Explicit formula for the $(\mathrm{GL}_{2},\mathrm{GL}_{2})$ theta lift via Bruhat decomposition Formule explicite pour la correspondance thêta $(\mathrm{GL}_{2},\mathrm{GL}_{2})$ via la décomposition de Bruhat

A similar computation using complexes to compoute equivariant cohomology was completed in [KS, §3], in a more general setting using a generalized Kronecker-Eisenstein series. Our calculation can be considered as a more explicated form of the calculation there, for the self-product of the universal elliptic curve with trivial coefficients, except that:

1. (1)

   we treat the entire action of $\mathrm{GL}_{2}$ at once, instead of just working with the equivariance under one subtorus at a time;

2. (2)

   we construct also a cocycle for a larger group with no geometric action on the fibers ($\mathrm{GL}_{2}(\mathbb{Q})$ instead of $\mathrm{GL}_{2}(\mathbb{Z})$, or $\mathrm{GL}_{2}(K)$ in the CM case);

3. (3)

   we do not rely on any stabilization or smoothing coming from a degree-zero torsion cycle.

We restrict to this setting both because it is particularly arithmetically rich, and allows for a considerably simpler and less technical exposition. Futhermore, by eschewing smoothing, the resulting formulas are more transparently related to classical objects such as classical holomorphic Eisenstein series or the period formulas of Siegel (as in [X4]), and corresponds directly to the theta kernel of integration of [BCG1]. Furthermore, our formulas can be specialized (with cohomological meaning) at arbitrary non-zero torsion points, thanks to Theorem 3.5.

As we noted in our previous article [X4] in a simpler setting, our same methods should produce analogous formulas with twisted coefficients corresponding to higher-weight automorphic sheaves; the resulting formulas will involve progressively more complicated combinations of higher-weight specializations of Kronecker-Eisenstein series. Another natural generalization would be to use the more complicated decomposition of parabolic subgroups for $\mathrm{GL}_{n}$ and $n$-fold products of elliptic curves; in this setting, our unsmoothed formulas grow in complexity very quickly with $n$, and it is not clear to us if there exists a symbol formalism that can help one understand this complexity. Furthermore, the direct analogue of Theorem 3.5 does not hold for $n>2$, so the ability to specialize at torsion sections is more limited. Nevertheless, in the smoothed setting, things are much simpler and can be captured by symbols coming from elementary linear algebra; we follow this approach for elliptic schemes in [X3].

I would like to thank first and foremost Nicolas Bergeron, to whose ideas I owe all of my work in this subject area; and who very kindly accommodated me in Paris in February 2023, listening to and helping to develop my ideas even with the limited energy he had at the time. This article is dedicated in his memory. I would also like to thank Romyar Sharifi, Marti Roset Julia, and Pierre Charollois for helpful consultations during the writing process.

We now define the main tool we need, in this setting: the weight-$2$ distributional Dolbeault complex. Let $X$ be a complex manifold of complex dimension $d$, and write $\mathcal{A}^{p,q}_{X}$ for the sheaves of smooth complex differentials on $X$ of holomorphic degree $p$ and antiholomorphic degree $q$; these sheaves are flabby on $X$, so we (a bit abusively) use this same notation for their global sections. These forms support the pair of anti-commuting Dolbeault differentials

which are square zero, satisfy the Leibniz rule, and sum to the usual exterior derivative on differential forms. For an integer $w\geq 0$, we recall that the holomorphic $(w,0)$-differentials are precisely the kernel of $\overline{\partial}$ inside $A^{w,0}_{X}$.

Ordinarily, the Dolbeault resolution of $\Omega^{2}_{X}$ is constructed from these sheaves, but instead, we dualize: first, write $\mathcal{A}^{p,q}_{X,c}$ for the space of compactly supported relative $(p,q)$-differentials on $X$, carrying the same differentials as the usual ones (though they do not form a sheaf). Then we define

which we call the space of $(p,q)$-currents on $X$. Unlike the compactly supported forms, these do form a sheaf, since they can be restricted along inclusions of opens in adjunction to the pushforward of compactly supported forms. Since these pushforwards are injective, the sheaf restrictions maps are surjective and thus each $\mathcal{D}_{X}^{p,q}$ is flabby. We therefore again abusively use the same notation for the sheaf and its global sections. For these properties of currents, the original reference is [dR].

The Dolbeault operators

are defined as the graded adjoints of the exterior derivative on forms, i.e.

where $\deg$ refers to the total degree (i.e. $p+q$ for a $(p,q)$-current).

There is a natural inclusion

given by

where $\eta$ is a compactly supported smooth $(d-p,d-q)$-form; the integral makes sense and is finite since the wedge product is also compactly supported, and of degree $(d,d)$. It is clear that $\nu_{X}$ commutes with the Dolbeault differentials $\partial$ and $\overline{\partial}$. Furthermore, since $(p,q)$-compactly supported forms have covariant functoriality by flat holomorphic maps and contravariant by proper holomorphic maps, by adjunction, $(p,q)$-relative currents can be pulled back by flat maps and pushed forward by proper ones. One can check that the push-pull formula for differential implies that $\nu_{X}$ is functorial for flat pullback and proper pushforward of complex manifolds.²²2For orientation-reversing maps, one has to be careful about this comparison map, since they introduce an extra sign; this was an issue we had to work with in [X4]. However, all of our maps will be morphisms of complex manifolds, and hence orientation-preserving.

We now come to the main cohomological result about these currents:

In particular, for $2$-dimensional $X$ and $w=2$, the complex

is functorial in $X$ (in the sense described previously) and computes the coherent cohomology of $\Omega^{2}_{X}$. (Here, we recall that $X$ is assumed to be relative dimension $2$, so there are no further terms.)

Having proved this key cohomological property, we now introduce an important class of currents: associated to closed complex submanifolds $Z\subset X$ of codimension $r$, we have a closed current of integration

defined by

Now consider a fixed squared elliptic fiber $T_{\tau}=E_{\tau}\times E_{\tau}$. Then the class of a closed current $\omega\in\mathcal{D}^{2,1}_{T_{\tau}}$ having residue $\mathcal{C}\in H^{0}(T_{\tau}[c])$ along the residue map

is equivalent to $d\omega=\delta_{\mathcal{C}}$ (where this is interpreted as a suitable linear combination of the currents of integration along points in the support of $\mathcal{C}$); see for example [X, (3.3)] from the author’s thesis.

Since the distributional Dolbeault complex is a functorial complex computing the coherent cohomology of holomorphic differentials, the coresponding equivariant coherent cohomology of $\Omega^{2}_{X}$ can thus be computed by the double complex $C^{\bullet}(\Gamma,\mathcal{D}^{2,\bullet}_{X})$, where $\Gamma\subset\mathrm{GL}_{2}(\mathbb{Z})$ acts on the currents by pushforward.

Analogously to the non-equivariant case, then, an element $\omega$ of this double complex for $T$ restricts to a representative of a class in $H^{1}_{\Gamma}(T_{\tau}-T_{\tau}[c],\Omega^{2}_{T_{\tau}-T_{\tau}[c]})$ with residue

if and only if the total differential of $\omega$ is $\delta_{T_{\tau}[c]}-c^{4}\delta_{0}\in C^{0}(\Gamma,\mathcal{D}^{2,2}_{T_{\tau}})$. In particular, if we can find such a class which is in the trace-fixed part $(\mathcal{D}^{2,\bullet}_{T_{\tau}})^{(0)}$, this notation meaning the part which is invariant by $[a]_{*}$ for all integers $a$ relatively prime to $c$, then it will represent the class

We now know that in principle, one can compute Eisenstein classes by finding suitable lifts inside the distributional de Rham complex. It remains to find currents realizing these lifts.

Recall the Kronecker-Eisenstein series [Weil]

for $\tau\in\mathcal{H}$, $z,u\in\mathbb{C}$, and $a\in\mathbb{Z}$; the superscript apostrophe denotes that the sum omits any term where $\omega+z=0$. This series is convergent for $\text{Re}\,s>1+a/2$, but has meromorphic continuation to all $s\in\mathbb{C}$, with simple poles possibly only at $s=0$ (when $z\in\mathbb{Z}+\mathbb{Z}\tau$ and $a=0$) $s=1$ (when $u\in\mathbb{Z}+\mathbb{Z}\tau$ and $a=0$). We define special notations, following [BCG1, §9], for the specializations we need:

Since these functions are manifestly $(\mathbb{Z}+\mathbb{Z}\tau)$-periodic in $z$ for fixed $\tau$, we can consider $E_{1}(\tau,z)$ and $E_{2}(\tau,z)$ as functions on $E_{\tau}$. Further, $E_{1}(\tau,z)$ is odd and $E_{2}(\tau,z)$ is even in $z$, for any $\tau$; in particular, we see that $E_{1}(\tau,0)=0$. As noted in [Weil], these two functions specialize at torsion points

to holomorphic modular forms in $\tau$ of weight $1$, respectively $2$, which are the classical Eisenstein series of those weights, of level corresponding to the stabilizer of $(\alpha,\beta)$ in $\mathrm{SL}_{2}(\mathbb{Z})$.

From [BCG1, (9.6)] together with [BCG1, Proposition 20], we observe that we have the distribution relations as functions

for any integer $a\in\mathbb{Z}\setminus\{0\}$. Further, [BCG1, Theorem 19] and [BCG1, §9.4] together imply that, considered as a current on $E_{\tau}$,

where here we give the formula for the normalized volume form on $E_{\tau}$. Away from the zero section (ignoring the current of integration $\delta_{0}$), this derivative holds on the level of functions. Furthermore, [Weil, (35)] says that

as functions (and hence currents) on $E_{\tau}$. Thus, both $E_{1}$ and $E_{2}$ are smooth away from the zero section, where $E_{2}$ is once-differentiable and $E_{1}$ has a singularity. Note that the formulas in [BCG1, §9.4] imply that on a punctured neighborhood of the zero section, $E_{1}$ looks like it has a simple log pole (even though its actual value at zero is $0$). It therefore makes sense to consider forms like $E_{1}(\tau,z)\,dz$ and $E_{2}(\tau,z)\,dz$ as $(1,0)$-currents on $E_{\tau}$ by considering them as kernels of integration as in the map $\nu_{X}$ we defined for smooth forms, since the corresponding integrals will converge for compactly supported smooth forms.

We now construct a certain element of total degree $1$ (or $3$, if one adds in the Hodge weight $2$)

whose total differential is the cocycle

To construct this lift, we must specify $\zeta_{\tau}^{0,1}\in C^{0}(\mathrm{GL}_{2}(\mathbb{Q}),(\mathcal{D}_{T_{\tau}}^{2,1})^{(0)})$ and $\zeta_{\tau}^{1,0}\in C^{1}(\mathrm{GL}_{2}(\mathbb{Q}),(\mathcal{D}_{T_{\tau}}^{2,0})^{(0)})$ such that $\overline{\partial}\zeta_{\tau}^{1,0}=\partial_{\mathrm{GL}_{2}(\mathbb{Q})}\zeta_{\tau^{0,1}}$ where the latter differential refers to the group coboundary map.

We therefore fix the choice

whose image under $\overline{\partial}$ is can be computed as

It remains therefore to find, for each $\gamma\in\mathrm{GL}_{2}(\mathbb{Q})$, a lift to $(\mathcal{D}_{T_{\tau}}^{2,0})^{(0)}$ of

We observe that there are no more choices to be made; these lifts are unique:

To find formulas for these lifts for a fully general matrix $\gamma$ is complicated if approached directly: the pushforward action on currents can yield expressions with arbitrarily many terms. Instead, we decompose our matrices to simplify the calculation.

The main observation we need is that we have the “telescoping” relation

reducing the problem of finding a lift for the product of matrices to the problem for the individual matrices. Thus, the problem of finding lifts can be reduced to a set of generators of $\mathrm{GL}_{2}(\mathbb{Q})$.

We recall that the Bruhat decomposition says that

where $B$ denotes the upper triangular Borel subgroup, and $w$ the antidiagonal Weyl element. We can make this more explicit by writing

whenever $c\neq 0$, for an arbitrary matrix $\gamma\in\mathrm{GL}_{2}(\mathbb{Q})$. Here, we have further decomposed the appearing upper triangular matrices by factoring them into a diagonal times a unipotent matrix. When $c=0$, our matrix is already in the Borel, so we can directly write

The point of this is that $w$, diagonal, and unipotent matrices all act in very computable ways on $\zeta^{0,1}_{T_{\tau}}$, enabling us to find lifts:

1. (1)

   All diagonal matrices act trivially on $\zeta_{\tau}^{0,1}$, by the isogeny properties of $E_{1}$ we proved earlier together with the fact that the volume form of a torus is isogeny-invariant. Hence the corresponding lifts are zero.

2. (2)

   For the element $w$, we can write $(w-1)\zeta_{T_{\tau}}$ as

   $$E_{1}(\tau,z_{2})\,dz_{2}\,\delta_{z_{1}=0}+E_{1}(\tau,z_{1})\,dz_{1}\wedge\text{vol}_{z_{1}=0}-E_{1}(\tau,z_{1})\,dz_{1}\,\delta_{z_{2}=0}-E_{1}(\tau,z_{2})\,dz_{2}\wedge\text{vol}_{z_{2}=0}$$

   $$=\overline{\partial}[E_{1}(\tau,z_{1})E_{1}(\tau,z_{2})\,dz_{1}\wedge dz_{2}]$$

   which is a very simple lift.

3. (3)

   For a unipotent matrix $\gamma_{u}$ with upper-right entry $u$, we find that the term with the current $\delta_{z_{2}=0}$ cancels in $(\gamma_{u}-1)\zeta_{T_{\tau}}$ (since the coordinate $z_{1}$ is fixed), and what remains is

   (3.12)

   $$\frac{2iu}{y}E_{1}(\tau,z_{2})\,dz_{1}\wedge dz_{2}\wedge d\overline{z_{2}}=\overline{\partial}\left[4uE_{2}(\tau,z_{2})\,dz_{1}\wedge dz_{2}\right]$$

Combining these with our explicit Bruhat decomposition, we find the following formulas: in the case when $c=0$, we obtain

In the case when $c\neq 0$, we use the full Bruhat decomposition to obtain the following:

Notice that here, we have to introduce factors coming from the determinants of the matrices to go from pushforwards of forms to pushforwards of functions.

We therefore have obtained full formulas for $\zeta_{T_{\tau}}$, whose $c$-stabilizations yield the kernel classes $Z_{\tau}^{(c)}$ for the Eisenstein theta lift after restriction to $\mathrm{GL}_{2}(\mathbb{Z})$.

For a general matrix $\gamma$, there does not appear to be a substantial further simplification of the preceding formulas, besides writing out the pushforwards as sums.

The actual “theta lift” for $(\mathrm{GL}_{2},\mathrm{GL}_{2})$ is generally taken to be valued after contraction pullback by torsion sections. Fix an arithmetic subgroup $\Gamma\subset\mathrm{GL}_{2}(\mathbb{Z})$; in order to compare our results with the cohomological theta lift, we will restrict $\zeta_{\tau}$ to $\Gamma$.

In this context, we wish to consider the image of $Z_{\tau}^{(c)}$ under the composite

where $D$ is any $\Gamma$-fixed torsion cycle $D$ disjoint from $T_{\tau}[c]$. (Here, we are slightly abusive in writing the pullback $D^{*}$; this is actually a sum of pullbacks over the various torsion sections in the support of $D$.) Analogous to the case of a single torsion section, we will write $\Theta_{\tau,D}^{(c)}$ for this image.

We wish to interpret this composite in terms of the explicit double complex representative $\zeta_{\tau}$; the main issue is that currents cannot in general be pulled back by closed immersions.

The technical tool we need to remedy this is the introduction of a variant of the distributional Dolbeault complex: For any $\Gamma\subset\mathrm{GL}_{2}(\mathbb{Q})$, let $H_{\Gamma}\subset T_{\tau}$ be the $\Gamma$-orbit of the lines $\{z_{1}=0\}$ and $\{z_{2}=0\}$ and their translates by $c$-torsion sections inside $T_{\tau}$; this is a union of infinitely many elliptic subschemes (and their $c$-torsion translates). For any finite subarrangement $H\subset H_{\Gamma}$, we define a complex $\mathcal{D}_{T_{\tau},H}^{2,\bullet}$ via the pullback square

which results in an identification of $\mathcal{D}_{T_{\tau},H}^{2,i}$ with the $(2,i)$-currents such that their restriction to $T_{\tau}-H$ are given by smooth $(2,i)$-forms. Here, the bottom horizontal map is the earlier-defined inclusion, and the right vertical map is the restriction dual to the pushforward of compactly-supported differential forms. We define then

where the limit runs over $H$ finite subarrangements of $H_{\Gamma}$, along the natural inclusion maps. The group $\Gamma$ permutes the pullback diagrams for each $H$ (sending it to that of $\gamma H$), and these assemble to give a pushforward action on $\mathcal{D}_{T_{\tau},H_{\Gamma}}^{2,\bullet}$. Further, because the bottom row in each pullback diagram is a quasi-isomorphism, we see that $\mathcal{D}_{T_{\tau},H_{\Gamma}}^{2,\bullet}$ computes the cohomology of $\Omega^{2}_{T_{\tau}}$, just as $\mathcal{D}_{T_{\tau}}^{2,\bullet}$ does. Furthermore, analogously to the full distributional de Rham complex, we have a left exact sequence

meaning that $\zeta_{\tau}$, considered in this more refined complex, is still uniquely determined. The important new phenomenon for us is that if $\Gamma$ fixes the torsion cycle $D\subset T_{\tau}$ disjoint from $H_{\Gamma}$, then there is a composite pullback map

which induces the composite (3.15).

We now concern ourselves with the case that $D$ is supported on $N$-torsion for an integer $N>1$. From the preceding discussion, we can conclude that

so long as $D$ is disjoint from $H_{\Gamma}$. Note that if $c\equiv 1\pmod{N}$, we can write the left-hand side as

The restriction on $D$ is rather irritating, as it depends on the arbitrary choice of coordinates $z_{1},z_{2}$ we used to choose our lift $\zeta_{\tau}^{0,1}$: for any given torsion section, we could simply start with a different lift to obtain formulas for the pullback. However, this would result in a somewhat unsatisfying lack of unity in our formulas. Luckily, we can use a trick to bypass this issue entirely, and make the formulas valid even for “bad” torsion sections:

The proof consists of “bootstrapping” from torsion points disjoint from $H_{\Gamma}$ to all of them. In order to do this, we will need the following lemma allowing us to “improve” our current-valued cocycles to be form-valued: