Let V be a smooth cubic surface over a p-adic field k with good reduction. Swinnerton-Dyer (1981) proved that R-equivalence is trivial on V(k) except perhaps if V is one of three special types--those whose R-equivalence he could not bound by proving the universal (admissible) equivalence is trivial. We consider all surfaces V currently known to have non-trivial universal equivalence. Beyond being intractable to Swinnerton-Dyer's approach, we observe that if these surfaces also had non-trivial R-equivalence, they would contradict Colliot-Thélène and Sansuc's conjecture regarding the k-rationality of universal torsors for geometrically rational surfaces. By devising new methods to study R-equivalence, we prove that for 2-adic surfaces with all-Eckardt reductions (the third special type, which contains every existing case of non-trivial universal equivalence), R-equivalence is trivial or of exponent 2. For the explicit cases, we confirm triviality: the diagonal cubic X3+Y3+Z3+ζ3T3=0 over Q2(ζ3)--answering a long-standing question of Manin's (Cubic Forms, 1972)--and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982). This is the first in a series of works derived from a year of interactions with generative AI models such as AlphaEvolve and Gemini 3 Deep Think, with the latter proving many of our lemmas. We disclose the timeline and nature of their use towards this paper, and describe our broader AI-assisted research program in a companion report (in preparation).
We investigate the extent to which the reduced minimal Weierstrass coefficients of an elliptic curve over Q may be computed from it's Frobenius traces. Decision tree models reveal that the first two reduced minimal Weierstrass coefficients can be recovered with perfect accuracy from the Frobenius traces at the primes 2 and 3, and the third by supplementing these two traces with the conductor parity. We subsequently prove explicit formulae for these coefficients using the Frobenius traces and conductor parity. These formulae appear to be new. In particular, we deduce that the first three reduced minimal Weierstrass coefficients of an elliptic curve are determined by its isogeny class.
Barinder S. Banwait, Xiaoyu Huang, Kyu-Hwan Lee +3
We propose a new frame work of p-adic neural network. Unlike the original p-adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as activation functions, we use a single injective p-adic character on the topological Abelian group Zp of p-adic integers as an activation function. We prove the p-adic universal approximation theorem for this formulation of p-adic neural network, and reduce it to the feasibility problem of polynomial equations over the finite ring of integers modulo a power of p.
The Fresnel wave surface governs the propagation of light in a transparent biaxial crystal. It is a special Kummer quartic, and we identify it exactly. Over the complex numbers the wave surface of a crystal is the Kummer surface of the Jacobian of an explicit genus-two curve branched at the signed square roots of the three principal permittivities. This Jacobian is isogenous, by an isogeny with kernel of order four, to a product of two elliptic curves. One elliptic curve carries the three permittivities, and the other carries the optic-axis angle. The physical family is Zariski dense in the locus of genus-two curves with an extra involution, and its automorphism strata are explicit. The identification instantiates a task-aware quotient, which identifies parameters that differ by a nuisance transformation and carries invariant coordinates and explicit strata. For biaxial crystals, two ratios of the permittivities form a complete invariant of the wave surface up to rotation and rescaling, and the four real nodes are given in closed form. At an interface the candidate transmitted waves are the roots of a quartic of exact degree four. Its real-root count, root order, and repeated-root events are decided by exact algebraic predicates, and along the generic single-node encounters of the paper its discriminant vanishes to second order. Floating-point solvers drop forward transmitted modes near the optic axes, and the certified solver does not. On exact equivalence classes, learned models on quotient coordinates are invariant and more accurate than models on raw tensors, while learned root-count predicates fail near the optic axes.