1.5. Inverse conjecture over finite fields 89
Now consider the other extreme, in which Q and R lie in general position.
Then, if we differentiate (1.49) in k, we obtain that one has
∂k∂hφ = ∂kQ∂hR + Q∂k∂hR + ∂kQ(∂k∂hR) + ∂kFh (Lh),
and then anti-symmetrising in k, h one has
0 = ∂kQ∂hR ∂hQ∂kR + (∂kQ ∂hQ)∂k∂hR + ∂kFh (Lh) ∂hFk (Lh).
If Q and R are unrelated, then the linear forms ∂kQ, ∂kR will typically be
in general position with respect to each other and with Lh, and similarly
∂hQ, ∂hR will be in general position with respect to each other and with
Lk. From this, one can show that the above equation is not satisfiable
generically, because the mixed terms ∂kQ∂hR−∂hQ∂kR cannot be cancelled
by the simpler terms in the above expression.
An interpolation of the above two arguments can handle the case in
which Qh does not depend on h. Now we consider the other extreme, in
which Qh varies in h, so that Qh and Qk are in general position for generic
h, k, and
similarly15
for Qh and Qh+k, or for Qk and Qh+k.
To analyse this situation, we return to the cocycle equation (1.46), which
currently reads
(1.50) Fh+k(Qh+k, Lh+k) = Fh(Qh, Lh) + T
hFk(Qk,
Lk).
Because any two of Qh+k,Qh,Qk can be assumed to be in general position,
one can show using equidistribution theory that the above equation can only
be satisfied when the Fh are linear in the Qh variable, thus
∂hφ = QhFh(Lh) + Fh (Lh)
much as before. Furthermore, the coefficients Fh(Lh) must now be (essen-
tially) constant in h in order to obtain (1.50). Absorbing this constant into
the definition of Qh, we now have
∂hφ = Qh + Fh (Lh).
We will once again pretend that Lh is just a single linear form Lh. Again
we consider two extremes. If Lh = L is independent of h, then by passing
to a bounded index subspace (the level set of L) we now see that
∂hφ is
quadratic, hence φ is cubic, and we are done. Now suppose instead that
Lh varies in h, so that Lh,Lk are in general position for generic h, k. We
look at the cocycle equation again, which now tells us that Fh (Lh) obeys
the quasicocycle condition
Qh,k + Fh+k(Lh+k) = Fh (Lh) + T
hFk
(Lk)
15Note though that we cannot simultaneously assume that Qh, Qk, Qh+k are in general
position; indeed, Qh might vary linearly in h, and indeed we expect this to be the basic behaviour
of Qh here, as was observed in the preceding argument.
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