1,243 research outputs found
K3 surfaces over finite fields with given L-function
The zeta function of a K3 surface over a finite field satisfies a number of
obvious (archimedean and l-adic) and a number of less obvious (p-adic)
constraints. We consider the converse question, in the style of Honda-Tate:
given a function Z satisfying all these constraints, does there exist a K3
surface whose zeta-function equals Z? Assuming semi-stable reduction, we show
that the answer is yes if we allow a finite extension of the finite field. An
important ingredient in the proof is the construction of complex projective K3
surfaces with complex multiplication by a given CM field.Comment: (v2: minor corrections, added numerical evidence by Kedlaya and
Sutherland
Characteristic classes for curves of genus one
We compute the cohomology of the stack M_1 with coefficients in Z[1/2], and
in low degrees with coefficients in Z. Cohomology classes on M_1 give rise to
characteristic classes, cohomological invariants of families of curves of genus
one. We prove a number of vanishing results for those characteristic classes,
and give explicit examples of families with non-vanishing characteristic
classes
The Carlitz shtuka
Recently we have used the Carlitz exponential map to define a finitely
generated submodule of the Carlitz module having the right properties to be a
function field analogue of the group of units in a number field. Similarly, we
constructed a finite module analogous to the class group of a number field.
In this short note more algebraic constructions of these "unit" and "class"
modules are given and they are related to Ext modules in the category of
shtukas.Comment: 9 page
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