The cohomological Kudla conjecture for unitary Shimura varieties

Fran\c{c}ois Greer; Salim Tayou

REVIEW 3 major objections 5 minor 1 cited by

The cohomological Kudla conjecture for unitary Shimura varieties

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For unitary Shimura varieties of signature (n+1,1), boundary-supported corrections make the generating series of special cycles a holomorphic Hermitian modular form of weight n+2 in all codimensions g≤n/2.

desk verdict Substantial and likely correct advance on the cohomological Kudla conjecture for U(n+1,1), but the central modularity criterion is deferred to forthcoming work and must be checked by the referee. read the letter →

arxiv 2507.13299 v2 pith:J3L2PUA5 submitted 2025-07-17 math.NT math.AG

François Greer , Salim Tayou This is my paper · ORCID

classification math.NTmath.AG MSC 11F2711F4611G1814C2514G35

keywords unitaryShimuravarietiesKudla–MillsongeneratingseriesHermitianmodularformsquasi-modularspecialcyclestoroidalcompactificationthetasl2-action

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to prove the cohomological version of the Kudla conjecture for toroidal compactifications of unitary Shimura varieties of signature (n+1,1): the generating series of special cycles, after adding cycles supported in the boundary, should be a holomorphic Hermitian modular form valued in cohomology. The authors establish this for all codimensions g up to n/2, and they also show that the uncorrected series of Zariski closures is a Hermitian quasi-modular form with an explicit non-holomorphic completion. If correct, this resolves a conjecture of Kudla and Bruinier–Rosu–Zemel in cohomology in that range and provides the first general construction of compactified-cycle modularity for arbitrary codimension. The restriction g≤n/2 is traced to a splitting lemma, and the authors conjecture the same results hold in all codimensions.

What carries the argument

The central mechanism is the sl2-action on the finite-dimensional spaces F_{n,g} of polynomials P:M_{n×g}(C)→C satisfying the equivariance P(U A)=|det(A)|^2 P(U). The lowering operator Δ, raising operator Λ, and weight operator H form an sl2-triple, so each polynomial decomposes into primitive (pluriharmonic) pieces $Λ^{{g-ℓ}}$$P^{{ℓ,ℓ}}$. A modularity criterion (Theorem 2.12) turns a polynomial P into a $\theta$ series ϑ_P(τ)=det(Y)^{-1} Σ_λ exp(-Δ/4π)(P)(λ·$Y^{{1/2}}$) $q^{{h(λ)}}$ e_λ that transforms like a Hermitian modular form after completion. On each boundary component the special cycles are represented by harmonic forms f^g(λ)=g! f(λ_1)∧⋯∧f(λ_g), and pairing these with a cohomology class produces an element of F_{n,g}; this map intertwines the sl2-actions, with Lefschetz on cohomology matching Λ/Δ on polynomials. The other load-bearing ingredient is the Splitting Lemma, which uses Hard Lefschetz on the normal bundle of each boundary divisor to express every cohomology class in degree 2g≤n as an interior class plus a boundary class, reducing modularity of the full series to a computation in the boundary abelian varieties E_M.

What would settle it

Compute the Fourier coefficients of the corrected series $eΦ^{1}$_L(τ) for the smallest nontrivial case, say a signature (3,1) lattice with k=Q(i) and g=1, using the explicit correction formula of Theorem 1.4, and compare them with the Fourier coefficients of the known weight-4 Hermitian Eisenstein series for U(1,1)(Z); a mismatch at any positive-definite matrix N would disprove the central modularity claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the non-compactness of a ball quotient does not destroy the modularity of Kudla–Millson generating series; it merely moves the series into a larger space. For each isotropic boundary line J, with boundary divisor B_J ≅ E ⊗_{O_k} M, the paper computes the restriction of special cycles to B_J, represents those cycles by harmonic forms f(λ)=u_λ^*ω_E, and uses an sl2-triple (Λ, Δ, H) on the space of polynomial weight functions to interpolate between primitive classes and their Lefschetz powers. The correction term in Theorem 1.4 is the explicit boundary cycle Σ_{J} Σ_{ℓ,i} (r_J/d_k) P_i^ℓ(ν,N) ι_{J,*}[W_i^ℓ ∪ $L^{{g-ℓ-1}}$], and Corollary 1.5 packages these corrections as the statement that eΦ^g_L(τ) transforms with weight n+2 and representation ρ_{L,g} under U(g,g)(Z). Theorem 1.3 asserts the same for the uncorrected series after a non-holomorphic completion.

Load-bearing premise

The load-bearing premise is Theorem 2.12 — the criterion that every polynomial-weighted theta series can be completed to a Hermitian modular form — which the paper states without proof, defers to forthcoming work of Ben Howard, and uses for every completion and correction.

Editorial extensions

If this is right

For every neat arithmetic group and every g≤n/2, the corrected generating series eΦ^g_L(τ) is a holomorphic Hermitian modular form of weight n+2 with representation ρ_{L,g}, giving the cohomological form of Conjecture 1.1 in the unitary case.
The uncorrected generating series of Zariski closures of special cycles is a Hermitian quasi-modular form, the first instance of Hermitian quasi-modularity appearing for special cycles.
The explicit correction formula makes the modularity effective: Fourier coefficients of the corrected series can be read from intersection numbers with explicit boundary cycles W_i^ℓ ∪ L^{g-ℓ-1}.
The authors conjecture the same statement holds in all codimensions, since the restriction g≤n/2 is an artifact of the splitting lemma rather than of the theta-lift mechanism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The sl2-projector mechanism is independent of the Splitting Lemma, so a finer boundary analysis could plausibly extend the result beyond the middle without changing the modularity criterion itself.
The explicit correction formula suggests a practical numerical test: for small n and g, pairing eΦ^g_L with test classes should reproduce Fourier coefficients of known Hermitian Eisenstein series, giving a check that does not depend on the deferred proof of Theorem 2.12.
The same Lefschetz-graded correction idea should apply to orthogonal Shimura varieties in higher codimensions, extending the known codimension-one case to a full generating series of compactified cycles.
If the deferred modularity criterion is supplied and the convergence issue for Chow-valued series is resolved, the same correction formulas would likely upgrade from cohomology to Chow groups, matching the original conjecture.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this is the first real push on the cohomological Kudla conjecture for unitary Shimura varieties in codimension up to the middle, and the machinery—Hermitian quasi-modular forms, the sl2 projectors, the boundary corrections—looks right. But the entire modularity result depends on Theorem 2.12, a weighted theta series criterion stated without proof and deferred to Ben Howard's forthcoming paper. That is the thing to check.

What's new: Bruinier–Zemel and EGT23 handled divisors, BRZ24 handled zero-cycles; this paper handles arbitrary g ≤ n/2 for U(n+1,1). The generating series of corrected Zariski closures is shown to be a holomorphic Hermitian modular form, and the Zariski closure series is a Hermitian quasi-modular form with explicit non-holomorphic completion. That is a genuine advance, and the explicit correction formula with boundary cycles and the sl2-action on polynomial weight functions is a nice piece of work.

What the paper does well: the strategy is coherent—reduce to the boundary via Greer's splitting lemma, compute harmonic representatives of boundary cycles, then correct the theta series using projectors onto primitive sl2 components. The paper is honest about the range restriction g ≤ n/2, and the corrections are constructed, not fitted; there is no circularity. The reliance on the Hodge conjecture for abelian varieties is fine; that's a theorem in this setting.

Soft spots, in order of seriousness. First, Theorem 2.12 is load-bearing and unproved. Every non-holomorphic completion and modular correction in Sections 4–5 passes through it. A normalization or weight error there would change the weight or representation in Theorems 1.2–1.5. The paper gives no internal check. This is an external-validation concern, not an internal contradiction, but it is real: the proof, as written, is conditional on that theorem. Second, there are notation mismatches between the introduction (Theorem 1.4, Corollary 1.5) and the proof in Section 5—the definition of P_i^ℓ(ν,N) and the φ formula differ in small but confusing ways. A final version should clean these up. Third, the Splitting Lemma (3.3) is what forces g ≤ n/2; the authors acknowledge it, and it's fine, but it does limit the result.

The reader's conditional verdict is right. If Theorem 2.12 holds, this is a major result. I didn't see any circular dependence on the main claim.

Bottom line: send it to a good referee. The referee's main job is to check Theorem 2.12 against Howard's work and to verify the boundary transfer. If that checks out, publish. I would cite it.

Referee Report

3 major / 5 minor

Summary. The paper proves a cohomological version of the Kudla conjecture for unitary Shimura varieties of signature (n+1,1) in codimensions g ≤ n/2. For a neat arithmetic group and toroidal compactification, the authors construct boundary-supported correction classes to the Kudla–Millson generating series of special cycles and prove that the corrected series is a holomorphic Hermitian modular form of weight n+2 with respect to the genus-g Weil representation. They also construct non-holomorphic completions, exhibiting the uncorrected series as a Hermitian quasi-modular form. The proof combines a splitting lemma for homology, an explicit analysis of the boundary divisors as abelian varieties, a new sl2-action on spaces of homogeneous polynomials, and weighted theta series with completions. The main results are stated as Theorems 1.2–1.4 and Corollaries 1.5–1.6.

Significance. If the proof is completed, this is a substantial advance: it provides the first general modularity result for compactified special cycles of arbitrary codimension in unitary Shimura varieties, going beyond the codimension-one and zero-cycle cases in the literature. The development of Hermitian quasi-modular forms and the explicit sl2-projector formalism are valuable technical contributions. The argument is not circular: the correction terms are constructed from sl2-projections and an external modularity criterion, not fitted to force the theorem. However, the central modularity criterion is deferred to unpublished work, and there are notation mismatches in the main formula that must be resolved before the paper can be accepted.

major comments (3)

[§2.5, Theorem 2.12] The paper's central modularity criterion is stated without proof and deferred to forthcoming work of Ben Howard. This theorem is used as the basis for the modularity of the boundary theta lift (Theorem 4.12), for the corrected boundary series (Theorem 4.19), and for the final transfer to X^tor in Section 5; a normalization or weight error in the determinant factor or the exponentiated Laplacian would change the weight or representation in Theorems 1.2–1.5. As written, the main theorems are conditional on an unpublished external result, and no internal consistency check (e.g., a low-genus or low-weight example) is supplied.
[§1.1, Theorem 1.4; §2.2; §5, Definition 5.1] The notation in the main correction formula is internally inconsistent. Theorem 1.4 and Corollary 1.5 write the boundary correction as ι_{J,*}[W_i^ℓ ∪ L^{g−ℓ−1}], whereas Definition 5.1 uses D_J^{g−ℓ−1}; Corollary 4.5 gives c1(N∨_{B_J}) = d_k/r_J D_J, so if L denotes the conormal bundle, the two formulas differ by powers of d_k/r_J. In addition, Theorem 1.4 states the summation condition [λ] = p_M^L(ν), but p_M^L maps from (M∨/M)^g to (L∨/L)^g; the correct projection of ν is p^M_L(ν). These mismatches make the theorem statement not directly usable.
[§4.3 vs §5, Definition 5.1] The symbol \tilde Z is used for two different corrections: in Section 4.3 (before Theorem 4.19) it denotes [Z(λ)] − Σ ... [W_i^ℓ ∪ D_J^{g−ℓ}], while Definition 5.1 defines the boundary correction for the compactified cycle as [Z(ν,N)] + Σ (r_J/d_k) ... [W_i^ℓ ∪ D_J^{g−ℓ−1}]. The sign and the exponent of D_J change between these formulas, and a reader cannot tell whether Theorem 4.19 is compatible with Theorem 1.4. Please state explicitly how the boundary correction of Section 4.3 is converted, via the Gysin map and excess intersection, into the correction in X^tor.

minor comments (5)

[§5, proof of Theorems 1.3–1.4] The proof refers to “Theorem 3.3” for the splitting of homology; the statement is Lemma 3.3.
[§5, proof of Theorems 1.3–1.4] The text says the pairings are Hermitian modular forms of “weight 1+n”; the correct weight is n+2, as stated in Theorems 1.2 and 3.1.
[§1.1 and §3.5] The introduction cites “Theorem 3.6” for the Hodge conjecture for the boundary abelian variety, but the actual statement is Remark 3.6.
[§1.1, Corollary 1.6 paragraph] “Thethetaseries” is a typo for “The theta series”.
[§4.3, definition of \tilde Z] The proof of Theorem 1.2 asserts the existence of cycles, but the proof in Section 5 defines cohomology classes; please clarify that the corrections are algebraic cycles with Q-coefficients and specify the coefficient field of the polynomials P_i^ℓ(ν,N).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the central modularity claim rests on an externally deferred modularity criterion (Thm 2.12), which is a rigor gap, not a self-referential reduction.

full rationale

The claimed derivation is not circular. Kudla--Millson modularity (Thm 3.1) is an external input; the boundary restriction (Thm 3.5), the excess intersection computation, and the sl2 projector machinery (Sections 2.3--2.4) are developed inside the paper. The correction terms in Thm 1.4 are explicitly constructed from primitive boundary cohomology classes via the raising operator Λ, and the corrected series eΦ is then proved modular in Thm 4.19 using the harmonic decomposition and the theta modularity criterion; modularity is a consequence, not a fitted input. Splitting Lemma 3.3, though credited to [Gre19], is proved here by Hard Lefschetz, so the self-citation is not load-bearing. The one serious caveat is Theorem 2.12 (Section 2.5): the weighted theta series criterion is stated without proof and deferred to forthcoming work of Ben Howard, and all completions and corrections in Sections 4--5 build on it. This makes the main theorems conditional on an externally unverified assertion, but that is a missing-support/rigor concern, not circularity: the theorem is not identified with the Kudla conjecture and is not derived from the paper's own conclusions. Accordingly, no circular step is exhibited, and the score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard results in Shimura varieties, toroidal compactifications, and theta correspondence, plus one deferred theorem (Theorem 2.12) whose proof is not included. No free parameters are fitted to data; the construction is parameter-free and the correction terms are determined by the geometry. The paper introduces no new postulated entities such as new particles, forces, or dimensions.

assumptions (7)

domain assumption Theorem 2.12 (Howard): weighted theta series of a homogeneous polynomial P in F_{n,g} transform like Hermitian modular forms of weight 2+n; proof deferred to upcoming work.
Stated in Section 2.5. The paper relies on this for all non-holomorphic completions and for the modularity of corrected series; no proof is included.
standard math Kudla-Millson theorem (KM90): the cohomology class of the generating series of special cycles on the open Shimura variety is a holomorphic Hermitian modular form.
Invoked as Theorem 3.1 and used to handle the interior part of pairings in Section 5.
standard math Hard Lefschetz theorem on the boundary abelian varieties E_M, applied to the ample conormal bundle.
Used in the proof of Splitting Lemma 3.3 to obtain the required surjectivity of -∩c_1(N_{B_J}).
standard math Hodge conjecture for the CM abelian variety E_M (Tate, Murasaki), so primitive Hodge classes are algebraic cycles.
Used in Remark 3.6 to realize the basis W^ℓ_i as classes of algebraic cycles and to construct the associated harmonic polynomials.
standard math Structure of toroidal compactifications for unitary Shimura varieties (AMRT10, How15), including boundary divisor B_J ≃ E ⊗_{O_k} M and the conormal bundle class computation.
Background for Section 3.3, where the boundary geometry is described.
domain assumption Reduction to neat arithmetic subgroups Γ (Remark 3.7) via finite covers.
The main theorems are proved for neat Γ; the reduction argument is stated and depends on standard pushforward formulas.
domain assumption Assumption 4.2: M is free over O_k with diagonal h; justified by isogeny reduction to E'_{M}.
Simplifies local computations in Section 4; the paper argues it suffices to check on an isogenous pullback.

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Pith. "Pith review of The cohomological Kudla conjecture for unitary Shimura varieties." pith.science (2026). https://pith.science/paper/J3L2PUA5

@misc{pith2026250713299,
  author       = {Pith},
  title        = {Pith review of: The cohomological Kudla conjecture for unitary Shimura varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3L2PUA5}},
  note         = {Machine review of arXiv:2507.13299}
}

abstract

We construct natural extensions of the Kudla--Millson generating series of cohomology classes of special cycles in compactified unitary Shimura varieties of signature $(n+1,1)$ and prove that they are holomorphic Hermitian modular forms. This proves the cohomological version of a conjecture of Kudla and Bruinier--Rosu--Zemel, in all codimensions up to the middle. We also develop the theory of Hermitian quasi-modular forms, with a particular focus on polynomial weighted theta functions, and prove that the generating series of Zariski closures of special cycles is a Hermitian quasi-modular form.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Weighted Cohomology, Hodge Theory and Intersection Cohomology of Shimura varieties
math.AG 2026-03 conditional novelty 6.0 of 10

On Shimura varieties, the top weight-graded cohomology of the reductive Borel-Serre compactification is canonically isomorphic to the intersection cohomology of the Baily-Borel compactification.

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Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 1 Pith paper

[1]

Smooth compactifications of locally symmetric varieties

Avner Ash , David Mumford , Michael Rapoport , and Yung-Sheng Tai . Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze . Cambridge: Cambridge University Press, 2010

work page 2010
[2]

W. L. Baily, Jr. and A. Borel. Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) , 84:442--528, 1966

work page 1966
[3]

Bruinier, Benjamin Howard, Stephen S

Jan H. Bruinier, Benjamin Howard, Stephen S. Kudla, Michael Rapoport, and Tonghai Yang. Modularity of generating series of divisors on unitary S himura varieties. Ast\' e risque , 421, Diviseurs arithm\' e tiques sur les vari\' e t\' e s orthogonales et unitaires de Shimura:7--125, 2020

work page 2020
[4]

Borcherds

Richard E. Borcherds. Automorphic forms with singularities on G rassmannians. Invent. Math. , 132(3):491--562, 1998

work page 1998
[5]

Borcherds

Richard E. Borcherds. The G ross- K ohnen- Z agier theorem in higher dimensions. Duke Math. J. , 97(2):219--233, 1999

work page 1999
[6]

Modularity of special 0-cycles on toroidal compactifications of Shimura Varieties

Jan Bruinier , Eugenia Rosu , and Shaul Zemel . Modularity of special 0-cycles on toroidal compactifications of Shimura Varieties . arXiv e-prints , page arXiv:2404.06254, April 2024

work page Pith review arXiv 2024
[7]

Kudla's modularity conjecture and formal Fourier - Jacobi series

Jan Bruinier and Martin Westerholt-Raum. Kudla's modularity conjecture and formal Fourier - Jacobi series. Forum Math. Pi , 3:30, 2015. Id/No e7

work page 2015
[8]

Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties

Jan Bruinier and Shaul Zemel . Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties . Math. Z. , 2021

work page 2021

Show all 27 references

[9]

J. W. Cogdell. Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus . J. Reine Angew. Math. , 357:115--137, 1985

1985
[10]

Mixed mock modularity of special divisors

Philip Engel , Fran c ois Greer , and Salim Tayou . Mixed mock modularity of special divisors . arXiv e-prints , page arXiv:2301.05982, jan 2023

2023 arXiv
[11]

Modularity of cycles in compactified orthogonal shimura varieties

Philip Engel, Fran c ois Greer, and Salim Tayou. Modularity of cycles in compactified orthogonal shimura varieties. In preparation , 2026

2026
[12]

Garc \' a

Luis E. Garc \' a . Kudla-Millson forms and one-variable degenerations of Hodge structure . arXiv e-prints , page arXiv:2301.08733, January 2023

2023 arXiv
[13]

Quasi-modular forms from mixed Noether - Lefschetz theory

Fran c ois Greer. Quasi-modular forms from mixed Noether - Lefschetz theory. Adv. Math. , 355:25, 2019. Id/No 106765

2019
[14]

Complex multiplication cycles and Kudla - Rapoport divisors

Benjamin Howard. Complex multiplication cycles and Kudla - Rapoport divisors. II . Am. J. Math. , 137(3):639--698, 2015

2015
[15]

Hirzebruch and D

F. Hirzebruch and D. Zagier. Intersection numbers of curves on H ilbert modular surfaces and modular forms of N ebentypus. Invent. Math. , 36:57--113, 1976

1976
[16]

Kudla and John J

Stephen S. Kudla and John J. Millson. Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables. Publ. Math., Inst. Hautes \'E tud. Sci. , 71:121--172, 1990

1990
[17]

Special cycles on unitary Shimura varieties

Stephen Kudla and Michael Rapoport. Special cycles on unitary Shimura varieties. II : Global theory. J. Reine Angew. Math. , 697:91--157, 2014

2014
[18]

Stephen S. Kudla. Special cycles and derivatives of E isenstein series. In Heegner points and Rankin (L)-series. Papers from the workshop on special values of Rankin (L)-series, Berkeley, CA , USA , December 2001 , pages 243--270. Cambridge: Cambridge University Press, 2004

2001
[19]

Kashiwara and M

M. Kashiwara and M. Vergne. On the Segal - Shale - Weil representations and harmonic polynomials. Inventiones mathematicae , 44(1):1--47, February 1978

1978
[20]

Arithmetic theta lifting and $L$ -derivatives for unitary groups

Yifeng Liu. Arithmetic theta lifting and $L$ -derivatives for unitary groups. I . Algebra Number Theory , 5(7):849--921, 2011

2011
[21]

Murasaki

T. Murasaki. On rational cohomology classes of type (p,p) on an Abelian variety. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A , 10:66--74, 1969

1969
[22]

Siegel theta series for indefinite quadratic forms

Christina Roehrig. Siegel theta series for indefinite quadratic forms. Res. Number Theory , 7(3):28, 2021. Id/No 45

2021
[23]

John T. Tate. Algebraic cycles and poles of zeta functions. Arithmetical algebraic Geom ., Proc . Conf . Purdue Univ . 1963, 93-110 (1965)., 1965

1965
[24]

Some cases of Kudla 's modularity conjecture for unitary Shimura varieties

Jiacheng Xia. Some cases of Kudla 's modularity conjecture for unitary Shimura varieties. Forum Math. Sigma , 10:31, 2022. Id/No e37

2022
[25]

The Gross - Kohnen - Zagier theorem over totally real fields

Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang. The Gross - Kohnen - Zagier theorem over totally real fields. Compos. Math. , 145(5):1147--1162, 2009

2009
[26]

Hermitian Jacobi Forms Having Modules as their Index and Vector-Valued Jacobi Forms

Shaul Zemel . Hermitian Jacobi Forms Having Modules as their Index and Vector-Valued Jacobi Forms . arXiv e-prints , page arXiv:2310.16508, October 2023

2023 arXiv
[27]

Modularity of generating functions of special cycles on Shimura varieties

Wei Zhang. Modularity of generating functions of special cycles on Shimura varieties . PhD thesis, Columbia University New York, 2009

2009

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Smooth compactifications of locally symmetric varieties

Avner Ash , David Mumford , Michael Rapoport , and Yung-Sheng Tai . Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze . Cambridge: Cambridge University Press, 2010

work page 2010

[2] [2]

W. L. Baily, Jr. and A. Borel. Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) , 84:442--528, 1966

work page 1966

[3] [3]

Bruinier, Benjamin Howard, Stephen S

Jan H. Bruinier, Benjamin Howard, Stephen S. Kudla, Michael Rapoport, and Tonghai Yang. Modularity of generating series of divisors on unitary S himura varieties. Ast\' e risque , 421, Diviseurs arithm\' e tiques sur les vari\' e t\' e s orthogonales et unitaires de Shimura:7--125, 2020

work page 2020

[4] [4]

Borcherds

Richard E. Borcherds. Automorphic forms with singularities on G rassmannians. Invent. Math. , 132(3):491--562, 1998

work page 1998

[5] [5]

Borcherds

Richard E. Borcherds. The G ross- K ohnen- Z agier theorem in higher dimensions. Duke Math. J. , 97(2):219--233, 1999

work page 1999

[6] [6]

Modularity of special 0-cycles on toroidal compactifications of Shimura Varieties

Jan Bruinier , Eugenia Rosu , and Shaul Zemel . Modularity of special 0-cycles on toroidal compactifications of Shimura Varieties . arXiv e-prints , page arXiv:2404.06254, April 2024

work page Pith review arXiv 2024

[7] [7]

Kudla's modularity conjecture and formal Fourier - Jacobi series

Jan Bruinier and Martin Westerholt-Raum. Kudla's modularity conjecture and formal Fourier - Jacobi series. Forum Math. Pi , 3:30, 2015. Id/No e7

work page 2015

[8] [8]

Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties

Jan Bruinier and Shaul Zemel . Special Cycles on Toroidal Compactifications of Orthogonal Shimura Varieties . Math. Z. , 2021

work page 2021