Abstract
Suppose X is a (smooth projective irreducible algebraic) curve over
a finite field k. Counting the number of points on X over all finite
field extensions of k will not determine the curve uniquely. Actually,
a famous theorem of Tate implies that two such curves over k
have the same zeta function (i.e., the same number of points over
all extensions of k) if and only if their corresponding Jacobians are
isogenous. We remedy this situation by showing that if, instead of
just the zeta function, all Dirichlet L-series of the two curves are
equal via an isomorphism of their Dirichlet character groups, then
the curves are isomorphic up to “Frobenius twists”, i.e., up to automorphisms
of the ground field. Because L-series count points on
a curve in a “weighted” way, we see that weighted point counting
determines a curve. In a sense, the result solves the analogue of
the isospectrality problem for curves over finite fields (also know
as the “arithmetic equivalence problem”): It states that a curve is
determined by “spectral” data, namely, eigenvalues of the Frobenius
operator of k acting on the cohomology groups of all ℓ-adic sheaves
corresponding to Dirichlet characters. The method of proof is to
show that this is equivalent to the respective class field theories of
the curves being isomorphic as dynamical systems, in a sense that
we make precise.
Original language | English |
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Pages (from-to) | 9669-9673 |
Number of pages | 5 |
Journal | Proceedings of the National Academy of Sciences of the United States of America |
Volume | 110 |
Issue number | 24 |
DOIs | |
Publication status | Published - 2013 |