different rank statements in the same announcement.
Selected fibers show higher rank lower bounds
When t is specialized to selected rational values, the fibration gives individual elliptic curves, known here as fibers. The preprint identifies examples with rank lower bounds of at least 25, 26, 27 and 28.
Those numbers need careful reading. The 25 and 26 cases are not given as exact ranks. Exactness for the 27 and 28 cases is conditional on GRH, the generalized Riemann hypothesis for number fields, rather than an unconditional conclusion. Without that condition, the supplied result for the 28 case is rank at least 28.
Changing the base adds new sections
One route to a higher rank is a quadratic base change, a degree-2 map to the original t-line. It comes from a (-2)-curve intersecting the fiber class twice, which the paper calls a rational quadratic section.
In the displayed case, the extra section P_C is independent of the pulled-back Mordell-Weil group. The base-changed fibration is reported to have rank at least 18, a lower bound rather than an exact rank.
Two selected quadratic base changes are combined into a biquadratic base change over a genus-1 curve E0/Q. The construction has two new sections and a reported Mordell-Weil rank of at least 19.
For the displayed example, E0 is reported to have rank 4, and the resulting point on it has infinite order.
The search behind the example reports no vectors of norm 6 among the cosets considered modulo 2, corresponding to a finite set of distinct quadratic base changes. Long polynomials, matrices and equations are supplied in machine-readable GP/PARI syntax for copying into computer algebra systems.
The result extends beyond one surface
At the broadest level, the paper states that infinitely many elliptic curves over Q have rank at least 19. The phrase “at least” is important because the result is not presented as an exact-rank statement.
Taken together, the preprint reports an exact rank for the generic surface, selected fibers with higher lower bounds, and base changes with ranks at least 18 and 19. For the fiber reported at rank at least 28, exact rank 28 remains conditional on GRH, while the fibers reported at least 25 and 26 remain without exact ranks in the supplied account.
Paper data and sources
Original title: An elliptic K3 surface X/Q(t) with Mordell-Weil rank 17, I: Formulas for X and base changes of rank 18 and 19
Authors: Noam D. Elkies
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text