Short Second Moment Bound for GL(2) $L$-functions in $q$-Aspect

The study of moments of $L$-functions at the central point, $s=\frac{1}{2}$, has been an important area of research in analytic number theory. While originally this interest grew due to connections with the Lindelöf Hypothesis, estimation of similar moments for a family of $L$-functions has now become an interesting point of study on its own.

One key result in this area is the following proposition, due to Iwaniec in 1978 (Theorem $3$ in [Iwa80]).

Proposition 1.1 is one of the first examples of an upper bound on a ‘short’ moment of $L$-functions. Most of the earlier results were concerned with estimating integrals of the type $\int_{0}^{T}\lvert\zeta\left(\frac{1}{2}+it\right)\rvert^{k}dt$, for some positive even integer $k$.

Note that (1.1) is consistent with the Lindelöf Hypothesis, and is an example of a Lindelöf-on-average upper bound. For level $1$ holomorphic cusp forms, an analogous result follows from Good [Goo82],

Equations (1.1) and (1.2) imply a Weyl-type subconvexity bound for the respective $L$ functions, $\zeta(\cdot)$, and $L(f,\cdot)$. In fact, for holomorphic cusp forms, [Goo82] was the first instance where such a result was obtained.

In their paper on the Weyl bound for Dirichlet $L$-functions [PY23], Petrow and Young, proved a $q$-aspect analogue of Proposition 1.1. A special case of Theorem $1.4$ (with $q=p^{3},d=p^{2}$) in [PY23] can be stated as follows.

We discuss this analogy in more detail in Section 1.4.

In this paper, following up on the ideas in [PY23], we derive a $q$-aspect analogue of Good’s result in Proposition 1.2. We prove the following theorem.

We note that, similar to Propositions 1.1, 1.2, 1.3, this is also a Lindelöf-on-average bound.

Also, even though Theorem 1.4 is stated for $q=p^{3}$, we expect this to generalise to a short moment exactly analogous to the Theorem $1.4$ [PY23]. While the authors in [PY23] needed the fully general result to prove the Weyl bound for all Dirichlet $L$-functions (see also [PY20]), we do not have any such demand. So, for the sake of simplicity, we choose to work with $q=p^{3}$.

Using a lower bound on the associated first moment, we can deduce from Theorem 1.4 the following result about non-vanishing of $L$-functions within this family.

Moments of a family of $L$-functions encode a lot of information about the individual members of the family. For one such example, the strength of the second moment bound in $\eqref{eq:MainThm}$ is enough to immediately deduce a Weyl-type subconvexity bound for individual $L$-functions in this family. This result was originally proven by Munshi and Singh in 2019 (Theorem $1.1$ in [MS19] with $r=1,t=0$). The authors use a completely different approach. They do not compute moments for an associated family; relying instead on a novel variant of the circle method, first introduced in [Mun14].

We state their result as a corollary.

We use standard conventions present in analytic number theory. We use $\varepsilon$ to denote an arbitrarily small positive constant. For brevity of notation, we allow $\varepsilon$ to change depending on the context.
The expression $F\ll G$ implies there exists some constant $k$ for which $\lvert F\rvert\leq k\cdot G$ for all the relevant $F$ and $G$. We use $F\ll_{\varepsilon}G$ to emphasize that the implied constant $k$ depends on $\varepsilon$ (it may also depend on other parameters). For error terms, we often use the big $O$ notation, so $f(x)=O(g(x))$ implies that $f(x)\ll g(x)$ for sufficiently large $x$. We use the term ‘small’ or ‘very small’ to refer to error terms which are of the size $O_{A}(p^{-A})$, for any arbitrarily large $A$.
By a dyadic interval, we mean an interval of the type $[2^{\frac{k}{2}}M,2^{\frac{k+1}{2}}M]$ for some $k\in\mathbb{Z}$, $M\in\mathbb{R}$. We also use $m\asymp M_{0}$ to mean $m$ ranges over the dyadic interval $[M_{0},2M_{0}]$.
We also use $\sideset{}{{}^{*}}{\sum}_{n}$ to denote $\smashoperator[]{\sum_{\begin{subarray}{c}n\\ \text{gcd}(n,p)=1\end{subarray}}^{}}$ . Similarly, $\sideset{}{{}^{*}}{\sum}_{a(\mathrm{mod}\ c)}$ is used for $\smashoperator[]{\sum_{\begin{subarray}{c}a(\mathrm{mod}\ c)\\ \text{gcd}(a,c)=1\end{subarray}}^{}}$ , and $\sideset{}{{}^{*}}{\sum}_{\chi(p)}$ for $\smashoperator[]{\sum_{\begin{subarray}{c}\chi(\mathrm{mod}\ p)\\ \chi\text{ primitive}\end{subarray}}^{}}$.
As usual, $e(x)=e^{2\pi ix}$. Also, $e_{p}(x)=e(\frac{x}{p})=e^{2\pi i\frac{x}{p}}$.

We give a brief sketch of Theorem 1.4 in this section.
In this sketch, we use the symbol ‘$\approx$’ between two expressions to mean the expressions on the left side can be written as an expression similar to the right side, along with an acceptable error term.

Using an approximate functional equation and orthogonality of characters, (see Section 3), it suffices to prove the following result on an associated shifted convolution problem.

The trivial bound on $S(N,\alpha)$ is $\frac{N^{2}}{p^{2}}p^{\varepsilon}\ll Np^{1+\varepsilon}$. To improve on this, we first note that $S(N,\alpha)$ displays a conductor dropping phenomenon, notably

(1.9)

$$\alpha(n+p^{2}l)\overline{\alpha(n)}=e_{p}(a_{\alpha}l\overline{n}).$$

for some non-zero $a_{\alpha}(\mathrm{mod}\ p)$.

Notice that, when $p\mid l$ in (1.7), $S(N,\alpha)$ does not have any cancellations. However, we are saved by the fact that the number of such terms is $O(Np^{\varepsilon})$. So for the rest of the sketch, we assume $(l,p)=1$.

In order to separate the variables $n$ and $(n+p^{2}l)$ we introduce a delta symbol (see Section 2.3). This introduces a new averaging variable $c$, and once again we split the resulting expression into whether gcd$(c,p)=1$, or not.
The former requires more work, and we focus on this term for the sketch. We have

(1.10)

$$S(N,\alpha)\approx\sideset{}{{}^{*}}{\sum}_{l\leq\frac{N}{p^{2}}}\sideset{}{{}^{*}}{\sum}_{c\leq\sqrt{N}}\ \sideset{}{{}^{*}}{\sum}_{a\negthickspace\negthickspace\negthickspace\pmod{c}}\int_{-\infty}^{\infty}g_{c}(v)e(-p^{2}lv)\\ \cdot\sum_{m\asymp N}\lambda_{f}(m)e\left(\frac{am}{c}\right)w_{N}(m)e(mv)\sideset{}{{}^{*}}{\sum}_{n\asymp N}\overline{\lambda}_{f}(n)e_{p}(a_{\alpha}l\overline{n})e\left(\frac{-an}{c}\right)w_{N}(n)e(-nv)\ dv.$$

Here, $g_{c}(v)$ is a smooth function which is small when $v\gg\frac{1}{c\sqrt{N}}$.

We can now use the Voronoi summation formula (see Sec. 2.5), for the $m$ and $n$ sums in (1.10). Note that, as there is an extra additive character $e_{p}(a_{\alpha}l\overline{n})$ in the $n$ sum, we need to use a modified Voronoi summation formula (see Prop 2.17) here. While this modification does introduce some extra terms to the final expression, these terms are pretty easily dealt with (see Section 6.1). We get that

(1.11)

$$S(N,\alpha)\approx\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}l\leq\frac{N}{p^{2}}\\ m\ll p^{\varepsilon}\\ n\ll p^{2+\varepsilon}\end{subarray}}\frac{\lambda_{f}(m)\overline{\lambda_{f}}(n)}{p^{2}}\sideset{}{{}^{*}}{\sum}_{c\leq\sqrt{N}}\frac{1}{c^{2}}S(\overline{p}^{2}n-m,-p^{2}l;c)\text{Kl}_{3}(-n\overline{c}^{2}a_{\alpha}l,1,1;p)I_{N}(c,l,m,n),$$

with $S(a,b;c)=\sum_{mn\equiv 1(\mathrm{mod}\ c)}e\left(\frac{am+bn}{c}\right)$, and $\text{Kl}_{3}(x,y,z;c)=\sum_{pqr\equiv 1(\mathrm{mod}\ c)}e\left(\frac{px+qy+rz}{c}\right)$ denoting the Kloosterman and hyper-Kloosterman sums respectively. Also, $I_{N}(\cdot)$ is an integral of special functions (see (4.24)) which trivially satisfies $I_{N}(c,l,m,n)\ll_{f,\varepsilon}\frac{1}{c}N^{\frac{3}{2}+\varepsilon}$.

Here, we should indicate that while the original sums were of length $N$ each (and $N\ll p^{3+\varepsilon}$), the dual sums are significantly shorter - one is $m\ll p^{\varepsilon}$, and the other $n\ll p^{2+\varepsilon}$. Even though this represents significant savings (by a factor of $\frac{N^{2}}{p^{2}}$), it is still not sufficient. If we use Weil’s bound for the Kloosterman sum, and Deligne’s bound for the hyper-Kloosterman sum, along with the trivial bound on $I_{N}(\cdot)$, we get (1.11) is bounded by $Np^{\frac{3}{4}+\varepsilon}$. While this beats the trivial bound for (1.7), it still falls short of the desired bound by a factor of $p^{-\frac{3}{4}}$.

In order to get more cancellations, we want to use a spectral decomposition for the $c$-sum. We do this by using the Bruggeman Kuznetsov formula (Prop. 2.18). Note that if gcd$(r,p)=1$, the hyper-Kloosterman sum can be decomposed as

$$\text{Kl}_{3}(r,1,1;p)=\frac{1}{\phi(p)}\sum_{\chi(p)}\tau(\chi)^{3}\chi(r).$$

Using this we rewrite (1.11) as

$$\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}l\leq\frac{N}{p^{2}}\\ m\ll p^{\varepsilon}\\ n\ll p^{2+\varepsilon}\end{subarray}}\frac{\lambda_{f}(m)\overline{\lambda_{f}}(n)}{\phi(p)p^{2}}\sum_{\chi(p)}\tau(\chi)^{3}\chi(-na_{\alpha}l)\sideset{}{{}^{*}}{\sum}_{c}\frac{1}{c^{2}}S(\overline{p}(n-p^{2}m),-pl;c)\overline{\chi}^{2}(c)I_{N}(c,l,m,n).$$

We now use the Bruggeman-Kuznetsov formula for $\Gamma_{0}(p)$ with central character $\overline{\chi}^{2}$ at the cusps $\infty$ and $0$. We use two different versions of the integral transforms, depending on whether the integral $I_{N}(c,l,m,n)$ has oscillatory behavior, or not (see Section 2.6). We focus on the non-oscillatory case here, the other one is similar. We have

(1.12)

$$S(N,\alpha)\approx\frac{N}{p^{2}}\sideset{}{{}^{*}}{\sum}_{\chi(p)}\chi(a_{\alpha})\frac{\tau(\chi)^{3}}{p^{3}}\sum_{t_{j}}\sum_{\pi\in\mathcal{H}_{it_{j}}(p,\chi^{2})}L(\tfrac{1}{2},\overline{\pi}\otimes\chi)\sum_{\begin{subarray}{c}m\ll p^{\varepsilon}\\ n\ll p^{2+\varepsilon}\end{subarray}}\frac{\overline{\lambda}_{f}(n)\overline{\lambda}_{\pi}(\lvert p^{2}m-n\rvert)\chi(-n)}{\sqrt{\lvert p^{2}m-n\rvert}}\\ +\text{(Holomorphic terms)}+\text{(Eisenstein terms)}.$$

Here $\mathcal{H}_{it}(n,\psi)$ denotes the set of Hecke-Maass newforms of conductor $n$, central character $\psi$, and spectral parameter $it$. Analysing the associated integral transforms, it suffices to restrict (1.12) to when $t_{j}\ll p^{\varepsilon}$.

The contribution of the holomorphic and Eisenstein terms are similar to the Maass form term written down here.
Using the Cauchy-Schwarz inequality, it then suffices to bound the sums

$$\sum_{t_{j}\ll p^{\varepsilon}}\sideset{}{{}^{*}}{\sum}_{\chi(p)}\sum_{\pi\in\mathcal{H}_{it_{j}}(p,\chi^{2})}\lvert L(\tfrac{1}{2},\overline{\pi}\otimes\chi)\rvert^{2}\ ,\ \ \sum_{t_{j}\ll p^{\varepsilon}}\sideset{}{{}^{*}}{\sum}_{\chi(p)}\sum_{\pi\in\mathcal{H}_{it_{j}}(p,\chi^{2})}\sum_{n\ll p^{2+\varepsilon}}\left|\frac{\overline{\lambda}_{f}(n)\overline{\lambda}_{\pi}(\lvert p^{2}m-n\rvert)\chi(-n)}{\sqrt{\lvert p^{2}m-n\rvert}}\right|^{2}.$$

Using the fact that $\overline{\pi}\otimes\chi\in\mathcal{H}_{it_{j}}(p^{2},1)$, we can get the required bounds by applying the spectral large sieve inequality on each of the two sums. The details can be seen in Section 5.

If we compare our proof and the proof of Prop 1.3, we can see that the authors in [PY23] do not use a delta symbol for the shifted convolution sum at the beginning, instead relying on an approximate functional equation-type formula for the divisor function. A more significant difference can be observed later, by considering the shape of the terms in (1.12) and equation (1.20) in [PY23]. Unlike our case, the authors get a product of three $L$-functions that they then bound using Holder’s inequality. This reduces their problem to bounding the fourth moment of L -functions, which is accomplished via the use of the spectral large sieve inequality.

Theorem 1.4 joins a growing list of results on short moments of families of $L$-functions. The general idea here is to work with a subfamily of $L$-functions that exhibit a conductor lowering phenomenon. Say, we start with a family $\mathcal{F}$ of analytic $L$-functions; we choose a subfamily $\mathcal{F}_{0}$, such that the quantity $C(\pi_{1}\otimes\overline{\pi_{2}})$, where $C(\pi)$ denotes the analytic conductor of $\pi$, is of a comparatively smaller size, when $\pi_{1},\pi_{2}$ are in $\mathcal{F}_{0}$. (See Section 1.5 in [PY23] for a more detailed discussion on this).

We see an archimedean version of this in Propositions 1.1 and 1.2. The full family of $L$-functions considered in Proposition 1.2, for example, is $\mathcal{F}=\{L(f\otimes\lvert\cdot\rvert^{it},\tfrac{1}{2}),T\leq t\leq\ 2T\}.$ Here $f$ is a level $1$ cusp form. The analytic conductor of $L(f\otimes\lvert\cdot\rvert^{it_{1}}\otimes\overline{f\otimes\lvert\cdot\rvert^{it_{2}}})$ is proportional to $\lvert t_{1}-t_{2}\rvert^{4}$. Now, for arbitrary $t_{1},t_{2}$ in $[T,2T]$, this can be as high as $T^{4}$. However, if we restrict to the subfamily, $\mathcal{F}_{0}=\{L(f\otimes\lvert\cdot\rvert^{it},\tfrac{1}{2}),T\leq t\leq\ T+T^{\tfrac{2}{3}}\}$ - as considered in (1.2), we have that $\lvert t_{1}-t_{2}\rvert^{4}\leq T^{\tfrac{8}{3}}$, which is significantly smaller than $T^{4}$. This leads to a conductor lowering analogous to (1.9).

For our work, we start with the family, $\mathcal{F}=\{L(f\otimes\chi,\tfrac{1}{2}),\chi(\mathrm{mod}\ q)\}$, where $q=p^{3}$. Using some of the results in [Li75], we know that the analytic conductor of $L\left((f\otimes\chi_{1}\otimes\overline{(f\otimes\chi_{2})},\tfrac{1}{2}\right)$ depends on the fourth power of the conductor of $(\chi_{1}\overline{\chi}_{2})$. Now, for arbitrary $\chi_{1}$, and $\chi_{2}$, this can be as high as $q^{4}$. To get a short moment in the level aspect, we choose a subfamily of $\mathcal{F}$ that lowers this significantly. We consider the subfamily $\mathcal{F}_{0}=\{L(f\otimes(\alpha\cdot\psi),\tfrac{1}{2}),\psi(\mathrm{mod}\ q^{\tfrac{2}{3}})\},$ for a fixed primitive character $\alpha(\mathrm{mod}\ q)$.Now, the analytic conductor of $L\left((f\otimes(\alpha\cdot\psi_{1}))\otimes\overline{(f\otimes(\alpha\cdot\psi_{2}))},\tfrac{1}{2}\right)$ (more accurately, of $L\left((f\otimes\overline{f}\otimes(\psi_{1}\overline{\psi}_{2}),\tfrac{1}{2}\right)$) depends on $\left(\text{conductor}(\psi_{1}\overline{\psi_{2}})\right)^{4}\leq q^{\frac{8}{3}}$, which is significantly lower than $q^{4}.$

Short moments in the level aspect, in a $GL(1)$ setting, have also been considered in [Nun21], and in [MW21]. Another similar application was considered along Galois orbits in [KMN16].

I am deeply grateful to my doctoral advisor, Matthew P. Young, for suggesting this problem, and for engaging in extensive discussions on this subject throughout the process of working on this paper. I would also like to thank Peter Humphries and Chung-Hang Kwan for some insightful discussions regarding this work.

We state a version of the approximate functional equation for analytic $L$-functions. For proofs, see Theorem $5.3$ and Proposition $5.4$ in [IK04].

As $\lvert\varepsilon\left(f,\tfrac{1}{2}\right)\rvert=1$, we have the following immediate corollary (using $s=\frac{1}{2},X=1$),

We will need to use a particular case of the Postnikov formula stated below.

This section is a brief review of the delta-symbol method from Section $20.5$ in [IK04].

Let $w(u)$ be a smooth, compactly supported function on $[C,2C]$ for some $C>0$. Additionally suppose that $\sum_{q=1}^{\infty}w(q)=1$. Then we can rewrite the Kronecker delta function at zero as

(2.4)

$$\delta(n)=\sum_{q\mid n}\left(w(q)-w\left(\frac{\lvert n\rvert}{q}\right)\right).$$

Here,

$$S(0,n;c)=\sideset{}{{}^{*}}{\sum}_{d\negthickspace\negthickspace\negthickspace\pmod{c}}e\left(\frac{dn}{c}\right),\ \text{and }\Delta_{c}(u)=\sum_{r=1}^{\infty}\frac{1}{cr}\left(w(cr)-w\left(\frac{\lvert u\rvert}{cr}\right)\right).$$

Since $\Delta_{c}(u)$ is not compactly supported, we multiply it by a smooth function $f$ supported on $[-2N,2N]$ ($N>0$) such that $f(0)$ = 1. We also assume, for any $a\in\mathbb{N}$, $f^{(a)}(u)\ll N^{-a}$.

Thus, we have

(2.6)

$$\delta(n)=\sum_{c=1}^{\infty}S(0,n;c)\Delta_{c}(n)f(n).$$

As $w$ is supported on $[C,2C]$ and $f$ is supported on $[N,2N]$, the sum on the right hand side is only upto $c\leq 2\text{ max }\left(C,\frac{N}{C}\right)=X$. The optimal choice would thus be to take $C=\ \sqrt{N}$, giving $X=2C=2\sqrt{N}$.

Thus, (2.6) becomes

(2.7)

$$\delta(n)=\sum_{c\leq 2C}S(0,n;c)\Delta_{c}(n)f(n).$$

We can get additional bounds on the derivatives of $\Delta_{c}(u)$ if we assume more conditions on $w$.

Finally, it is often useful to express $\Delta_{c}(n)f(n)$ in terms of additive characters, by considering its Fourier transform.
Let

(2.11)

$$g_{c}(v)=\int_{-\infty}^{\infty}\Delta_{c}(u)f(u)e(-uv)du,$$

be the Fourier transform of $\Delta_{c}(u)f(u)$. By the Fourier inversion formula we have

(2.12)

$$\Delta_{c}(u)f(u)=\int_{-\infty}^{\infty}g_{c}(v)e(uv)dv.$$

Using this in (2.7) , we have

(2.13)

$$\delta(n)=\sum_{c\leq 2C}S(0,n;c)\int_{-\infty}^{\infty}g_{c}(v)e(nv)dv.$$

we have the following bounds on $g_{c}(v)$:

We mention some properties of inert functions in this section. Inert functions are special families of smooth functions characterised by certain derivative bounds. See Sections $2$ and $3$ in [KPY19], and Section $4$ in [KrY21], for proofs.

We will often denote the sequence of constants $C(j_{1},j_{2},\dots,j_{d})$ associated with this inert function as $C_{\mathcal{F}}$.

We note that the requirements for the functions to be supported on dyadic intervals can be easily achieved by applying a dyadic partition of unity.

We give one simple example to highlight how such families can be constructed.

The following propositions can be checked immediately:

Suppose that $w_{T}(x_{1},\cdots,x_{d})$ is $X$-inert and is supported on $x_{i}\asymp X_{i}$. Let

(2.17)

$$\widehat{w}_{T}(t_{1},x_{2},\cdots,x_{d})=\int_{-\infty}^{\infty}w_{T}(x_{1},\cdots,x_{d})e(-x_{1}t_{1})dx_{1},$$

denote its Fourier transform in the $x_{1}$-variable.

We state the following result (Prop 2.6 in [KPY19]) regarding the Fourier transform of inert functions.

We can similarly describe the Mellin transform of such functions as well. We state the following result (Lemma 4.2 in [KrY21]).

The following is a restatement of Lemma 3.1 in [KPY19].

The previous result has a natural generalisation (Main Theorem in [KPY19]).