Abstract
Let F be a number field, let \(\mathbb {A}_F\) be its ring of adeles, and let \(g_1,g_2,h_1,h_2 \in \mathrm {GL}_2(\mathbb {A}_F)\). We provide an absolutely convergent geometric expression for
where the sum is over isomorphism classes of cuspidal automorphic representations \(\pi \) of \(\mathrm {GL}_2(\mathbb {A}_F)\). Here \(K_{\pi }\) is the typical kernel function representing the action of a test function on the space of \(\pi \).
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1 Introduction
Let F be a number field, let G be a connected reductive group over F, and let \(A_G \le G(F_\infty )\) be the connected component in the real topology of the \(\mathbb {R}\)-points of the greatest \(\mathbb {Q}\)-split torus in the center of \(\mathrm {Res}_{F/\mathbb {Q}}G\). For \(f \in C_c^\infty (A_G \backslash G(\mathbb {A}_F))\), let
be the usual operation induced by the action of \(G(\mathbb {A}_F)\) on the right. We let
be the kernel of R(f) restricted to the cuspidal subspace. Here the sum is over isomorphism classes of cuspidal automorphic representations \(\pi \) of \(A_G \backslash G(\mathbb {A}_F)\) (i.e., cuspidal automorphic representations of \(G(\mathbb {A}_F)\) trivial on \(A_G\)). Moreover, \(K_{\pi (f)}(x,y)\) is the unique smooth function with \(L^2\)-expansion
where \(\mathcal {B}(\pi )\) is an orthonormal basis of the space of \(\pi \).
The starting point of any trace formula is a geometric expansion for \(K^{\mathrm {cusp}}_{f}(x,y)\), namely
where \(K_{f}(x,y):=\sum _{\gamma \in G(F)} f(x^{-1}\gamma y)\) is the kernel function for R(f) and \(K_f^{\mathrm {Eis}}(x,y)\) is the contribution from Eisenstein series (which we will not explicate). Integrating \(K^{\mathrm {cusp}}_f(x,y)\) along various subgroups of \((A_G G(F) \backslash G(\mathbb {A}_F))^{\times 2}\) yields various trace formulae. The canonical example is integration along the diagonal copy of \(A_G G(F) \backslash G(\mathbb {A}_F)\); this leads to the usual trace formula.
Other subgroups can be used, and this leads to trace formulae that isolate representations having particular properties. For example, integrating along a twisted diagonal isolates representations isomorphic to their conjugates under an automorphism, and by Jacquet’s philosophy that has been made more precise in work of Sakellaridis and Venkatesh [14], integration along spherical reductive subgroups ought to isolate representations that are functorial lifts from smaller groups.Footnote 1
There are natural limits to what sort of representations can be isolated via these methods. As mentioned above, integration along a twisted diagonal isolates representations whose isomorphism class is invariant under a cyclic subgroup of the group \(\mathrm {Out}_F(G)\) of outer automorphisms of G, and integrating along a pair of spherical subgroups seems usually to detect representations whose isomorphism class is invariant under a pair of involutory automorphisms, which can at most generate a dihedral subgroup of \(\mathrm {Out}_F(G)\). As the author has advocated in [5, 8], for applications to nonsolvable base change, it would be useful to develop trace formulae that isolate representations whose isomorphism class is invariant under nonsolvable subgroups of \(\mathrm {Out}_F(G)\).
Let \(f_1,f_2 \in C_c^\infty (A_G \backslash G(\mathbb {A}_F))\) and let \(g_1,g_2,h_1,h_2 \in G(\mathbb {A}_F)\). One way of approaching the problem of isolating representations invariant under more automorphisms is to build geometric expressions for
where the sum is over isomorphism classes of cuspidal automorphic representations \(\pi \) of \(A_G \backslash G(\mathbb {A}_F)\) and \(w(\pi ) \in \mathbb {C}\) is some weight factor. One could then integrate this kernel over two twisted diagonals and isolate representations whose isomorphism class is invariant under a subgroup of \(\mathrm {Out}_F(G)\) generated by two elements. We recall that any finite simple nonabelian group is generated by two elements (see [9, Theorem 1.6] and the paragraph after it), so this is a quite general setup.
In this paper, we develop a geometric expression for (1.1) for a particular weight \(w(\pi )\) in the special case where \(G=\mathrm {GL}_2\).
1.1 The case at hand
We now let \(G:=\mathrm {GL}_2\) and \(A:=A_G\). Let S be a set of places of F including the infinite places such that \(F/\mathbb {Q}\) is unramified outside of S and \(\mathcal {O}_F^S\) has class number 1.
Let \(f_1,f_2 \in C_c^\infty (A \backslash \mathrm {GL}_2(F_S))\), let \(M_\ell ,M_r \le \mathrm {GL}_2\) be the split tori whose points in a \(\mathbb {Z}\)-algebra R are given by
let
and let
where the sum is over isomorphism classes of cuspidal automorphic representations of \(A \backslash G(\mathbb {A}_F)\). Let \(\mathfrak {gl}_n\) denote the affine \(\mathbb {Z}\)-scheme of \(n \times n\) matrices and let
viewed as affine schemes over \(\mathbb {Z}\). We also let \(\mathcal {W} \subset \mathbb {G}_m \times \mathcal {V}'\) denote the closed subscheme whose points in a \(\mathbb {Z}\)-algebra are given by
We note that there is an action
given on points by
This action preserves \(\mathcal {W}\). By forgetting the \(\mathbb {G}_m\) factor, we also obtain an action of \(\mathrm {GL}_2^{\times 2} \times M_\ell \times M_r\) on \(\mathcal {V}\) that preserves \(\mathcal {V}'\).
Define
Writing \(T=(t_{ij})\), we give \(\mathcal {V}(\mathbb {A}_F)\) the additive Haar measure
where \(\hbox {d}x=\prod _v\,\hbox {d}x_v\) is the Haar measure on \(\mathbb {A}_F\) such that
In general, all Haar measures in this paper are normalized as in [8, §2, (3.1.1)]. We let \(\psi :=\psi _{\mathbb {Q}} \circ \mathrm {tr}_{F/\mathbb {Q}}\) where \(\psi _{\mathbb {Q}}=\psi _\infty \prod _{p}\psi _p\) is the unique additive character of \(\mathbb {Q}\backslash \mathbb {A}_\mathbb {Q}\) that is trivial on \(\widehat{\mathbb {Z}}\) and satisfies \(\psi _\mathbb {R}(x)=e^{-2\pi i x}\) (it is given explicitly in loc. cit.). Our choice of measure is not self-dual with respect to \(\psi \), which leads to the appearance of powers of \(d_F^{1/2}\) in our formulae.
For \(\beta =(b,\alpha ) \in \mathcal {W}(F_S)\), \(f_1,f_2 \in C_c^\infty (A \backslash \mathrm {GL}_2(F_S))\), and \(V \in C_c^\infty ((0,\infty ))\), we define an integral transform
The convergence of this integral is proven in Propositions 3.1 and 3.2 below.
Remark
The function V has no relation to the scheme \(\mathcal {V}\).
Let
The following is the main theorem of this paper:
Theorem 1.1
If \(R(f_1)\) and \(R(f_2)\) have cuspidal image, then
where the sum on c is over a set of representatives for the nonzero ideals of \(\mathcal {O}_F^S\). This sum converges absolutely.
Remark
The assumption that \(R(f_1)\) and \(R(f_2)\) have cuspidal image is only invoked to simplify the spectral side of our expression. In fact, the only place in the paper where the assumption is used is in the assertion (1.7). In principle, this assumption should be no loss of generality spectrally [13].
Originally, we hoped to integrate this kernel over an appropriate subgroup of \(A\mathrm {GL}_2(F) \backslash \mathrm {GL}_2(\mathbb {A}_F)^{\times 4}\) to isolate representations whose isomorphism class is invariant under a simple nonabelian subgroup of \(\mathrm {Aut}_\mathbb {Q}(F)\). This would necessarily involve some truncation and some sort of version of the Rankin–Selberg method. Later we found an alternate expression for the kernel that will make this process easier [6]. However, Theorem 1.1 can be put to other use immediately. As a concrete example, let \(\chi _1,\chi _2,\chi _3,\chi _4:F^\times \backslash \mathbb {A}_F^\times \rightarrow \mathbb {C}^\times \) be a quadruple of characters. By integrating over appropriate split tori and twisting over characters, the formula could be used to study the asymptotics of sums of products of L-functions of the form
as the analytic conductor of \(\pi \) increases. W. Zhang has also pointed out to the author the possibility of using the main theorem to prove a new Waldspurger-type formula (compare [15, §4.2]) involving products of L-functions as above.
Remark
In [8], the authors provided an absolutely convergent geometric expansion of a trace formula that isolates representations whose isomorphism class is invariant under a simple nonabelian group. However, it is not clear how to write the resulting trace formula as a sum of terms that factor along places of \(G(\mathbb {A}_F)\). We hope that the present approach and its refinement in [6] will allow us to work around this difficulty.
1.2 Outline of the proof
For \(m \in \mathcal {O}_F^S\) let
be the usual unramified Hecke operators. For \(? \in \{\emptyset , \mathrm {cusp}\}\) let
where the sums on a and m are over a set of representatives for the (principal) nonzero ideals of \(\mathcal {O}_F^S\).
The sums over a and m are finite for each X, and sum over \(y_1,y_2 \in F^\times \) of the integrals over \(F \backslash \mathbb {A}_F\) is part of the Fourier expansion of the smooth function
evaluated at \((t_1,t_2)=0\); hence, the sums over \(y_1,y_2\) are rapidly decreasing.
Remark
The motivation for introducing this partial Fourier expansion is that it has no effect on the cuspidal part of the kernel (compare Proposition 5.1), but eliminates the nongeneric spectrum. If the nongeneric spectrum were included, its contribution would be of size O(X), whereas the cuspidal terms in which we are interested are of size O(1) (since we are dividing by X). In principle, this should be unnecessary, as our assumption that \(R(f_1)\) and \(R(f_2)\) have cuspidal image also eliminates the nongeneric spectrum, but we do not know how to use the assumption that \(R(f_1)\) and \(R(f_2)\) have cuspidal image when working with the geometric side of the formula.
By our assumption that \(R(f_1),R(f_2)\) have cuspidal image, we have
and the proof of Theorem 1.1 boils down to computing the limit as \(X \rightarrow \infty \) of both sides of this expression. Regarding \(\Sigma ^{\mathrm {cusp}}(X)\), the function of taking a sum over m and taking a limit as \(X \rightarrow \infty \) is to isolate the pairs of representations \(\pi _1,\pi _2\) occurring in the product
such that \(\pi _2 \cong \pi _1^{\vee }\). We use Rankin–Selberg theory to make the precise and compute \(\lim _{X \rightarrow \infty } \Sigma ^{\mathrm {cusp}}(X)\) in Sect. 5.
The majority of this paper is devoted to computing \(\lim _{X \rightarrow \infty }\Sigma (X)\) geometrically. The sum \(\Sigma (X)\) is divided into two contributions in Sect. 2, namely \(\Sigma _1(X)\), corresponding to the first Bruhat cell, and \(\Sigma _2(X)\), corresponding the second Bruhat cell (both in the second kernel). We analyze \(\Sigma _2(X)\) in Sect. 3. The main result is Theorem 3.6. The limit \(\lim _{X \rightarrow \infty }\Sigma _1(X)\) turns out to be zero, as proven in Sect. 4. We note that the reason we can execute the limit of the geometric side is that the relevant exponential sums have the same length as the relevant modulus, just as in the beyond endoscopy approach to Rankin–Selberg L-functions exposed in [11] [compare the remark after (3.2) below]. In fact, one could probably use the geometric estimates of \(\Sigma (X)\) we give in this paper to prove, independently of Rankin–Selberg theory, that a certain sum of Rankin–Selberg L-functions has an analytic continuation to \(\mathrm {Re}(s)>1-\delta \) for some \(\delta >0\), but we do not pursue this as it is not our purpose here.
2 First manipulations with the geometric side
We consider, for \(X \in \mathbb {R}_{>0}\),
Let \(B=MN \le \mathrm {GL}_2\) be the standard Borel subgroup of upper triangular matrices, with M the diagonal matrices and N the unipotent radical. The Bruhat decomposition is
where \(w_0 =\left( {\begin{matrix} &{} 1 \\ 1 &{} \end{matrix}} \right) \). Here B(F) (resp. \(N(F) w_0B(F)\)) is referred to as the first (resp. second) Bruhat cell. We apply this to write \(\Sigma (X)=\Sigma _{1}(X)+\Sigma _{2}(X)\) as the sum of the term corresponding to the first Bruhat cell:
and the second Bruhat cell:
There is a somewhat confusing point hidden in these formulae which we now elucidate. First
To arrive at the expressions for \(\Sigma _1(X)\) and \(\Sigma _2(X)\) above one uses this and then a change of variables \(\delta \mapsto a\delta \).
We will compute the limit of these expressions as \(X \rightarrow \infty \) in the following sections, starting with (2.2). Throughout the remainder of this paper, any unspecified constants are allowed to depend on the quantities \(F,S,f_1,f_2,V,g_1,g_2,h_1,h_2\).
3 The second Bruhat cell
We study the contribution (2.2) of the second Bruhat cell. Under the action of \(N \times N\) on \(\mathrm {GL}_2\) via \( (n_1,n_2)\cdot g:=n_1^{-1}gn_2\), the stabilizers of elements in the second Bruhat cell are trivial, and each is in the orbit of a unique element of \(w_0 M(F)\). We therefore have that
We write \(\delta =\left( {\begin{matrix} &{} b/c \\ c &{} \end{matrix}} \right) \) and take a change of variables \((t_1,t_2) \mapsto (c^{-1}t_1,c^{-1}t_2)\) for each c to obtain
Substituting this into (3.1) and solving for m, we have
We hope the use of the symbol B for an element of \(\mathfrak {gl}_2(F)\) and for the Borel subgroup of \(\mathrm {GL}_2\) does not cause confusion. Here we have extended the domain of \(\frac{V(t)}{t}\) from \((0,\infty )\) to \(\mathbb {R}\) by taking it to be zero outside of \((0,\infty )\) (it remains smooth upon extension because \(V \in C_c^\infty ((0,\infty ))\).
Remark
Assume for simplicity that \(g_1=g_2=h_1=h_2=I\). The moduli in the sum above are the c. Considering the support of \(f_2\), we see that \(|c|_S \ll \sqrt{X}\) and the sum on \(B \in \mathfrak {gl}_2(\mathcal {O}_F^S)\) is over matrices whose entries \(b_{ij}\) satisfy \(|b_{ij}|_S \ll \sqrt{X}\). Thus, Poisson summation in B has a chance of being profitable, and indeed it is, as we will see in the next subsection.
3.1 Poisson summation in B
For \(n \ge 1\) and \(\Psi \in C_c^\infty (\mathfrak {gl}_n(\mathbb {A}_F))\) let
be the Fourier transform of \(\Psi \). We use the analogous notation in the local setting. Note in particular that
For \(g_1,g_2 \in \mathrm {GL}_n(\mathbb {A}_F)\) and \(a \in \mathbb {A}_F^\times \), the Fourier transform of
is
We also note that the Poisson summation formula holds in the form
We apply Poisson summation in \(B \in \mathfrak {gl}_2(F)\) to (3.2) to see that \(\Sigma _2(X)\) is equal to
Here \(\mathcal {V}\) is defined as in (1.2) and \(\langle \cdot , \cdot \rangle \) is defined as in (1.4).
Taking a change of variables
the above becomes
where we have set \(g=(g_1,g_2,h_1,h_2)\) and (as in the introduction)
3.2 Bounds on archimedian orbital integrals
In Sect. 3.5, we will apply Poisson summation in \(c \in F^\times \) to \(\Sigma _2(X)\). In order to work with the resulting sum, we require some bounds that are collected in this section. The archimedian bounds in Proposition 3.1 and the nonarchimedian bounds in Proposition 3.2 are obtained via the stationary phase method.
If w is a place of F, let
Moreover, if \(a \in \mathfrak {gl}_2(F_\infty )\) or \(a \in F_\infty \) let
Proposition 3.1
Let \(f_1 \in C_c^\infty (\mathrm {GL}_2(F_\infty ))\) and \(f_2 \in C_c^\infty (A \backslash \mathrm {GL}_2(F_\infty ))\). Assume \(b \in F_\infty ^\times \) satisfies \(|b|_\infty \asymp 1\). The function
vanishes if b lies outside a compact subset of \(F_\infty ^\times \) depending only on \(f_1,f_2\) and the bound on \(|b|_\infty \).
Moreover, for any \(N \in \mathbb {Z}_{\ge 0}\), \(\beta \in \mathbb {R}_{>0}\), (unitary) character \(\chi :F^\times _\infty \rightarrow \mathbb {C}^\times \), and \(s \in \mathbb {C}\) with \(\beta>\mathrm {Re}(s)>-3\) the integral
is bounded by a constant depending on \(f_1,f_2,N,\beta \) times
where \(\alpha =(B,y_1,y_2) \in V'(F_\infty )\).
In the proposition, \(C(\chi ,t)\) is the analytic conductor of \(\chi \) normalized as in [3, §1].
Remark
To clarify the assumptions in the proposition, note that if F has more than one infinite place then the set of \(b \in F_\infty ^\times \) with \(|b|_\infty \asymp 1\) is noncompact.
Proof
Choose \(\widetilde{f}_2 \in C_c^\infty (\mathrm {GL}_2(F_\infty ))\) such that \(\int _{A}\widetilde{f}_2(zg)\hbox {d}z^\times =f_2(g)\). Then taking a change of variables \((t_1,t_2) \mapsto z^{-1}(t_1,t_2)\), we see that the function in the proposition is equal to
when evaluated at \((z^{-1}t,y_1,y_2,B)\). The function \(\widetilde{f}_2\) is evaluated on an element of determinant \(-z^2b \det T\), and hence for the integral to be nonzero b and z must lie in a compact set depending only on \(f_1,\widetilde{f}_2\). We may therefore fix b and z and drop the integral over A in the ensuing argument and consider instead of (3.8) the integral
where \(\alpha _z:=(zB,y_1,y_2)\).
We now employ an idea from [12] to rewrite this as an integral to which we can apply the stationary phase method. Write
it is a partial Fourier transform of \(\widetilde{f}_2\). Let
By Fourier inversion, we have that (3.9) is equal to
Here we have renamed variables (so \(\hbox {d}v=\hbox {d}x_1\,\hbox {d}x_2\,\hbox {d}T\)). Thus, we are tasked with bounding, for each \(w|\infty \) and each character \(\chi :F_w^\times \rightarrow \mathbb {C}^\times \), the integral
Let \(D:=t\frac{\partial }{\partial t}\) and, if w is complex, \(\overline{D}:= \overline{t} \frac{\partial }{\partial \overline{t}}\). We view these as differential operators on \(F_w^\times \). Let \(f_4 \in C_c^\infty (\mathfrak {gl}_2(F_w))\). Suppose that for all \(N \ge 0\), \(i \ge 0\) (and if w is complex \(j \ge 0\)) one has
for all t in the support of \(f_4\left( {\begin{matrix} -x_1 &{} x_3 \\ t &{} x_2\end{matrix}} \right) \) (this is a compact subset of \(F_w\)). Here we take \(j=0\) if w is real. Assuming this is the case a repeated application of integration by parts in t (and \(\overline{t}\) when w is complex) implies that (3.10) is convergent for \(\mathrm {Re}(s)>-3\) and moreover that (3.10) is bounded by
for \(\beta>\mathrm {Re}(s)>-3\), and this implies the proposition. But, since \(f_1\) and \(f_4\) were arbitrary, it is not hard to see that the estimate (3.11) follows in general from the special case when \(i=j=0\). In other words, we have reduced the proposition to proving that for all \(N \ge 0\) one has
for each \(t \in F_w^\times \).
We now apply the stationary phase method to estimate this sum. We view it as a family of phase integrals indexed by \(x_3\). We will estimate, for each \(x_3 \in F_w\), the integral
Let \(D_{x_3} \subset \mathcal {V}(F_w)\) be the singular locus of \(\mathcal {F}_{x_3}(v)\). We have
So \(D_{x_3}\) is empty if \(x_3=0\) and otherwise \(D_{x_3}\) consists of the single point
and the determinant of the Hessian matrix of \(\mathcal {F}_{x_3}(v)\), evaluated at the only point in \(D_{x_3}\), is \(\pm x_3^6z^8b^4\).
Now if \(D_{x_3}\) is not in the support of \(f_1(T ) f_4\left( {\begin{matrix} -x_1 &{} x_3 \\ t &{} x_2\end{matrix}} \right) \), then (3.13) can be estimated using the Riemann–Lebesgue lemma. On the other hand, if \(D_{x_3}\) is in the support, we obtain a bound on (3.13) of the form \(O_{f_1,f_4}\left( |\frac{t^3}{x_3^3z^4b^2}|_w\right) \) by the stationary phase method. Thus, we obtain a bound
where \(f_5\) is a Schwartz function on \(F_w\). This in turn is bounded by
\(\square \)
3.3 Bounds in the ramified nonarchimedian case
For this subsection, let w be a finite place of F. We omit it from notation, writing \(F:=F_w\). We write \(\varpi \) for a uniformizer of \(\mathcal {O}_F\) and set \(q:=|\varpi |^{-1}\). We let \(\delta \in \mathcal {O}_F\) be a generator of the absolute different \(\mathcal {D}_F\) of \(\mathcal {O}_F\). Finally for ideals \(\mathfrak {m}\subsetneq \mathcal {O}_F\) we write
and \(\mathcal {O}_F^\times (\mathcal {O}_F)=\mathcal {O}_F^\times \). For \(m \in \mathcal {O}_F-0\) we also write \(\mathcal {O}_F^\times (m):=\mathcal {O}_F^\times (m\mathcal {O}_F)\). Write
In this subsection, we prove the following proposition:
Proposition 3.2
Let \(\chi :F^\times \rightarrow \mathbb {C}^\times \) be a (unitary) character, let \(f_1,f_2 \in C_c^\infty (\mathfrak {gl}_2(F))\) and assume that \(b \in F^\times \). The integral
is absolutely convergent for \(\mathrm {Re}(s)>0\). For \(\alpha \in \mathcal {V}'(F)\) in the expression
the integral over \(F^\times \) is bounded in absolute value by a constant depending on \(f_1,f_2,|b|,w\) times \(1+q^{6\mathrm {min}(v(b_{ij}),v(y_i))}\sum _{n=2}^\infty q^{-n(3+s)}\) for \(\mathrm {Re}(s)>-3\). Moreover, it vanishes if \(|y_1|,|y_2|,|B|\) or the absolute norm of the conductor of \(\chi \) is sufficiently large in a sense depending only on \(f_1\), \(f_2\), and |b|.
Proof
It is not hard to see that the integral over \(F^\times \times \mathcal {V}(F)\) in (3.16) is absolutely convergent for \(\mathrm {Re}(s)>0\). We therefore assume that \(\mathrm {Re}(s)>0\) until otherwise stated to justify the ensuing manipulations.
Consider
We claim that if \(u \in \mathcal {O}_F^{\times }(\varpi ^k)\) and \(k \ge 1\) is large enough in a sense depending only on \(f_1,f_2\) then
is equal to (3.17). Indeed, this follows from a change of variables \(v \mapsto uv\). We conclude that the integral (3.16) vanishes if \(\chi |_{\mathcal {O}_F^\times (\varpi ^k)}\) is nontrivial for k sufficiently large in a sense depending only on \(f_1\) and \(f_2\), in other words, if the conductor of \(\chi \) is sufficiently large in a sense depending only on \(f_1,f_2\).
After a change of variables \(v \mapsto tv\) in (3.17), we see that it is equal to
Notice that
is invariant under \(v \mapsto v+\varpi ^kv'\) for any \(v' \in \mathcal {V}(\mathcal {O}_F)\), provided that k is sufficiently large in a sense depending only on \(f_1,f_2\) and |b|. Thus, (3.17) (and hence (3.16)) vanishes if \(|y_1|\), \(|y_2|\) or |B| is sufficiently large in a sense depending only on \(f_1,f_2\) and |b|.
Thus, we are left with proving the bound claimed in the proposition. As in the proof of Proposition 3.1, we employ a partial Fourier transform as in [12], writing
Let \(d_F \in \mathbb {Z}_{>0}\) be the absolute discriminant of F. By Fourier inversion, we have that (3.16) is equal to \(d_F^{-1}\) times
We can assume that \(f_1=\mathbbm {1}_{\gamma \varpi ^{-m}+\varpi ^{k}\mathfrak {gl}_2(\mathcal {O}_F)}\) and \(f_2=\mathbbm {1}_{\beta \varpi ^{-m}+\varpi ^{k}\mathfrak {gl}_2(\mathcal {O}_F)}\) for some \(\gamma , \beta \in \mathfrak {gl}_2(\mathcal {O}_F)\) and \(m,k \ge 0\). Thus, the above becomes
Applying a change of variables \((v,x_3,t) \mapsto \varpi ^{-m}(v,x_3,t)\), we arrive at
The factor \(\chi (\varpi ^{-m})q^{(7+s)m}\) is inessential for our purposes so we drop it.
Now let \(\ell \ge 0\) be the smallest integer such that \(\varpi ^{\ell }b \in \mathcal {O}_F\). We can then write the above as
To bound this integral, we can and do assume \(\chi =1\). Write \(\gamma =(\gamma _{ij})\), \(\beta =(\beta _{ij})\). To ease notation, let
where \(\delta \in \mathcal {O}_F\) is a generator for the absolute different of F. Then, (3.18), in the special case \(\chi =1\), is
We first observe that if \(\varpi ^{k+m} \not \mid \beta _{21}\) then \(|t|=|\beta _{21}|\) for all t in the support of the integrand, and it is easy to obtain the bound asserted in the lemma in this case. If \(\varpi ^{k+m} \mid \beta _{21}\), then the integral above is equal to
We claim that is bounded by a constant depending on \(k,m,\ell ,|b|,\beta ,\gamma \) times
for \(\mathrm {Re}(s)>-3\); establishing this claim will complete the proof of the proposition. To prove the claim, it suffices to show that
is bounded by a constant depending on \(k,m,\ell ,|b|,\beta ,\gamma \) times \(|t|^3\), provided that \(w(t) \ge \max (2k,2)\) (for \(0 \le w(t) \le \max (2k,2)\) we can just bound the integral trivially).
For this, we can apply the stationary phase method in the nonarchimedian setting as developed in [4]. In more detail, let p be the rational prime lying below w and let
where \((\nabla \mathcal {F}_{x_3})\) is the ideal generated by the entries of the gradient \(\nabla \mathcal {F}_{x_3}\). This gradient is
Since \(w(t)>\max (2k,m)\), we have
Therefore, by [4, Theorem 1.8(a)] we have that (3.19) is bounded by a constant times
where \(a=\lfloor \frac{w(t)+2m+\ell }{2}\rfloor \). But
so we deduce the proposition. \(\square \)
3.4 The unramified computation
Fix a (finite) place \(w \not \in S\) of F. As in the previous subsection, in this subsection we omit w from notation, write \(F:=F_w\), let \(\varpi \) be a uniformizer of F and set \(q:=|\mathcal {O}_F/\varpi |\). We fix \(b \in \mathcal {O}_F^\times \) for the section. Let \(\chi :F^\times \rightarrow \mathbb {C}^\times \) be a (unitary) character. Moreover, let
In this section, we prove the following proposition:
Proposition 3.3
The integral
is absolutely convergent for \(\mathrm {Re}(s)>-3\). It vanishes if \(\chi \) is ramified. If \(\chi \) is unramified, it is equal to
We start with two preparatory lemmas:
Lemma 3.4
For \(t \in \mathcal {O}_F\) one has
Proof
Assume first that \(w(t)=1\). Then writing \(v=\left( \left( {\begin{matrix} x_1 &{} x_2 \\ x_3 &{} x_4 \end{matrix}} \right) ,z_1,z_2 \right) \),
Let \(\mathbb {P}\mathcal {V}\) be the projectivization of the \(\mathcal {O}_F\)-module \(\mathcal {V}\). Then grouping elements of \(\mathcal {V}(\mathcal {O}_F/\varpi )\setminus 0\) according to their image in \(\mathbb {P}\mathcal {V}(\mathcal {O}_F/\varpi )\) and evaluating the resulting Ramanujan sums, we see that the above is equal to
To count the points
satisfying
we observe that there are \(q^4\) points with \(z_1 \ne 0\), \(q^3\) points with \(z_1=0,x_1 \ne 0\), \(q^2\) points with \(z_1=x_1=0\), \(x_2 \ne 0\), and \(\frac{q^3-1}{q-1}\) points with \(z_1=x_1=x_2=0\). Thus, we end up with
points. We deduce that (3.20) is equal to \(q^{-3}\).
We now consider the case \(w(t){>}1\). Let
be the gradient of P(b, v) and let
be the Hessian matrix of P(b, v). Let p be the rational prime below w and let
where \((\nabla P(b,v))\) is the ideal generated by the entries of \(\nabla P(b,v)\). Thus, D is a closed affine subscheme of \(\mathbb {A}^{6[F:\mathbb {Q}_p]}\) that is étale over \(\mathbb {Z}_p\) since \(H(b,v) \in \mathrm {GL}_6(\mathcal {O}_F)\). Applying [4, Theorem 1.4] (a result which the authors attribute to Katz), one has
where
Now \(D(\mathbb {Z}_p)=0\), and for \(2 \not \mid w(t)\), one has
Thus, altogether we deduce that (3.21) is equal to \(|t|^3\) as claimed. \(\square \)
Lemma 3.5
Assume that \(x \in \mathcal {O}_F^\times \). The integral
vanishes unless \(\alpha \in \mathcal {V}(\mathcal {O}_F)\) and \(\chi \) is unramified, in which case it is equal to
Proof
It is easy to see that the integral vanishes unless \(\alpha \in \mathcal {V}(\mathcal {O}_F)\). We henceforth assume \(\alpha \in \mathcal {V}(\mathcal {O}_F)\). We observe that
where f is the \(\mathcal {O}_F\)-linear isomorphism
Thus,
Invoking Lemma 3.4, we see that this is equal to
\(\square \)
Proof of Proposition 3.3
One has
We take a change of variables \(t \mapsto \varpi ^kt\) to arrive at
We now invoke Lemma 3.5 to see that this vanishes if \(\chi \) is ramified and otherwise it is equal to
\(\square \)
3.5 Poisson summation in c
Recall that (3.5) gives us the following equality:
where we have set \(g=(g_1,g_2,h_1,h_2)\). In this subsection, we apply Poisson summation in c to this expression and asymptotically evaluate the resulting sum. The main result follows:
Theorem 3.6
The limit \(\lim _{X \rightarrow \infty } \Sigma _2(X)\) exists and is equal to the absolutely convergent sum
Here \(I_S(f,\beta )\) is defined as in (1.5).
Proof
We apply Poisson summation in \(c \in F^\times \) to (3.25) to arrive at
Here the sum on \(\chi \) is over \((AF^\times \backslash \mathbb {A}_F^\times )^{\wedge }\). A convenient reference for this multiplicative version of Poisson summation is [1, §2]. We warn the reader that there is a difference of measure; if \(\hbox {d}x^\times _{BB}\) is the measure used in [1, §2], then
where \(\hbox {d}x^\times \) is our measure. We will discuss justifying this application of Poisson summation in just a moment. The nonarchimedian integral
was computed in Proposition 3.3; it vanishes unless \(\chi \) is unramified outside of S, in which case it is equal to
where the sum on c is over the nonzero ideals of \(\mathcal {O}_F^S\) and \(P^{\vee }(b,\alpha ):=P(b^{-1},\alpha )\).
For \(\alpha \in \mathcal {V'}(F_S)\) let
and
We note that for \(\mathrm {Re}(s)>-3\) the transform \(I_S(f,(b,\alpha ),\chi ,s)\) is rapidly decreasing as a function of \(\alpha \), \(\chi \) and the analytic conductor of \(\chi |\cdot |^s\) in a sense made precise by propositions 3.1 and 3.2.
Let
where \(z^{[F:\mathbb {Q}]^{-1}}\) is embedded diagonally. Taking a change of variables
using the A-invariance of \(f_1,f_2\), and bearing in mind the nonarchimedian computation just mentioned, we see that
where the sum on \(\chi \) is over characters of \(F^\times \backslash \mathbb {A}_F^\times /\widehat{\mathcal {O}}_F^{S \times }\). We note that we have used the fact that \(\chi _S(a_S^{-1})=\chi ^S(a)\) to simplify the expression above. We now can justify our application of Poisson summation in c by noting that
for \(\chi ,b,\alpha \) contributing to the sum and applying the estimates of Propositions 3.1 and 3.2.
We now move the contour of the integral over s to the line \(\mathrm {Re}(s)=-\tfrac{5}{2}\). The integral \(D_{\chi ,b,\alpha }(s)\) is absolutely convergent in this range unless \(P(b^{-1},\alpha )=0\) (which occurs if and only if \(P(b^{-1}\det g_1^{-1}g_2h_1h_2^{-1},g.\alpha )=0\)). On the other hand, if \(P(b^{-1},\alpha )=0\), which is to say that \((b,\alpha ) \in \mathcal {W}(F)\), then
which is meromorphic in \(\mathrm {Re}(s)>-3\), in fact holomorphic apart from a possible simple pole at \(s=-2\). The simple pole only occurs if \(\chi \) is trivial, in which case it has residue
Thus, we \(\Sigma _2(X)\) is equal to the contribution of the residues plus a remainder term. The contribution of the residues is
where we have set
as in the introduction. Here we are using the fact that \(\hbox {d}t_{S {\setminus } \infty }=\zeta _{FS{\setminus } \infty }(1)^{-1}\hbox {d}t_{S{\setminus } \infty }^{\times }\).
This is precisely the expression for \(\lim _{X \rightarrow \infty }\Sigma _2(X)\) asserted in Theorem 3.6. The sum on \(c,b,\alpha \) is absolutely convergent by Propositions 3.1 and 3.2. Thus, to complete the proof of Theorem 3.6 it suffices to prove that the remainder term mentioned above is indeed a remainder term. This remainder term is \(\frac{d_F^{-5/2}|\det h_1h_2^{-1}|}{2\pi i\mathrm {Res}_{s=1}\zeta _F^\infty (s)}\) times
Notice that the sum over a causes no problems since it only appears via the factor \(\frac{\chi ^S(a)}{|a|_S^{4+s}}\) and \(\sum _{a} |a|_S^{-4-s}\) converges absolutely for \(\mathrm {Re}(s)=-\frac{5}{2}\). Moreover, as mentioned above,
is rapidly decreasing as a function of \(b,\alpha ,c\) and the analytic conductor of \(\chi |\cdot |^s\) is a sense made precise in Propositions 3.1 and 3.2. As for the factor \(D_{\chi ,b,\alpha }\), we observe that in view of (3.27) there is an \(A>0\) such that if \(b,\alpha ,\chi \) contribute to the sum above and \(P(b^{-1},\alpha ) \ne 0\) and \(\mathrm {Re}(s)=-\tfrac{5}{2}\) then
If \(P(b,\alpha )=0\), one has \(D_{\chi ,b,\alpha }(s)=L^S(s+4,\chi )^{-1}L(s+3,\chi )\) as mentioned above. Combining these observations, we easily deduce that (3.30) is \(O_{f,V,\varepsilon }(X^{\varepsilon -1/4})\) for any \(\varepsilon >0\). This completes the proof of Theorem 3.6. \(\square \)
4 The first Bruhat cell
In this section, we study the contribution of the first Bruhat cell:
The main result of this section, Lemma 4.1, asserts that \(\lim _{X \rightarrow \infty }\Sigma _1(X)=0\).
Under the action of \(N(F) \times N(F)\), every element of B(F) is in the orbit of a unique element of the form
and the stabilizer of such an element is
We give \(N_{\gamma }(\mathbb {A}_F)\) a Haar measure via the isomorphism
One says that \(\delta \) is relevant if the character
is trivial on this stabilizer; thus, \(\delta \) is relevant if and only if \(-bc^{-1}y_1=y_2\). Only relevant elements can contribute to \(\Sigma _1(X)\). Thus,
We compute
Substituting this into the expression (4.2) for \(\Sigma _1(X)\) and simplifying, we arrive at
Let \(S_0 \supseteq S\) be a finite set of places such that \(g_1^{S_0},g_2^{S_0},h_1^{S_0},h_2^{S_0} \in \mathrm {GL}_2(\widehat{\mathcal {O}}_F^{S_0})\). Then the above is equal to
Notice that multiplying our representative a for an ideal in \(\mathcal {O}_F^{S}\) by an element of \(\mathcal {O}_F^{S \times }\) does not affect the sum, as it simply permutes the c, b, y and B sums. Therefore, we can and do assume that our representatives a for ideals in \(\mathcal {O}_F^S\) are chosen so that
for \(w|\infty \) and
for \(w \in S {\setminus } \infty \).
Lemma 4.1
For any \(\varepsilon >0\) one has \(\Sigma _1(X) \ll _\varepsilon X^{\varepsilon -\tfrac{1}{2}}\).
Proof
Choose \(\widetilde{f}_2 \in C_c^\infty (\mathrm {GL}_2(F_S))\) such that \(\int _A \widetilde{f}_2(zg)\hbox {d}z^\times =f_2(g)\). Then,
Here \(\Delta \) is defined as in (3.28).
By considering determinants, we see that in order for the integrand to be nonzero we must have
For a, B to contribute to \(\Sigma _1(X)\) we must have \(|a^2 \det B|_S \asymp X\), and hence, for z in the support of the integrand in (4.5) one has
for \(w|\infty \). Thus, we can essentially ignore the integral over z. It also follows similarly that
for all \(w|S_0\), so we may fix b and ignore the sum over b.
We also observe that for c to contribute to \(\Sigma _1(X)\) we must have
for all \(w \in S_0\). There are at most \(O\left( \frac{\sqrt{X}}{|a|_S}\right) \) such \(c \in \mathcal {O}_F^{S_0}-0\).
The integral (4.5) vanishes if \(|y|_w \gg |a|_w\) for \(w \in S_0-\infty \). Applying integration by parts in \(t_w\) for \(w|\infty \) to (4.5) implies that for all \(N \ge 0\) it is bounded by a constant depending on N times
for an appropriate differential operator D.
Given our conventions (4.3) and (4.4), we deduce that if \(\varepsilon >0\) and \(|a|_S \le X^{1/2-\varepsilon }\) then for any \(N>0\) one has
Here we are using (4.6) to obtain a bound of \(\sqrt{X}\) for the number of contributing c.
We are left with the case \(|a|_S \ge X^{1/2-\varepsilon }\). For this case, we observe that any B contributing to the sum satisfies
for \(w|\infty \) and \(|B|_w \ll 1\) for \(w|S_0-\infty \). Here \(|B|_w\) is defined as in (3.6).
Let the box norm \(|| \cdot ||_\infty \) be defined as in (3.7). Then, for a small enough nonzero ideal \(\mathfrak {N} \subseteq \mathcal {O}_F\) the contribution of these a to \(\Sigma _1(X)\) is bounded by a constant times
where one factor of \(\frac{\sqrt{X}}{|a|_S}\) comes from the sum over c and the other factor of \(\frac{\sqrt{X}}{|a|_S}\) comes from the bound (4.7), which has also been used to bound the sum on y. \(\square \)
5 The cuspidal contribution and the proof of Theorem 1.1
Theorem 1.1 follows immediately upon combining (1.7), Theorem 3.6, Lemma 4.1, and the following proposition:
Proposition 5.1
The limit \(\lim _{X \rightarrow \infty } \Sigma ^{\mathrm {cusp}}(X)\) exists and is equal to the absolutely convergent sum
Here the sum on \(\pi \) is over isomorphism classes of cuspidal automorphic representations of \(A \backslash \mathrm {GL}_2(\mathbb {A}_F)\).
Proof
We first observe that since \(K_{f_2}^{\mathrm {cusp}}(x,y)\) is cuspidal
is the Fourier expansion of \(d_FK^{\mathrm {cusp}}_{f_2\mathbbm {1}_{a,a}*\mathbbm {1}_m}\left( \left( {\begin{matrix}1 &{} x_1 \\ &{} 1 \end{matrix}}\right) h_1,\left( {\begin{matrix}1 &{} x_2 \\ &{} 1 \end{matrix}}\right) h_2\right) \) evaluated at \(x_1=x_2=0\). Thus,
Here the sum on \(\pi _1,\pi _2\) is over pairs of cuspidal automorphic representations of \(A \backslash \mathrm {GL}_2(\mathbb {A}_F)\). By Mellin inversion and standard preconvex estimates [3, (10)], there is a \(\delta _1>0\) such that
where \(C(\pi _1 \times \pi _2):=C(\pi _1 \times \pi _2,0)\) is the analytic conductor of \(\pi _1 \times \pi _2\) (compare, e.g., [2, §3]). By Rankin–Selberg theory, the residue is nonzero only if \(\pi _1 \cong \pi _2^\vee \), in which case it is bounded by \(C(\pi \times \pi ^{\vee })^{\delta _2}\) for some \(\delta _2>0\) [3, (10)]. We also recall that
by [3, (8)].
Thus, by dominated convergence, to complete the proof it suffices to show that
is bounded for any \(N>0\). By a standard argument (compare the proof of [7, Theorem 3.1]), to prove this it suffices to show that for any \(N>0\) and \(f \in C_c^\infty (A \backslash \mathrm {GL}_2(\mathbb {A}_F))\) the sum
is bounded. Here \(f^*(g):=\overline{f}(g^{-1})\). But this is implied by [10, (15’)] (stated in adelic language in [7, Theorem 3.5]) and the fact that \(\mathrm {tr}\,\pi (f*f^*)\) is rapidly decreasing as a function of \(C(\pi )\) [5, Lemma 4.4] (compare the proof of [7, Theorem 3.1]). \(\square \)
Acknowledgements
The author thanks P. E. Herman for comments on a preliminary draft of this paper that improved the exposition, and H. Hahn for her constant encouragement and help with editing. He also thanks the anonymous referee for a careful reading; in particular, the proof of Proposition 3.1 is now clearer due to his or her comments.
The author is thankful for partial support provided by NSF Grant DMS-1405708. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Notes
Many important cases have been worked out, some by Jacquet himself, but the literature is too extensive to adequately cite here.
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Getz, J.R. A four-variable automorphic kernel function. Res Math Sci 3, 20 (2016). https://doi.org/10.1186/s40687-016-0069-6
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DOI: https://doi.org/10.1186/s40687-016-0069-6