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.