# An exponential sum of Conrey and Iwaniec

In their remarkable paper on the third moment of special values of twisted $L$-functions of modular forms, Conrey and Iwaniec require (among many ingredients!) a best-possible estimate (square-root cancellation) for the exponential sums $S(\chi,\eta)=\sum_{u,v}{\chi(uv(u+1)(v+1))\eta(uv-1)}$
where $\chi$ and $\eta$ are non-trivial multiplicative characters modulo a prime $p$. Using a combination of the formalism of such exponential sums (including the consequences of the rationality of the corresponding $L$-functions over finite fields and of the Riemann Hypothesis of Deligne) together with some elementary averaging arguments inspired by ideas of Bombieri, and special treatment of the case $\chi=\eta$, they succeed in proving that $S(\chi,\eta)\ll p,$
which is the expected estimate.

I looked at this sum recently because I wondered if (like the other sum of Friedlander and Iwaniec mentioned at the end of my previous post) it would turn out to be a special case of the correlation sums that feature in my recent paper with Fouvry and Michel. As far as I can see, it isn’t of this type, but this led me to attempt to find a proof of the estimate of Conrey and Iwaniec based on the philosophy of reducing a character sum in more than one variable to a one-parameter sum which involves more complicated summands than character values.

This does indeed work. I’ve written a short note explaining this (it is not intended for publication, so rather terse at some points). The outline is rather straightforward: we write $S(\chi,\eta)=\sum_{u}R(u)T(u)$
where $R(u)=\chi(u(u+1))$
is a character and $T(u)=\sum_{v}{\chi(v(v+1))\eta(uv-1)},$
which is a more “complicated” summand, but still a one-variable character sum.

It turns out that $R(u)$ and $-T(u)$ are both among these nice algebraic trace functions that occur in the paper with Fouvry and Michel. In particular they have the crucial irreducibility property that make such functions quasi-orthogonal, which means here that we get $\sum_u R(u)T(u)\ll q$
unless $\overline{R(u)}$ and $T(u)$ are proportional for all $u$. (The size $q$ on the left-hand side comes from the fact that $T(u)$ is of weight $1$, i.e., a sum of a bounded numbers of algebraic numbers of modulus $\sqrt{q}$; the one-variable sum over $u$ gives square-root cancellation — by the Riemann Hypothesis — for the bounded summands $q^{-1/2}R(u)T(u)$.)

One should feel that it is unlikely that $\overline{R(u)}$ and $T(u)$ are proportional, and indeed it is easy to show during the construction that they are not (the algebraic reason is that one is the trace of a representation of degree $1$, whereas the other is a trace of a representation of degree $2$…). To finish the proof, one must still be careful to control the implied constant in the estimate above, since it is a priori dependent on the characteristic of the finite field involved. A true dependency like this would be catastrophic, but one can show (in different ways) that this constant is in fact bounded uniformly (the basic reason is that the invariants measuring the complexity of the functions $R(u)$ and $T(u)$ are bounded independently of $p$).

This argument is quite clean in outline (and requires no special consideration of special cases like $\chi=\eta$) but it uses rather deep tools. In addition to three applications of the Riemann Hypothesis (one to understand the summand $T(u)$, another to perform the sum over $u$, and the last as explained in a second), there is a fair amount of deep formalism involved in checking that $T(u)$ has the desired properties for a nice algebraic function. There is one last nice ingredient: in order to ensure quasi-orthogonality, one needs to know that $T(u)$ is irreducible in some precise sense. This requirement turns out to translate (using a last time the Riemann Hypothesis) to a nice diophantine property of these sums, namely that $\lim_{|k|\rightarrow +\infty}\frac{1}{|k|^2}\sum_{u}{|T_k(u)|^2}=1,$
where $k$ runs over finite extensions of $\mathbf{F}_q$ and, denoting by $N$ the form from $k$ to $\mathbf{F}_q$, the sum $T_k(u)=\sum_{v\in k}{\chi(N(v(v+1)))\eta(N(uv-1))},$
is the natural companion of $T$ to such extensions. The proof of this limit formula is a nice exercise in manipulating finite sums in the tradition of analytic number theory…