Modular Local Theta Correspondence

• Modular local theta correspondence is a framework for studying smooth representations over positive characteristic fields via the modular Weil representation and theta lifts.
• It distinguishes type I dual pairs—with bijective correspondences under large-characteristic conditions—from type II pairs, where multiplicities are governed by combinatorial symmetric group theory.
• Recent developments extend the theory to support-level correspondences, rationality and descent, and geometric Iwahori-level formulations, while also identifying inherent limitations.

Searching arXiv for papers on modular local theta correspondence, type II, and rationality/descent of the Weil representation. First, I’ll retrieve recent work directly on the modular local theta correspondence and related foundational papers. Using arXiv search now. Modular local theta correspondence is the study of local theta lifting for smooth representations with coefficient fields of positive characteristic, typically of characteristic $\ell\neq p$ over a non-archimedean local field of residual characteristic $p$. Its basic objects are the modular Weil representation and the resulting big and small theta lifts. In the modern literature, the subject divides sharply by the type of dual pair. For non-quaternionic type I pairs, recent work establishes a bijective correspondence when the coefficient characteristic is sufficiently large compared to the size of the pair. For type II pairs, by contrast, the modular theory is generally not multiplicity free, and its multiplicities are controlled by combinatorics of partial permutations and modular representation theory of symmetric groups (Trias, 15 Jul 2025, Droschl, 18 Jan 2026).

1. Foundational construction

The foundational setup begins with a field $F$ that is either finite or non-archimedean local, of characteristic different from $2$, together with a finite-dimensional symplectic $F$-space $W$, a coefficient field $R$ of characteristic $\ell\neq p$, and a nontrivial smooth additive character $\psi:F\to R^\times$. In this setting, the Stone–von Neumann theorem remains valid for representations with coefficients in $R$, yielding a unique irreducible smooth representation of the Heisenberg group $p$0 with central character $p$1. This produces a projective representation of $p$2, which lifts to a genuine smooth representation of a central extension of $p$3 by $p$4; this is the modular Weil representation of the metaplectic group. For any dual pair $p$5, their lifts to the metaplectic group may split or not according to the case at stake, and the biggest isotypic quotient of the modular Weil representation is then used to define the $p$6-lift (Trias, 2020).

This construction differs from the classical complex theory in two structural respects. First, the metaplectic extension is naturally built as an $p$7-extension, and only in the non-archimedean local case with $p$8 does one recover a nontrivial central extension by $p$9 on the derived subgroup. Second, the theory is formulated so that scalar extension and reduction modulo $F$0 can be treated intrinsically. These features make the modular theory a distinct framework rather than a formal restatement of the characteristic-zero one (Trias, 2020).

2. Type I dual pairs and modular Howe duality

For type I reductive dual pairs, the modular local theta correspondence is formulated in terms of three properties of the big theta lift $F$1: finite length, irreducibility or vanishing of the cosocle $F$2, and uniqueness of the source representation when small theta lifts coincide. In the setting of symplectic–orthogonal or unitary–unitary dual pairs over a $F$3-adic field, with coefficients in an algebraically closed field $F$4 of characteristic $F$5, recent work proves that one obtains a bijective correspondence as long as the characteristic of the coefficient field is large enough compared to the size of the dual pair (Trias, 15 Jul 2025).

The strongest explicit condition is a “strongly banal” hypothesis, expressed by

$F$6

where $F$7 is the order of $F$8 in $F$9. Under this condition, together with a doubling hypothesis denoted $2$0, the big theta lift has finite length, its cosocle has length at most $2$1, and common nonzero small theta lifts force equality of the source representation. The proof uses a modular reconstruction of the classical apparatus: a modular Weil representation, a modular MVW involution, Rallis and Kudla filtrations, first occurrence, cuspidal support control, and a boundary analysis modeled on Gan–Takeda-type arguments. In characteristic $2$2 for the base field $2$3, the paper further proves that these conclusions hold for all but finitely many $2$4 (Trias, 15 Jul 2025).

Small characteristic phenomena are not merely technical obstructions. The same work gives a counterexample in non-banal characteristic: for the dual pair

$2$5

if $2$6, then

$2$7

with $2$8. This shows that irreducibility and uniqueness can fail in the modular setting, even for first lifts, and that large-$2$9 hypotheses are structurally meaningful rather than merely convenient (Trias, 15 Jul 2025).

3. Type II dual pairs and multiplicity formulas

For type II, the relevant dual pair is $F$0, and the theta kernel is the natural representation on compactly supported functions on matrix space,

$F$1

In characteristic $F$2, the type II local theta correspondence is multiplicity one. In the modular setting over $F$3, however, multiplicity one and uniqueness fail in a precise and computable way. The central theorem identifies the multiplicity

$F$4

with a symmetric-group multiplicity attached to a recursively defined skeleton map

$F$5

namely

$F$6

Here $F$7 is the multiplicity of $F$8 in the permutation representation of $F$9 on the set of partial permutations $W$0 (Droschl, 18 Jan 2026).

This reformulation turns the modular theta problem into a branching problem for modular representations of symmetric groups. The basic combinatorial object is

$W$1

with orbit decomposition by rank. A key corollary is that if $W$2 is the order of the residue cardinality $W$3 in $W$4 and $W$5, then

$W$6

Thus even the trivial representation can occur with multiplicity greater than $W$7. In the range $W$8, the extracted partitions have size $W$9, and the multiplicities are governed by explicit algorithms coming from Pieri’s Formula. This makes the type II modular correspondence both more intricate and, in a controlled range, more explicit than its characteristic-zero analogue (Droschl, 18 Jan 2026).

4. Families, Bernstein centers, and support-level correspondence

A different modular perspective for type II replaces representation-level theta lifting by a support-level correspondence on Bernstein centers. For a commutative $R$0-algebra $R$1, the type II Weil representation is

$R$2

with the usual left-right matrix action. In the complex case, and more generally over algebraically closed fields of banal characteristic, one has the classical injective map on irreducibles. In non-banal characteristic, this fails already for $R$3: if $R$4, then

$R$5

so there is no well-defined injective correspondence on irreducibles. The paper argues that the correct invariant in families is therefore supercuspidal support or central character rather than individual irreducible representations (Moss et al., 2023).

The main construction is a canonical ring homomorphism

$R$6

where $R$7 is the Bernstein center of the category of smooth $R$8-modules. Its defining property is that the kernel of

$R$9

is generated by

$\ell\neq p$0

This map is compatible with scalar extension, finite, and surjective for any $\ell\neq p$1-algebra $\ell\neq p$2. In the complex and banal settings, the induced morphism on spectra recovers the explicit support-level theta map. In non-banal settings it still controls the central character of every constituent of a theta isotypic quotient, even when a representation-level correspondence fails. A plausible implication is that, for type II in families, modular local theta correspondence is most naturally formulated as a correspondence on central supports rather than a bijection on simples (Moss et al., 2023).

5. Rationality, descent, and perfect fields

A separate line of development concerns the rationality of the Weil representation and the field of definition of theta correspondence. Recent work proves that the Weil representation over a non-archimedean local field can be realized with coefficients in a number field, and that the same descent methods apply over $\ell\neq p$3-modular coefficient fields and coefficient rings such as rings of integers. It also proves that for a perfect coefficient field $\ell\neq p$4, the validity of the theta correspondence over $\ell\neq p$5 is equivalent to its validity over the algebraic closure $\ell\neq p$6. Here “validity” means exactly the package consisting of finite length of big theta lifts, irreducibility or vanishing of small theta lifts, and uniqueness of the source when small theta lifts agree. Taken together, these results show that the classical local theta correspondence is rational in the precise sense that it can be formulated and verified over smaller perfect coefficient fields once the Weil representation has descended there (Trias, 22 Jan 2026).

This descent theory is especially relevant for modular questions. In the modular setting $\ell\neq p$7 with $\ell\neq p$8, the even and odd parts of the Weil representation still exist, though they may fail to be irreducible. The paper shows that their character fields are again fields of realization, with no Schur-index obstruction because Wedderburn’s theorem forces central division algebras over finite fields to split. It also proves Galois compatibility of theta lifts: $\ell\neq p$9 These statements do not classify modular theta lifts, but they provide a rationality and descent infrastructure for the subject (Trias, 22 Jan 2026).

6. Geometric and Iwahori-level formulations

A geometric version of local theta correspondence at Iwahori level has been developed for type II pairs in the framework of geometric Langlands. Over

$\psi:F\to R^\times$0

with $\psi:F\to R^\times$1 of characteristic $\psi:F\to R^\times$2, and for the type II pair

$\psi:F\to R^\times$3

the geometric analogue of the $\psi:F\to R^\times$4-fixed vectors in the Weil representation is the category

$\psi:F\to R^\times$5

where $\psi:F\to R^\times$6. This category carries commuting geometric Hecke actions, and the corresponding work gives a partial geometric description of the category, including a filtration compatible with Hecke action, a geometric realization of the first term of Kudla’s filtration, and geometric Jacquet functors at Iwahori level commuting with suitable Hecke actions (Farang-Hariri, 2013).

A later development refines this picture and relates it to geometric local Arthur–Langlands functoriality. For a morphism of dual groups

$\psi:F\to R^\times$7

the paper constructs a bimodule over affine extended Hecke algebras that is conjectured to realize geometric local theta correspondence at Iwahori level. In the case $\psi:F\to R^\times$8, this conjectural description is proved: the Grothendieck group of the geometric theta category is identified with the $\psi:F\to R^\times$9-theory of a Springer-fiber-type stack, and the bimodule is computed explicitly. These geometric results do not address mod-$R$0 smooth representation theory directly, but they provide a categorical and positive-characteristic model of tamely ramified local theta phenomena (Farang-Hariri, 2015).

7. Limitations, failures, and current scope

The existing theory is powerful but highly stratified. For type I pairs, the current modular Howe duality results exclude quaternionic dual pairs, rely on $R$1, and in practice require either explicit large-$R$2 hypotheses or an “all but finitely many $R$3” statement over $R$4-adic fields of characteristic $R$5. For type II pairs, full representation-level uniqueness is not the correct general expectation: multiplicity one fails, and for general $R$6 the multiplicities $R$7 are reduced to modular symmetric-group problems that are themselves largely open (Trias, 15 Jul 2025, Droschl, 18 Jan 2026).

The support-level theory in families also has clear limits. It gives a canonical and surjective map on Bernstein centers and controls the supercuspidal support of all constituents of theta isotypic quotients, but it does not produce a bijection on irreducible smooth $R$8-representations in non-banal characteristic. The geometric Iwahori-level theory is likewise partial: it provides categorical avatars, Hecke bimodules, and explicit low-rank descriptions, but not a full geometric classification for all type II pairs (Moss et al., 2023, Farang-Hariri, 2015).

Taken together, these developments suggest that “modular local theta correspondence” is not a single theorem but a family of closely related structures. In type I, the central issue is when a genuine modular Howe duality survives. In type II, the issue shifts toward exact multiplicities, support-level correspondences, and combinatorial control by symmetric groups. Around both theories lie rationality, integral models, and geometric Iwahori-level realizations, which together define the present landscape of the subject.