A Hasse principle for the higher Chow groups of curves over a global field

2025/07/30 by Toshiro Hiranouchi, Hiranouchi, Toshiro
Mathematics · #math.NT

paper · pdf · doi:10.48550/arxiv.2507.22319

Abstract

Let X be a smooth projective curve over a global field F, and let V(X) denote the kernel of the push-forward map CH²(X,1)→ F^×. We study the mod-l structure of V(X) by combining Bloch's exact sequence with a Hasse principle in Galois cohomology associated with the mod-l representation of the Jacobian J of X. We obtain an exact sequence that describes the kernel and cokernel of the boundary map in terms of local reduction data and the coinvariant quotient J[l]_GF. As a consequence, if End _F(J)=ℤ and J has semistable reduction of toric dimension one at some place of F, then the mod-l boundary map is an isomorphism for all but finitely many primes l\neqchar(F). We also give explicit computations for elliptic curves.

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