The Hodge conjecture for Fermat fourfolds of degree at most 100, and the semi-standard obstruction at degree 70
Abstract
Let X_m^4 = {x_0^m + ... + x_5^m = 0} in P^5 be the Fermat fourfold of degree m. Combining Shioda's inductive structure, Ran's and Aoki's explicit cycles, Aoki's structure theorem for the module of Hodge characters, and an exhaustive machine census with independently checkable certificates, we verify the Hodge conjecture for X_m^4 for every m <= 100 except m = 70. For m = 70 we show that the remaining Hodge classes form a single Galois orbit which generates the 2-torsion quotient B_70/(S_70 + D_70) = Z/2 of Kubert–Aoki, that it is equivalent to Aoki's semi-standard class xi_35 = (1,2,16,17,21,22,30,31) on the Fermat sixfold X_35^6, that xi_35 x xi_35 is algebraic, and that xi_35 is the graph of a Hodge isomorphism (a similitude of the intersection forms) between two rank-24 Hodge structures of K3 type with maximal complex multiplication by Q(zeta_35). We also determine, from Jacobi sums, the Kummer extension K_theta over which any cycle representing this class must be defined, prove that K_theta is not abelian over Q (so no cycle defined over a cyclotomic field, at any level, can represent the class), identify the quadratic part of the Kummer generator as a Gauss period of mu_7 evaluated at a primitive 10-th root of unity, show that the Kummer class of any split-cover datum is the product prod c_i^{a_i} of its constants, and prove that no datum with at most five curves of degree <= 9 at level 70, and none with at most six surfaces of degree <= 4 at level 35 (six surfaces: the 23 special members are handled by the local class groups along their singular curves), can represent the class for any value of the constants (Newton polytopes, Gröbner bases, class groups of the special members and a Galois-eigencharacter criterion on their singular points). All computations are reproducible from the accompanying scripts.
Related
- Integral Brauer--Manin Obstruction for Fermat Near Misses
- The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds
- Ambient Chow Rings of Debarre--Voisin Fourfolds and the Beauville--Voisin Conjecture
- The Tate conjecture for abelian fourfolds over finite fields
- The Multiplicative Persistence Conjecture: Resolving the 2-Adic Obstruction for Nonzero Even Targets
BibTeX
@misc{rinaldi2026hodge,
title = {{The Hodge conjecture for Fermat fourfolds of degree at most 100, and the semi-standard obstruction at degree 70}},
author = {Oana Unciuleanu Rinaldi},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05756263v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hodge conjecture for Fermat fourfolds of degree at most 100, and the semi-standard obstruction at degree 70
AU - Oana Unciuleanu Rinaldi
PY - 2026
DA - 2026/09/20
PB - HAL
UR - https://hal.science/hal-05756263v1
AB - Let X_m^4 = {x_0^m + ... + x_5^m = 0} in P^5 be the Fermat fourfold of degree m. Combining Shioda's inductive structure, Ran's and Aoki's explicit cycles, Aoki's structure theorem for the module of Hodge characters, and an exhaustive machine census with independently checkable certificates, we verify the Hodge conjecture for X_m^4 for every m <= 100 except m = 70. For m = 70 we show that the remaining Hodge classes form a single Galois orbit which generates the 2-torsion quotient B_70/(S_70 + D_70) = Z/2 of Kubert–Aoki, that it is equivalent to Aoki's semi-standard class xi_35 = (1,2,16,17,21,22,30,31) on the Fermat sixfold X_35^6, that xi_35 x xi_35 is algebraic, and that xi_35 is the graph of a Hodge isomorphism (a similitude of the intersection forms) between two rank-24 Hodge structures of K3 type with maximal complex multiplication by Q(zeta_35). We also determine, from Jacobi sums, the Kummer extension K_theta over which any cycle representing this class must be defined, prove that K_theta is not abelian over Q (so no cycle defined over a cyclotomic field, at any level, can represent the class), identify the quadratic part of the Kummer generator as a Gauss period of mu_7 evaluated at a primitive 10-th root of unity, show that the Kummer class of any split-cover datum is the product prod c_i^{a_i} of its constants, and prove that no datum with at most five curves of degree <= 9 at level 70, and none with at most six surfaces of degree <= 4 at level 35 (six surfaces: the 23 special members are handled by the local class groups along their singular curves), can represent the class for any value of the constants (Newton polytopes, Gröbner bases, class groups of the special members and a Galois-eigencharacter criterion on their singular points). All computations are reproducible from the accompanying scripts.
ER -