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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

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