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<p^{1/2}), 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 Deligne–Katz–Laumon theory of the \ell-adic Fourier transform…

Algebraic twists of modular forms, I

In the next few posts, I want to describe some aspects of the paper “Algebraic twists of modular forms and Hecke orbits” that I just finished with Étienne Fouvry and Philippe Michel. Today, I will describe some cases of the main result (I was thinking of continuing dans la foulée with a high-level sketch of the proof, but it seems better to delay this until the next time…)

I’ll start with one of the simplest cases, which was also one of our original challenges. Let f be a classical holomorphic modular form, say of level N and weight k\geq 2, with Fourier expansion

f(z)=\sum_{n\geq 1}{n^{(k-1)/2}\rho_f(n)e(nz)},\quad\quad\quad\text{ where } \quad e(z)=e^{2i\pi z}

for z in the upper half-plane. If in doubt, you may assume that

f(z)=e(z)\prod_{n\geq 1}{(1-e(nz))^{24}}

is the Ramanujan \Delta-function, with N=1 and k=12.

The (normalized) Fourier coefficients \rho_f(n) are essentially bounded (they are bounded in mean-square norm over n\leq x, by Rankin-Selberg theory) and their signs oscillate quite randomly. In particular, one expects that sums of the type

\frac{1}{x}\sum_{n\leq x}{\rho_f(n)K(n)}

should be rather small, unless K(n) is itself somehow related to \overline{\rho_f(n)}.

Our question was: can we prove that this is the case when we take x=p, a prime number, and

K(n)=e\Bigl(\frac{\bar{n}}{p}\Bigr),\quad\quad\quad K(p)=0,

(where we denote by \bar{n} the multiplicative inverse modulo p of some integer n\geq 1 coprime with p)?

One can indeed ask this as a challenge: considering that we feel that we know something about Fourier coefficients of modular forms, shouldn’t we be able to prove that they do not correlate with e(\bar{n}/p)? (In addition, there is a geometric motivation, which I will defer to another post.)

It is a very simple and special case of our results that, indeed, we can do something with such sums. Precisely, we prove that

\sum_{n\leq p}{\rho_f(n)e\Bigl(\frac{\bar{n}}{p}\Bigr)} \ll p^{15/16+\varepsilon}

for any prime p and for any \varepsilon>0, where the implied constant depends on f and \varepsilon.

The more general statements are all of the following type: for a prime p, we have some “weight” K(n) defined for n modulo p, which we extend to all of \mathbf{Z} by composing with reduction modulo p. Then we want to bound from above the sum

\sum_{1\leq n\leq p}{\rho_f(n)K(n)},

for a fixed f. Our results are of the form

\sum_{1\leq n\leq p}{\rho_f(n)K(n)}\ll \|K\|_{alg}\ p^{15/16+\varepsilon},

where the implied constant depends only, as before, on f and \varepsilon>0, and where the dependency on K is contained entirely in the quantity \|K\|_{alg}, which I will only define precisely in later instalments, but which roughly speaking measures how difficult it is to express K as a linear combination of special coefficients which I will call here simply “of algebraic origin”.

In fact, we do not express the result in this manner. A technical difference is that we use smoothed sums, proving bounds like

\sum_{n\geq 1}{\rho_f(n)K(n)V(n/p)}\ll \|K\|_{alg}\ p^{7/8+\varepsilon},

where V is a smooth function compactly supported on [0,1] and the implied constant now also depends on V, but in very controlled ways. Moreover, we concentrate on the special coefficients of algebraic origin, leaving the general case to the triangle inequality. For instance, our previous weight

K(n)=e\Bigl(\frac{\bar{n}}{p}\Bigr),\quad\quad\quad K(p)=0,

is of algebraic origin for every prime p, with \|K\|_{alg} absolutely bounded, and hence the first result is a consequence of the more general case. Another more general type of examples (which remains far from the most general!) is given by

K(n)=\chi(\phi_1(n))e\Bigl(\frac{\phi_2(n)}{p}\Bigr)

for integral polynomials \phi_1, \phi_2, and for a Dirichlet character \chi modulo p. All of these are “of algebraic origin”, and when (\phi_1,\phi_2) are fixed, the norm \|K\|_{alg} is uniformly bounded as p varies. Note in particular that K(n)=\chi(n) is possible, and it is certainly among the most natural examples. Indeed, the smoothed form of our bounds leads by classical means to a subconvex estimate for twisted L-functions, which is of the type

L(f\otimes\chi,1/2)\ll p^{7/8+\varepsilon}.

This is not new, but it illustrates the range and depth of our results. (Such a bound, with a weaker exponent, was first proved by Duke-Friedlander-Iwaniec, and the exponent 1-1/8 was first obtained by Blomer and Harcos, refining a method of Bykovski.)

Before finishing for today, here is one example that suggests how one may go well beyond those previously considered. For p prime and a, b coprime with p, let

S(a,b;p)=\sum_{x\in\mathbf{F}_p^{\times}}{e\Bigl(\frac{ax+b\bar{x}}{p}\Bigr)}

be the classical Kloosterman sum. Then the weights

K_1(n)=-\frac{S(n,n;p)}{\sqrt{p}},\quad\quad K_2(n)=\frac{S(n,n;p)^2}{p}-1,

are “of algebraic origin”. Applying our methods to the second one, we end up with the question discussed in the previous post…

A bijective challenge

Étienne Fouvry, Philippe Michel and myself have just finished a new paper, which is available on my web page and will soon be also on arXiv. This was quite an extensive project, which also opens many new questions. I will discuss the general problem we consider, and the techniques we use, in other posts, but today I want to discuss a by-product that we found particularly nice (and amusing). It can be phrased as a rather elementary-looking challenge: given a prime number p, and an element t of \mathbf{Z}/p\mathbf{Z} which is neither 0, 4 or -4 modulo p, let N_p(t) be the number of solutions (x,y)\in (\mathbf{Z}/p\mathbf{Z}-\{0\})^2 of the congruence

x+\frac{1}{x}+y+\frac{1}{y}+t=0.

The challenge is to prove, bijectively if possible, that

N_p(t)=N_p\Bigl(\frac{16}{t}\Bigr)

and that

N_p(t)=N_p\Bigl(\frac{4t-16}{t+4}\Bigr)=N_p\Bigl(\frac{4t+16}{t-4}\Bigr)

if p\equiv 1\pmod{4}.

This sounds simple and elegant enough that an elementary proof should exist, but our argument is a bit involved. First, the number N_p(t) is the number of points modulo p of the curve with equation above, whose projective (smooth) model is an elliptic curve, say E_t, over \mathbf{F}_p. Then we checked using Magma that E_t and E_{16/t} are isogenous over \mathbf{F}_p, and this is well-known to imply that the two curves have the same nunmber of points modulo p. The other two cases are similar, except that for

\gamma(t)=\frac{4t-16}{t+4}\text{ or } \frac{4t+16}{t-4},

the relevant isogenies are between E_t and \tilde{E}_{\gamma(t)}, where \tilde{E}_t denotes the quadratic twist of E_t by -1. Hence the number of points are the same when -1 is a square modulo p.

In the first case, the isogeny is of degree 4, and the others are of degree 8, so the formulas which define them are rather unwieldy, at least in the equivalent Weierstrass model.

The best explanation of this has probably to do with the relation between the family of elliptic curves and the modular curve Y_0(8) (a relation whose existence follows from Beauville’s classification of stable families of elliptic curves over \mathbf{P}^1 with four singular fibers, as C. Hall pointed out), but we didn’t succeed in getting a proof of all our statements using that link. In fact, we almost expected to find the identities above already spelled out in some corner or other of the literature on modular curves and universal families of elliptic curvers thereon, but we did not find anything.

In the next post, I’ll come back to this to explain the link with our paper, which ostensibly is about estimates for sums of Fourier coefficients of modular forms multiplied with functions “of algebraic origin”. Kloosterman sums will enter the picture to make the connection (in more ways than one!), and we’ll see a rather elegant formula of Beltrami…

P.S. Here is a link to a transcript of a Magma session proving the existence of the isogenies which imply our formulas, and ending with the formula of the 4-isogeny, written in terms of the original curve.

Jules Verne, précurseur de Zazie?

There is undeniably a certain form a humor in the books of Jules Verne, but of a rather inoffensive kind, and the distracted geograph Paganel would probably be dismissed rather curtly by such a lively girl as Queneau’s Zazie. Nevertheless, while re-reading Les enfants de Capitaine Grant, I found a magnificent sentence that, I think, even Zazie would approve:

Les petits garçons et les petites filles, plus rageuses surtout, s’administraient des taloches superbes avec un entrain féroce.
(Les enfants du Capitaine Grant, 2ème partie, Chapitre XVI)

This is basically untranslatable; the literal meaning is something like

The boys and girls, even fiercer, exchanged superb blows with extreme alacrity.

but English words fail me to convey the finer meaning of taloche…

And I was reading this book because, believe it or not, Jules Verne is now a Pléiade author! Of course, grudgingly, since only four of his novels were deemed worthy of this supreme honor of French letters. In addition to Les enfants…, we have 20000 lieues sous les mers and L’île mystérieuse, a trilogy, and Le sphinx des glaces, but obviously some strong reactionary faction must have resisted any attempt in adding De la terre à la lune, or Kéraban le Têtu, or Hector Servadac, or…

I also had not realized before the embarrassing chronological problems of the trilogy: Les enfants… happens in 1864–1865; 20000 lieux… begins in 1866; but then L’île mystérieuse, which is supposed to take place twelve years after the first part, begins in 1865…