Algebraic twists of modular forms, II

I continue here the discussion, begun in my previous posts, of my recent work with Fouvry and Michel. So, recall that we want to estimate a sum of the type

$S(f,K)=\sum_{n\geq 1}{\rho_f(n)K(n)V(n/p)}$

were $\rho_f(n)$ denotes the Fourier coefficients of a fixed modular form $f$, $K(n)$ is some complex-valued function defined modulo a prime $p$ and extended to all integers by $p$-periodicity, and $V$ is a test function, compactly supported in $]0,1[$

I will present the strategy rather roughly first, and then refine it. For the moment, nothing will be assumed of $K(n)$.

• [Analytic step]. Using (rather sophisticated) techniques from the analytic theory of modular forms (in particular, amplification and the Kuznetsov formula), we show that one can bound from above the sum $S(f,K)$ using some auxiliary sums. Precisely, we first need the discrete Fourier transform of $K$, which we normalize by putting $\hat{K}(t)=\frac{1}{\sqrt{p}}\sum_{x\in\mathbf{F}_p}{K(x)e\Bigl(\frac{tx}{p}\Bigr)}$

for $t$ modulo $p$ (a unitary normalization). Then, for $\gamma\in \mathrm{PGL}_2(\mathbf{F}_p)$, we let

$C(K;\gamma)=\sum_{z\in\mathbf{F}_p}{\overline{\hat{K}(\gamma\cdot z)}\hat{K}(z)}.$

We call these sums correlation sums. Using an auxiliary parameter $L\geq 1$, their relation to our problem can be first expressed by

$|S(f,K)|^2 \ll p^2L^{-1}+pL\times \max_{\gamma\in \mathrm{PGL}_2(\mathbf{F}_p)} |C(K;\gamma)|$

(up to terms like $p^{\varepsilon}$ and logarithms). In fact, we do not write this down, because one can see that it useless as is: for $\gamma=1$, the correlation sum $C(K;1)$ is typically as large as $p$, so the right-hand side is at least of size $p^2$, which is essentially the trivial bound (that one can get from Rankin-Selberg theory, for $K$ bounded)…

• [Pruning]. This first objection to the method is however not significant. Indeed, it is clear from the proof that not all correlations sums $C(K;\gamma)$ play a role when estimating $|S(f,K)|^2$. In particular, provided $L$ is not too large (say $L), it is essentially immediate that one can replace the bound above with

$|S(f,K)|^2 \ll p^2L^{-1}+pL\times \max_{\gamma\notin B_p} |C(K;\gamma)|$

where $B_p$ is the stabilizer of $\infty$ in the action of $G_p=\mathrm{PGL}_2(\mathbf{F}_p)$ on the projective line $\mathbf{P}^1(\mathbf{F}_p)$, i.e.

$B_p=\Bigl\{\begin{pmatrix}a& b\\\ 0&d\end{pmatrix}\Bigr\}\subset G_p.$

This means that we do not mind if some correlation sums are big, provided these “bad” sums are of specific shape, here, upper-triangular.

• [Estimating]. Suppose we want to use the previous bound heuristically first for a weight $K(n)$ which is generic (or random), and bounded by $1$. Then we can expect that the unitary Fourier transform will be basically bounded (by the philosophy of square root cancellation — for which, by the way, I would have expected that there existed a better informal introduction on the internet than in this four-year-old post), and will also be rather random. But then the correlation sum is also an oscillating sum of length $p$, and optimistically, should be of size $\sqrt{p}$ for $\gamma\not=1$. We can therefore hope to get

$|S(f,K)|^2\ll p^2L^{-1}+p^{3/2}L.$

The optimal choice of $L$ is $L=p^{1/4}$, and this gives
$|S(f,K)|^2 \ll p^{2-1/4},\quad\quad\text{ or } \quad\quad S(f,K)\ll p^{1-1/8}.$

This is the limit of the method.

But can we really apply this approach to concrete functions $K(n)$? The reader will see that the quality of the bound here is exactly (up to epsilons) what I stated in the last post for cases like

$K_1(n)=\chi(n),\quad\quad K_2(n)=e\left(\frac{\bar{n}}{p}\right).$

However, these two cases are in fact rather different. In the second case, spelling out the definition of $C(K_2,\gamma)$, it is easy to see that the correlations sums are, up to a factor $1/p$, additive exponential sums in three variables (when $\gamma$ is anti-diagonal, it is a Kloosterman sum in three variables). For these, we show that there exists $M\geq 1$ such that

$|C(K_2,\gamma)|\leq Mp^{1/2}$

for all $\gamma\not=1$, so that the argument above applies (this is a case of the Riemann Hypothesis over finite fields with optimal cancellation for sums in three variables; when $\gamma$ is restricted to anti-diagonal matrices, the fundamental result of Deligne on hyper-Kloosterman sums shows that one can take $M=4$).

But for $K_1(n)$, things are different. Here it is easy to analyze the correlation sums because the Fourier transform of $\chi(n)$, for a non-trivial character modulo $p$, is a multiple of $\bar{\chi}(t)$. Hence $C(K_1,\gamma)$ is a one-variable multiplicative character sum, and can be analyzed using Weil’s methods. One finds that there exists an absolute constant $M\geq 1$ such that

$|C(K_1,\gamma)|\leq Mp^{1/2}$

except if either $\chi$ is non-real, and $\gamma$ is diagonal, or if $\chi$ is real, and $\gamma$ is either diagonal or anti-diagonal (and these are genuine exceptions, if $M$ is supposed to be an absolute constant).

The first case lies within the realm of the previous discussion, but when $\chi$ is a real character, this is not true anymore.

Similarly, if

$K_3(n)=\frac{S(n,n;p)^2}{p}-1,$

as at the end of the previous post, one finds that the Fourier transform $\hat{K}_3(t)$ is exactly the coefficient $a_p(t)$ such that

$|E_t(\mathbf{F}_p)|=p+1-a_p(t),$

where $E_t$ is the elliptic curve which was discussed in the first post in this series. The relations between $|E_t(\mathbf{F}_p)|$ and $|E_{\gamma\cdot t}(\mathbf{F}_p)|$ mentioned there imply that the correlation sums $C(K_3,\gamma)$ are of size $p$ for $\gamma$ in a certain dihedral group $D$ of order $8$, namely the setwise stabilizer of $\{-4,4,0,\infty\}$ in $\mathrm{PGL}_2$. Furthermore, one shows that there exists an absolute constant $M\geq 1$ such that

$|C(K_3,\gamma)|\leq Mp^{1/2}$

for all $\gamma\notin D$. But $D$ is not upper-triangular, and hence the first argument we sketched is also insufficient here…

I will stop here for now; in the next post, we have two obvious questions to discuss: (1) how does one deal with the complications which arose for $K_2(n)$ and $K_3(n)$? and (2) even in the most favorable case, how exactly can one show that the correlation sums are small? As can be expected already from today’s discussion, it is the Riemann Hypothesis over finite fields which is crucial. In fact, to attain the generality of our results, we need to apply it twice; in both cases we require the very deep statements proved by Deligne in the amazing achievement which people call Weil 2, but in one case, this is encapsulated in the DeligneKatzLaumon theory of the $\ell$-adic Fourier transform…