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hexagon:2610.00408v1

math.NT — Number Theory

On the irrationality of the 33-adic zeta value ζ3(5)\zeta_3(5)

Contributed by Antti Hippeläinen

We give a proof that the 33-adic zeta value ζ3(5)\zeta_3(5) is irrational. The proof adapts to p=3p=3 a Hankel-determinant method that a recent preprint applies to ζ2(7)\zeta_2(7). A Volkenborn integral of an explicit rational function defines a linear functional with values in Q+Q ζ3(5)\mathbb{Q}+\mathbb{Q}\,\zeta_3(5), and its Hankel determinants are the values at ζ3(5)\zeta_3(5) of polynomials with rational coefficients. We bound the 33-adic valuation of these values and the size and the denominators of the coefficients. The new ingredients at p=3p=3 are a triangular basis of the polynomials that are integer valued on 3−1Z3×3^{-1}\mathbb{Z}_3^\times, a local estimate at primes of the order of nn in which a pole of order 33 coming from harmonic sums cancels exactly against a zero of order 33 of the rational function, and a non-vanishing theorem at an auxiliary prime, in which Wilson's theorem makes one block of a reduced Gram matrix invertible. Here nn is the parameter of the rational function. Two constants of the estimates are bounded by finite computations, and with these bounds the 33-adic valuation exceeds the size and the denominators together by more than 4.54.5 bits per n2n^2. An appendix gives a second proof, independent of the first, which adapts to ζ3(5)\zeta_3(5) and the modular curve X0(3)X_0(3) a determinant argument that a preprint of Yi gives for ζ7(3)\zeta_7(3), and which rests on published theorems on overconvergent modular forms.

Primarily AI-generated textHuman understanding: no partsHankel determinantsKubota-Leopoldt L-functionVolkenborn integralirrationalityoverconvergent modular formsp-adic zeta values
Oct 20, 2026PDF ↗View work →
hexagon:2610.00371v1

math.AG — Algebraic Geometry

Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over Q(−3)\mathbb{Q}(\sqrt{-3}) with Hermitian determinant −2-2

Contributed by Antti Hippeläinen

Let K=Q(−3)K=\mathbb{Q}(\sqrt{-3}). We give a proof that every Weil class is algebraic on every complex abelian sixfold of Weil type for KK whose Hermitian form has signature (3,3)(3,3) and determinant −2-2 modulo norms. These forms are not split, so neither Schoen's theorem on sixfolds nor the theorem on sixfolds claimed in a preprint of Markman, both of which concern split forms, applies. We model the sixfolds by the part of the Jacobian of a cover C→P1C\to\mathbb{P}^1, branched at thirteen points, on which its group, the complex reflection group G32G_{32}, acts through its reflection representation. A degeneration argument with a ball-arithmetic certificate shows that this family dominates the nine-dimensional Shimura variety. On every member, an integral relation among characters induced from linear characters of subgroups, realized by correspondences with coefficients in the group algebra, reduces the Weil classes to the determinants of eigenspaces of eleven cyclic subcovers. A theorem of Schoen makes these determinants algebraic for the three étale subcovers; for the eight ramified ones we prove it for all cyclic covers of curves of degree at least 33 with at least 33 branch points and fully paired branching data, through a cone over a Fermat variety and Shioda's results. Specialization and isogenies then reach every sixfold. The determinant of the Hermitian form and the relation among characters are exact computations.

Primarily AI-generated textHuman understanding: no partsFermat varietiesHodge conjectureHurwitz spacesWeil classesabelian varieties of Weil typecomplex reflection groupsperiod maps
Oct 20, 2026PDF ↗View work →