Milne's rationality conjecture asserts that a rational Hodge cycle on an abelian variety with good reduction has rational intersection pairings with complementary products of divisors on the reduction, with the same value in the different cohomological realizations. The divisors on the reduction need not lift to the original abelian variety. This assertion concerns the value of a pairing and does not itself assert that the Hodge cycle is algebraic.
More precisely, let 
be a 
-dimensional abelian variety
over 
with good reduction 
over 
at a chosen 
-adic place. For a rational Hodge
cycle 
of codimension 
and Cartier divisors 
,
..., 
on 
,
the proposed value is a single 
satisfying
 |
Here 
is the specialized class, 
is the first Chern class,
and the trace is normalized to send a zero-cycle to its degree. The equality is required
for every prime 
in 
and for the crystalline realization at 
after extension to the completed algebraic
closure of the local field. For 
, the empty divisor product is 1.
