The Schloss Bisection and the Schloss Twist: Rank Jumps from the Pole Divisor of the Isosceles Pedal Correspondence, Two Elliptic K3 Surfaces, Additive Fibres at the Dynamical Thresholds, and the Exact Regulator
Year: 2026
Type: article
Abstract: The differential omega_t = ds/(2 s L_t(s) y) inherited from the isosceles pedal correspondence has four simple poles: two rational, two conjugate over Q(t)(sqrt d) with d = (3t-1)(t+1). This paper turns that marked geometry into arithmetic. THE SCHLOSS BISECTION. The non-rational pole pair is a bisection R = (r_t, sqrt(d)/(2t)), r_t = -(1-t)/(2t). Because the hyperelliptic involution is the group inverse, R is ANTI-INVARIANT under the Galois involution: sigma(R) = -R. Hence R defines a rational point of the quadratic twist E^(d) (the SCHLOSS TWIST), with POLYNOMIAL coordinates R^(d) = ( 3(3t-1)(t^3-t^2+6t-4), -108 t (3t-1)^2 ). Infinite order is certified by an exact Nagell-Lutz computation at t = -5, where I(-5) = 1729 and the fibre is Y^2 = X^3 - 46683X + 3887082: there R = (138,270) and 2R = (39369/400, 3966597/8000) is non-integral. RANK JUMPS. The +/- eigenspaces of Galois are orthogonal for the height pairing, so R is independent of the two known sections WITHOUT any height computation. With rank MW = 2 for the base surface this gives rank E^(d)(Q(t)) >= 1 and rank MW >= 3 after base change. The conic z^2 = 3t^2+2t-1 is rational, parametrised by t(m) = (m^2-4m+5)/(m^2-3), z(m) = -2(m-1)(m-3)/(m^2-3), and specialisation yields rank E_{t(m)}(Q) >= 3 > 2 for all but finitely many rational m, with three explicit independent points. For every integer m >= 6 the parameter t(m) lies INSIDE the Jonah interval (1/2,1). The three degenerate parameters have a complete dictionary: m=1 (t=-1) gives R = O; m=2 (t=1) gives R = P; m=3 (t=1/3) gives R = Q = -2P. TWO ELLIPTIC K3 SURFACES. The base change along the conic has fibres 2I_2 + 2I_6 + 8I_1 (sum of Euler numbers 24, sum (m_v-1) = 12). The Schloss twist has fibres I_2 + 4I_1 + 2I_0* + I_6 (sum 24, sum (m_v-1) = 14). Its two ADDITIVE fibres sit exactly at t = 1/3 and t = -1 — precisely the parameters where the pole divisor of omega_t degenerates, t = 1/3 being the common zero of the two multipliers of the Jonah saddle (both carry the factor 3t-1). This is the SCHLOSS MARKED-DEGENERATION THEOREM: the marked geometry of a plane dynamical system becomes additive reduction of an arithmetic surface. Consistency: the ramified quadratic base change at I_0* gives good reduction, so 4+8+12+0+0 = 24 matches the base-change model. EXACT REGULATOR. All four local contributions of R^(d) are determined: (c_2, c_*, c_*', c_6) = (0, 1, 0, 3/2). At t=0 the point is (12,0), the smooth third root of (X+6)^2(X-12); at t=-1 it is (144,1728), a smooth point of the cuspidal fibre Y^2 = X^3; at t=1/3 it is the cusp (0,0). The component index at I_6 is FORCED to be Theta_3 — the same component as the equilateral section — by combining Galois orthogonality with the integrality and non-negativity of intersection numbers: among the five possible indices only k=3 yields admissible epsilons. The exact intersection factorisations X'_R - X'_P = -36(m-2)(m^2-3)^2/(m-1)^2, X'_R - X'_E = -18(m^2-3)^3/(m-1)^2 give (R.O) = 1 and a single transverse meeting with P at m=2. Hence h(R^(d)) = 3/2, h(R) = 3 on the base-change K3, and the height matrix of (P, E, R) is diag(1/3, 1, 3), of DETERMINANT 1. Three independent consistency checks agree (twist/base-change height relation, sum of contributions, positivity). SCOPE. Statements are labelled [D] proved, [E] exact computation, [L] literature not verified against the primary record, [C] open. Quadratic twists, eigenspace decompositions, Neron-Tate heights, Shioda-Tate, the local contribution tables, Tate's algorithm, Nagell-Lutz and specialisation theorems are classical and cited; rank jumps in families are an established subject and are NOT claimed as new. What is offered as new is the SOURCE of the extra class: the pole divisor of a differential canonically attached to a plane geometric dynamical system, the coincidence of the additive fibres with the dynamical thresholds, and the resulting explicit regulator. The designations "Schloss bisection", "Schloss twist", "Schloss conic", "Schloss marked-degeneration theorem", "Schloss rank-jump theorem" and "Schloss regulator" are proposed labels within this series, not established terminology; each is defined by an explicit formula in a named-object index so it can be located independently of the name. Priority has not been verified and the bibliography of rank jumps in elliptic surfaces should be searched before any claim of precedence. No physical, quantum or thermodynamic claim is made, and no claim relating this material to the Riemann hypothesis is made. OPEN. The exact rank and torsion of either K3; the index k of the span of (P,E,R), constrained by k^2 | 144 to k in {1,2,3,4,6,12}; whether rank jumps exceed 3; the arithmetic of the degree-8 polynomial W; the analogous construction for the period-two multisections, which are points of E_t over the CM sextic fields of the trace-descent paper. REPRODUCIBILITY. schloss_bisection_exact.py (SymPy, exact rational arithmetic, no floating point) recomputes the bisection and twist identities, the polynomial coordinates of R^(d), the conic parametrisation and its derivative, both K3 discriminants and their factorisations, the intersection factorisations, the specialisation certificate at t=-5, the two elementary local contributions, and the forced-index table at I_6.
Author: Schloss Jonah
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225784817
FWCI:
Cites:
Cited by:
Related to: