On very general abelian varieties of dimension ≥4 (and genus-4 Jacobians), D^2 = 0 in CH^2 forces D torsion; consequently all rational sections of the genus-4 Kummer fibration are rational multiples of the Griffiths–Pirola section.
A differential approach to Ax-Schanuel, I
2 Pith papers cite this work. Polarity classification is still indexing.
2
Pith papers citing it
abstract
In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by the uniformizers. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of simple projective structure on curves including unifomizers of any Fuchsian group of the first kind and any genus. Combining our techniques with those of Ax, we give a strong Ax-Schanuel result for the combination of the derivatives of the j-function and the exponential function. In the general setting of Shimura varieties, we obtain an Ax-Schanuel theorem for the derivatives of uniformizing maps. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields.
years
2026 2representative citing papers
Classifies f-bialgebraic sets for Böttcher coordinates of polynomials and proves Ax-Lindemann-Weierstrass and Ax-Schanuel analogs when the Julia set is disconnected.
citing papers explorer
-
On the Chow ring of very general abelian varieties and a question of Pirola
On very general abelian varieties of dimension ≥4 (and genus-4 Jacobians), D^2 = 0 in CH^2 forces D torsion; consequently all rational sections of the genus-4 Kummer fibration are rational multiples of the Griffiths–Pirola section.
-
Bialgebraic geometry of B\"ottcher coordinates
Classifies f-bialgebraic sets for Böttcher coordinates of polynomials and proves Ax-Lindemann-Weierstrass and Ax-Schanuel analogs when the Julia set is disconnected.