Subconvex bounds on ${{\mathrm{GL}}}_3$ via degeneration to frequency zero

Abstract

For a fixed cusp form $\pi$ on ${{\mathrm{GL}}}_3(\mathbb {Z})$ and a varying Dirichlet character $\chi$ of prime conductor q, we prove that the subconvex bound

$$\begin{aligned} L\left( \pi \otimes \chi , \tfrac{1}{2}\right) \ll q^{3/4 - \delta } \end{aligned}$$

holds for any $\delta < 1/36$. This improves upon the earlier bounds $\delta < 1/1612$ and $\delta < 1/308$ obtained by Munshi using his ${{\mathrm{GL}}}_2$ variant of the $\delta$-method. The method developed here is more direct. We first express $\chi$ as the degenerate zero-frequency contribution of a carefully chosen summation formula à la Poisson. After an elementary “amplification” step exploiting the multiplicativity of $\chi$, we then apply a sequence of standard manipulations (reciprocity, Voronoi, Cauchy–Schwarz and the Weil bound) to bound the contributions of the nonzero frequencies and of the dual side of that formula.

Appendices

Appendix A: Correlations of Kloosterman sums

The estimates recorded here are unsurprising, but we were unable to find references containing all cases that we require (compare with e.g. [4, 5, 17]).

Lemma 1

Let s be a natural number. Let $a,b,c,d \in \mathbb {Z}/s$ be congruence classes for which $(d,s) = 1$. For each prime $p \mid s$, let $\mathcal {X}_0(p) \subseteq \mathbb {Z}/p$ be a subset of cardinality $p - {\text {O}}(1)$. Let $\mathcal {X}$ denote the set of elements $x \in \mathbb {Z}/s$ for which

• the class of x modulo p belongs to $\mathcal {X}_0(p)$ for each $p \mid s$, and

• $(c x + d,s) = 1$.

Define $\phi : \mathcal {X} \rightarrow \mathbb {Z}/s$ by

$$\begin{aligned} \phi (x) := x \frac{a x + b}{c x + d}. \end{aligned}$$

Then the exponential sum $\Sigma := s^{-1} \sum _{x \in \mathcal {X}} e_s(\phi (x))$

satisfies

$$\begin{aligned} |\Sigma | \leqslant 2^{{\text {O}}(\omega (s))} \frac{(a,b,s)}{s^{1/2} (a,s)^{1/2}}, \end{aligned}$$

where $\omega (s)$ denotes the number of prime divisors of s, without multiplicity.

Proof

We may assume that $s = p^n$ for some prime p. For $n=0$, there is nothing to show. For $n=1$, we appeal either to the Weil bound, to bounds for Ramanujan sums, or to the trivial bound according as $(a,p) = 1$, or $(a,p) = p$ and $(a,b,p) = 1$, or $(a,b,p) = p$. We treat the remaining cases by induction on $n \geqslant 2$.

If $(a,b,p) > 1$, then the conclusion follows by our inductive hypothesis applied to s / p, a / p, b / p, c, d. We may thus assume that $(a,b,p) = 1$.

A short calculation gives the identities of rational functions

$$\begin{aligned} \phi '(x) = \frac{a c x^2 + 2 a d x + b d}{(c x + d)^2}, \quad \phi ''(x) = \frac{2 a + 2 c \phi '(x)}{c x + d}. \end{aligned}$$

(A.1)

Write $n = 2 \alpha$ or $2 \alpha + 1$, and set $\mathcal {R} := \{x \in \mathcal {X}/p^\alpha : \phi '(x) \equiv 0 ~(p^\alpha )\}$. Then by p-adic stationary phase [11, Sect. 12.3],

$$\begin{aligned} \Sigma \ll s^{-1/2} \sum _{x \in \mathcal {R}} (\phi ''(x), p)^{1/2}. \end{aligned}$$

If $(a,p) > 1$, then $(b,p) = 1$ and $\phi '(x) \equiv b d / (c x + d)^2 ~(p)$, so $(\phi '(x), p) = 1$. Thus $\mathcal {R} = \emptyset$ and $\Sigma = 0$. Assume otherwise that $(a,p) = 1$. For $x \in \mathcal {R}$, we have $\phi ''(x) \equiv 2 a / (c x + d) ~(p^\alpha )$, so that

$$\begin{aligned} x \in \mathcal {R} \implies (\phi ''(x), p) = (2 a, p^\alpha ) = (2,p^\alpha ) \ll 1. \end{aligned}$$

(A.2)

Thus $\Sigma \ll s^{-1/2} \# \mathcal {R}$ and, by Hensel’s lemma, $\# \mathcal {R} \ll 1$. The proof of the required bound is then complete. $\square$

Lemma 2

Let $s_1,s_2$ be natural numbers. Let $a_1,a_2,b_1,b_2$ be integers with $(b_1,s_1) = (b_2,s_2) = 1$. Set $\ell _i := a_i/b_i \in \mathbb {Z}/s_i$. Set

$$\begin{aligned} \Delta := \frac{s_2^2 b_2 a_1 - s_1^2 b_1 a_2}{(s_1,s_2)^2}. \end{aligned}$$

• (i) Let $\xi$ be an integer. Set

  $$\begin{aligned} \Sigma := \frac{1}{[s_1,s_2]} \sum _{x([s_1,s_2])} K_{s_1}(\ell _1 x) \overline{K_{s_2}(\ell _2 x)} e _{[s_1,s_2]}(\xi x) \end{aligned}$$

  Then

  $$\begin{aligned} |\Sigma | \leqslant 2^{{\text {O}}(\omega ([s_1,s_2]))} \frac{(\Delta ,\xi ,s_1,s_2)}{ [s_1,s_2]^{1/2} (\xi , s_1, s_2)^{1/2}}. \end{aligned}$$

  (A.3)

  In particular,

  $$\begin{aligned} |\Sigma | \leqslant 2^{{\text {O}}(\omega ([s_1,s_2]))} \frac{(\Delta ,\xi ,s_1,s_2)^{1/2}}{ [s_1,s_2]^{1/2}}. \end{aligned}$$

  (A.4)

• (ii) Let $V : \mathbb {R} \rightarrow \mathbb {C}$ be a smooth function satisfying $x^m \partial _x^n V(x) \prec 1$ for all fixed $m,n \in \mathbb {Z}_{\geqslant 0}$. Let $X > 0$.

  Assume that $s_1, s_2 = {\text {O}}(q^{{\text {O}}(1)})$. Then

  $$\begin{aligned} \sum _{n} V (\frac{n}{X}) K_{s_1}(\ell _1 n) \overline{K_{s_2}(\ell _2 n)} \prec X \frac{(\Delta , s_1, s_2)^{1/2}}{[s_1,s_2]^{1/2}} + [s_1,s_2]^{1/2}. \end{aligned}$$

  (A.5)

Remark

These estimates are not sharp if either $(a_1,s_1)$ or $(a_2,s_2)$ is large, but that case is unimportant for us. In fact, we have recorded (A.3) only for completeness; the slightly weaker bound (A.4) is the relevant one for our applications. We note finally that $K_{s}$ is real-valued.

Proof

We begin with (i). Each side of (A.3) factors naturally as a product over primes, so we may assume that $s_i = p^{n_i}$ for some prime p. By the change of variables $x \mapsto b_1 b_2 x$, we may reduce further to the case $b_1 = b_2 = 1$, so that $\ell _i = a_i$.

In the case that some $\ell _i$ is divisible by p, the quantity $K_{s_i}(\ell _i x)$ is independent of x, has magnitude at most $s_i^{-1/2}$, and vanishes if $n_i > 1$. The required estimate then follows in the stronger form $\Sigma \ll (s_1 s_2)^{-1/2}$ by opening the other Kloosterman sum and executing the sum over x. We will thus assume henceforth that $\ell _1$ and $\ell _2$ are coprime to p.

Write $w_i := s_i / (s_1,s_2)$, so that $w_1 s_2 = s_1 w_2 = [s_1,s_2]$ and $\Delta = w_2^2 b_2 a_1 - w_1^2 b_1 a_2$. By opening the Kloosterman sums and summing over x, we obtain

$$\begin{aligned} \Sigma = \frac{1}{\sqrt{s_1 s_2}} \mathop { \sum _{x_1(s_1)^*} \, \sum _{x_2(s_2)^*} } _{w_1 \ell _2 x_2^{-1} \equiv w_2 \ell _1 x_1^{-1} + \xi \, ([s_1,s_2])} e_{[s_1,s_2]} (w_2 x_1 - w_1 x_2). \end{aligned}$$

(A.6)

Consider first the case $s_1 = s_2 =: s$, so that $w_1 = w_2 = 1$ and $\Delta = \ell _1 - \ell _2$ and $[s_1,s_2] = (s_1,s_2) = s$. The subscripted identity in (A.6) then shows that $x_2$ is determined uniquely by $x_1 =: x$ and, after a short calculation, that

$$\begin{aligned} \Sigma = \frac{1}{s} \sum _{x(s)^*} e_s\left( x \frac{\xi x + \Delta x}{\xi x + \ell _1}\right) . \end{aligned}$$

By the previous lemma, it follows that

$$\begin{aligned} \Sigma \ll \frac{(\Delta ,\xi ,s)}{s^{1/2} (\xi ,s)^{1/2}}, \end{aligned}$$

as required.

Suppose now that $s_1 \ne s_2$. Without loss of generality, $s_1 < s_2$. Then $w_1 = 1$ and $w_2 = s_2/s_1$; in particular, $w_2$ is divisible by p. The summation condition in (A.6) shows that $\Sigma = 0$ unless $(\xi ,p) = 1$, as we henceforth assume. Since $(\ell _1 \ell _2, p) = 1$, we have $(\Delta , p) = 1$, so our goal is to show that $\Sigma \ll s_2^{-1/2}$. We introduce the variable

$$\begin{aligned} y := \xi x_1 + w_2 \ell _1. \end{aligned}$$

Then

$$\begin{aligned} x_1 = \frac{y - w_2 \ell _1}{\xi }, \quad x_2 = \frac{w_1 \ell _2 }{y} \frac{y - w_2 \ell _1}{\xi }, \end{aligned}$$

and as y runs over $(\mathbb {Z}/s_1)^*$, the pair $(x_1, x_2)$ traverses the set indicated in (A.6). A short calculation gives

$$\begin{aligned} w_2 x_1 - w_1 x_2 = - \frac{\Delta }{\xi } + \frac{w_2}{\xi } \left( y + \frac{\ell _2 \ell _1}{y}\right) , \end{aligned}$$

hence

$$\begin{aligned} \Sigma = \frac{1}{\sqrt{s_1 s_2}} e_{s_2} \left( - \frac{\Delta }{\xi } \right) \underbrace{ \sum _{y(s_1)^*} e_{s_1} \left( \frac{1}{\xi } \left( y + \frac{\ell _2 \ell _1}{y}\right) \right) }_{ \sqrt{s_1} K_{s_1}\left( \ell _2 \ell _1/\xi ^2\right) }. \end{aligned}$$

The required conclusion then follows from the Weil bound.

To prove (ii), we first apply Poisson summation to write the LHS of (A.5) as

$$\begin{aligned} X \sum _{\xi } \hat{V} \left( \frac{\xi }{[s_1,s_2]/X}\right) \frac{1}{[s_1,s_2]} \sum _{x([s_1,s_2])} K_{s_1}(\ell _1 x) \overline{K_{s_2}(\ell _2 x)} e_{[s_1,s_2]}(\xi x), \end{aligned}$$

(A.7)

where $\hat{V}$ satisfies estimates analogous to those assumed for V. We then apply (A.4). The $\xi = 0$ term in (A.7) then contributes the first term on the RHS of (A.5), while an adequate estimate for the remaining terms follows from the consequence

$$\begin{aligned} \sum _{\xi \ne 0} |\hat{V}| \left( \frac{\xi }{[s_1,s_2]/X}\right) (\Delta ,\xi ,s_1,s_2)^{1/2} \prec [s_1,s_2]/X \end{aligned}$$

of the divisor bound. $\square$

Appendix B: Comparison with Munshi’s approach

We outline Munshi’s approach [17, 21] to the sums $\Sigma$ arising as in Sect. 3.1 after a standard application of the approximate functional equation, and compare with our own treatment. For simplicity we focus on the most difficult range $N \approx q^{3/2}$.

1.1 B.1. Averaged Petersson formula

Munshi employs the following decomposition of the diagonal symbol:

$$\begin{aligned} \delta (m,n)&= \frac{1}{B^\star }\sum _{b\in \mathcal {B}} \sum _{\psi (b)} (1-\psi (-1)) \sum _{f \in S_k(b,\psi )} {w_f}^{-1} \overline{\lambda _f(m)} \lambda _f(n) \nonumber \\&\quad - 2 \pi i^{-k} \frac{1}{B^\star }\sum _{b\in \mathcal {B}} \sum _{\psi (b)} (1-\psi (-1)) \sum _{c \equiv 0 (b)} \frac{S_\psi (m,n,c)}{c} J_{k-1} \left( \frac{4 \pi \sqrt{m n}}{c}\right) . \end{aligned}$$

(B.1)

Here $\mathcal {B}$ is a suitable set of natural numbers, $\psi$ runs over a suitable collection of odd Dirichlet characters modulo $b \in \mathcal {B}$, and $B^\star$ denotes the appropriate normalizing factor.

1.2 B.2 Munshi’s initial transformations

Set $A(n) := \lambda (1,n)$. Munshi writes¹

$$\begin{aligned} \sum _{n \sim N} A(n) \chi (n) \approx \frac{1}{S} \sum _{s \sim S} \sum _{n \sim N} A(n) \sum _{r \sim NS} \chi \left( \frac{r}{s}\right) \delta (r,ns) \end{aligned}$$

(B.2)

where s runs over primes of size S. Munshi applies (B.1) to $\delta (r,n\ell )$ with $\mathcal {B} = \{t q : t \sim T\}$, where t runs over primes of size T, and the characters $\psi$ are taken to be trivial modulo q. The use of (B.1) produces two main contributing terms, $\mathcal {F}^M$ from the sum of Fourier coefficients and $\mathcal {O}^M$ from the sum of Kloosterman sums, given roughly by

$$\begin{aligned} \mathcal {F}^M \approx \frac{1}{T^2S}\sum _{s} \sum _{t} \sum _{\psi (t)} \sum _{n \sim N} \sum _{r \sim NS} A(n) \chi \left( \frac{r}{s}\right) \sum _{f \in S_k(t q, \psi )} {\omega _f}^{-1} \overline{\lambda _f(r)}\lambda _f(n s) \end{aligned}$$

(B.3)

and

$$\begin{aligned} \mathcal {O}^M \approx \frac{1}{T^2 S}\sum _{s} \sum _{t} \sum _{\psi (t)}\sum _{n \sim N} \sum _{r \sim NS} A(n) \chi \left( \frac{r}{s}\right) \sum _{c \ll \sqrt{q} S/T} \frac{1}{c t q}S_\psi (r, n s;c t q) \end{aligned}$$

(B.4)

which Munshi then works to balance with the appropriate choices of S and T. (The superscripted M has been included to disambiguate from the closely related expressions defined in Sect. 3.4 of this paper.) In (B.4) we sum over moduli c up to the transition range of the resulting J-Bessel function, which we do not display for notational simplicity. (For the analogous problem in spectral or t-aspects, the J-Bessel function plays an important analytic role; cf. forthcoming work of Yongxiao Lin.)

1.3 B.3 Outline of Munshi’s method

We now present a brief outline of Munshi’s treatment of $\mathcal {F}^M$ and $\mathcal {O}^M$ (see [17] for details).

1.3.1 B.3.1 Treatment of $\mathcal {F}^M$

• (1) Dualize the n-sum via the ${{\mathrm{GL}}}_3 \times {{\mathrm{GL}}}_2$ functional equation.

• (2) Dualize the r-sum via the ${{\mathrm{GL}}}_2 \times {{\mathrm{GL}}}_1$ functional equation.

• (3) Sum over f via the Petersson trace formula. The diagonal contribution is negligible. The off-diagonal contribution is a c-sum over Kloosterman sums of the form $S_\psi (t^2qn,rs;ctq)$ with $c \ll \sqrt{q} T^2$.

• (4) Factor the Kloosterman sums modulo t and modulo cq. This yields Gauss sums modulo t; evaluate them. Sum over $\psi$ modulo t. Factor the remaining Kloosterman sum modulo c and modulo q. The mod q contribution gives a Ramanujan sum equal to $-1$.

• (5) The n-sum now oscillates only modulo c. Apply ${{\mathrm{GL}}}_3$ Voronoi and reciprocity.

• (6) Dualize the c-sum modulo r via Poisson. Only the zero dual frequency contributes. It remains to estimate sums of the form

  $$\begin{aligned} \frac{\sqrt{q}}{T^4} \sum _{t\sim T} \sum _{s \sim S} \sum _{r \sim \sqrt{q}T/S} \sum _{n \sim T^3} A(n) \overline{\chi }\left( \frac{r s}{t}\right) S\left( - \frac{n q}{t}, 1; r s\right) . \end{aligned}$$

  (B.5)

• (7) Pull the n, r sums outside and apply Cauchy-Schwarz.

• (8) Conclude via Poisson in n.

Such a treatment produces the following bound

$$\begin{aligned} \mathcal {F} \ll N \left[ \frac{T}{q^{1/4}S^{1/2}}+\left( \frac{T S}{q^{1/2}}\right) ^{1/4}+ noise _\mathcal {F}\right] , \end{aligned}$$

(B.6)

where $noise _\mathcal {F}$ comes from all of the other technical aspects resulting from working outside of the transition ranges and appropriately setting up the remaining object for each step of the above proof.

1.3.2 B.3.2. Treatment of $\mathcal {O}^M$

• (1) Factor the Kloosterman sums modulo t and cq. Evaluate the sum over $\psi$; this simplifies the Kloosterman sums modulo t to additive characters. Apply reciprocity. One now has oscillations only modulo cq.

• (2) Apply Poisson to the r sum. Only the zero frequency contributes non-negligibly to the dual sum. One is now left with estimating sums of the form

  $$\begin{aligned} \frac{1}{TS \sqrt{q}} \sum _{t\sim T} \sum _{s \sim S} \sum _{c \ll \sqrt{q} S/T} \sum _{n\sim N} A(n) \chi \left( \frac{t c}{s}\right) \mathcal {D}\left( \frac{n s}{ t c};q\right) , \end{aligned}$$

  (B.7)

  where

  $$\begin{aligned} \mathcal {D}(u;q):=\sum _{\begin{array}{c} b(q) \\ (b(b-1),q)=1 \end{array}} \overline{\chi }(b-1) e_q((b^{-1} - 1) u) \end{aligned}$$

  (B.8)

• (3) Apply Cauchy–Schwarz with the n-sum outside.

• (4) Conclude via Poisson in n.

Such a treatment produces the following bound

$$\begin{aligned} \mathcal {O} \ll N \left[ \frac{q^{1/4}}{T}+\frac{S}{T}+ noise _\mathcal {O}\right] \end{aligned}$$

(B.9)

where $noise _\mathcal {O}$ comes from all of the other technical aspects resulting from working outside of the transition ranges and appropriately setting up the remaining object for each step of the above proof.

1.3.3 B.3.3. Optimization

Ignoring the contributions from $noise _\mathcal {F}$ and $noise _\mathcal {O}$ in (B.6) and (B.9), one first restricts $S<q^{1/4}$, sets

$$\begin{aligned} \frac{T}{q^{1/4}S^{1/2}}=\left( \frac{TS}{q^{1/2}}\right) ^{1/4} \end{aligned}$$

to get that $S=Tq^{-1/6}$, and then sets

$$\begin{aligned} \frac{T}{q^{1/4}S^{1/2}}=\frac{q^{1/4}}{T} \end{aligned}$$

to get that $T=q^{5/18}$ and $S=q^{2/18}$ which would produce a combined bound of

$$\begin{aligned} \sum _{n \sim N} A(n)\chi (n) \ll N \left[ q^{-1/36}+ noise _{\mathcal {F}+\mathcal {O}}\right] . \end{aligned}$$

(B.10)

Therefore, the best possible bound that one could hope to achieve is a saving over the convexity bound of size $q^{-1/36}$. However, due to all of the technical obstacles that present themselves in the course of the proof, Munshi’s original approach [21] produced a saving of $q^{-1/1612}$, improved in the preprint [17] to $q^{-1/308}$.

1.4 B.4 Discovering the key identity (3.5)

After a topics course taught by the first author in the Fall of 2016 and subsequent discussions with the second author in June 2017, the key identity in this paper was discovered hidden within Munshi’s work. Indeed, starting from (B.5) in the treatment of $\mathcal {F}^M$, if one were to now apply Voronoi summation in the n sum followed by an application of reciprocity for the resulting additive characters, then one would need to instead analyze sums of the form

$$\begin{aligned} \frac{1}{T^2} \sum _{t\sim T} \sum _{s \sim S} \sum _{r \sim \sqrt{q}T/S} \sum _{n \sim q^{3/2}} \overline{A}(n) \overline{\chi }\left( \frac{r s}{t}\right) e_q\left( -\frac{n t}{r s}\right) . \end{aligned}$$

(B.11)

Viewing $-t/rs$ as the u in (3.5), we see that an application of Poisson summation in r returns us to the dual of our original object of interest (from the $h=0$ frequency of the dual) plus a sum which is the “${{\mathrm{GL}}}_3$ dual” of $\mathcal {O}^M$ (from the dual non-zero h frequencies) as expressed in (B.7)

$$\begin{aligned} \frac{1}{TS\sqrt{q}}\sum _{t\sim T} \sum _{s \sim S} \sum _{h \ll \sqrt{q}S/T} \sum _{n \sim q^{3/2}} \overline{A}(n) S_{\overline{\chi }}\left( \frac{h t}{s},n,q\right) . \end{aligned}$$

(B.12)

By “${{\mathrm{GL}}}_3$ dual,” we mean that Voronoi summation in n applied to (B.12) returns one to objects of the form (B.7). This observation led to the simplification presented in this paper whereby many of the initial steps of Munshi’s argument, as outlined above, are eliminated.