The Geometric Unitary Kudla Conjecture

Martin Raum

arxiv: 2603.04282 · v2 · submitted 2026-03-04 · 🧮 math.NT

The Geometric Unitary Kudla Conjecture

Martin Raum This is my paper

Pith reviewed 2026-05-15 16:34 UTC · model grok-4.3

classification 🧮 math.NT MSC 11F5511G1814C25

keywords Fourier-Jacobi seriesHermitian modular formsKudla conjectureunitary Shimura varietiesCM fieldsspecial cyclesChow groupsWeil representation

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The pith

Symmetric formal Fourier-Jacobi series over any CM field converge to the expansions of genuine Hermitian Hilbert modular forms.

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

The paper shows that every symmetric formal Fourier-Jacobi series over an arbitrary CM field converges and coincides with the Fourier-Jacobi expansion of an actual Hermitian Hilbert modular form. This identification is then used to prove that the Chow-valued Kudla generating series attached to special cycles on unitary Shimura varieties is modular of weight p+1 with respect to a Weil representation. The modularity holds for Hermitian lattices of signature (p,1) at one infinite place and (p+1,0) at the others, thereby establishing the geometric unitary Kudla Conjecture in every codimension and removing the modularity hypothesis from an earlier arithmetic inner-product formula.

Core claim

Over an arbitrary CM field, every symmetric formal Fourier-Jacobi series converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As a direct consequence, the Chow-valued Kudla generating series of special cycles on the associated unitary Shimura varieties is modular of weight p+1 for a Weil representation, proving the geometric unitary Kudla Conjecture in arbitrary codimension.

What carries the argument

The symmetry condition imposed on formal Fourier-Jacobi series together with the fixed signature conditions on the underlying Hermitian lattices at infinity, which together guarantee convergence and identification with genuine modular forms.

If this is right

The Chow-valued Kudla generating series is modular of weight p+1 for a Weil representation.
The geometric unitary Kudla Conjecture holds in arbitrary codimension for unitary Shimura varieties over CM fields.
The modularity hypothesis can be removed from the arithmetic inner-product formula of Li-Liu.
Special-cycle intersection numbers on these varieties become accessible through modular-form techniques.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same convergence technique may extend to Fourier-Jacobi series attached to other Shimura varieties of PEL type.
Explicit computations of generating series for small p could now be verified directly against known modular forms.
The result supplies a missing analytic step that could be combined with existing arithmetic geometry tools to produce new height formulas.

Load-bearing premise

The formal series under consideration must be symmetric and the Hermitian lattices must satisfy the stated signature conditions at the infinite places.

What would settle it

An explicit symmetric formal Fourier-Jacobi series over a CM field that fails to converge or whose coefficients do not match those of any Hermitian Hilbert modular form.

read the original abstract

We prove that, over an arbitrary CM field, every symmetric formal Fourier-Jacobi series converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties for Hermitian lattices over CM fields of signature $(p,1)$ at one infinite place and $(p+1,0)$ at all others is modular of weight $p+1$ for a Weil representation, establishing the geometric unitary Kudla Conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula by Li-Liu.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Raum proves the geometric unitary Kudla conjecture outright for arbitrary CM fields by showing every symmetric formal Fourier-Jacobi series converges to a genuine Hermitian Hilbert modular form.

read the letter

Raum proves the geometric unitary Kudla conjecture outright for arbitrary CM fields by showing every symmetric formal Fourier-Jacobi series converges to a genuine Hermitian Hilbert modular form. From there the Chow-valued Kudla generating series on the unitary Shimura varieties of signature (p,1) at one place and (p+1,0) elsewhere becomes modular of weight p+1 for the Weil representation, and this holds in any codimension. The immediate payoff is that the modularity hypothesis Li-Liu needed for their arithmetic inner product formula is now removed. Previous results were limited to special cases or carried extra assumptions; this version is unconditional once the series is symmetric and the lattice signatures at infinity are fixed. The argument applies standard facts about Hermitian modular forms to the general CM setting without new circular steps or fitted parameters. The symmetry condition and signature restrictions are stated clearly up front, so the scope is transparent. A minor soft spot is the dependence on earlier convergence results for Hermitian forms; if those have any subtle edge cases when the base field is arbitrary CM, they would need checking, but nothing in the claim structure suggests an internal gap. This is a paper for people already working on the Kudla program, special cycles, and arithmetic intersections on Shimura varieties. Anyone who has cited Li-Liu or needs a clean modularity statement for generating series will find it directly usable. It deserves a serious referee. The central claim is precise enough that experts can focus on verifying the convergence step and the application.

Referee Report

0 major / 2 minor

Summary. The paper proves that every symmetric formal Fourier-Jacobi series over an arbitrary CM field converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As an application, the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties for Hermitian lattices of signature (p,1) at one infinite place and (p+1,0) at the others is modular of weight p+1 for a Weil representation, establishing the geometric unitary Kudla Conjecture in arbitrary codimension and removing the modularity hypothesis from the arithmetic inner product formula of Li-Liu.

Significance. If the central claims hold, the result advances the Kudla program by providing unconditional modularity for generating series of special cycles on unitary Shimura varieties over general CM fields. This strengthens arithmetic intersection theory and removes a key hypothesis from Li-Liu's inner product formula, with potential implications for higher-codimension cycle classes and automorphic forms on Hermitian groups.

minor comments (2)

[Abstract] The abstract states the convergence result for symmetric formal series but does not define 'symmetric' explicitly; a brief reminder or reference to the relevant definition in §2 would aid readers.
[§4] The signature conditions at infinity are stated as (p,1) at one place and (p+1,0) elsewhere; confirm that the proof in §4 handles the transition between these signatures without additional local conditions at finite places.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The referee's summary accurately captures the main theorem on convergence of symmetric formal Fourier-Jacobi series and its application to the geometric unitary Kudla conjecture.

read point-by-point responses

Referee: The paper proves that every symmetric formal Fourier-Jacobi series over an arbitrary CM field converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As an application, the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties for Hermitian lattices of signature (p,1) at one infinite place and (p+1,0) at the others is modular of weight p+1 for a Weil representation, establishing the geometric unitary Kudla Conjecture in arbitrary codimension and removing the modularity hypothesis from the arithmetic inner product formula of Li-Liu.

Authors: We thank the referee for this precise summary of our central results. The convergence statement for symmetric formal Fourier-Jacobi series is indeed the key technical step, and the application to modularity of the Chow-valued Kudla generating series follows directly, removing the hypothesis from Li-Liu's formula. We have no disagreement with the description and no revision is required on this point. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation applies prior independent results

full rationale

The paper establishes convergence of symmetric formal Fourier-Jacobi series to genuine Hermitian Hilbert modular forms over CM fields by invoking standard techniques from automorphic forms and Shimura varieties. It then deduces modularity of the Chow-valued Kudla generating series of weight p+1, removing a hypothesis from Li-Liu. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the symmetry and signature assumptions are explicit inputs, and the argument remains self-contained against external benchmarks without renaming known results or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms of algebraic geometry and automorphic forms over CM fields; no free parameters or invented entities are introduced in the abstract. The proof likely invokes existence of Hermitian modular forms and properties of Chow rings as background.

axioms (2)

standard math Existence and basic properties of Hermitian Hilbert modular forms over CM fields
Invoked to equate formal series with genuine expansions
domain assumption Standard properties of Chow groups and special cycles on Shimura varieties
Used to define the Kudla generating series

pith-pipeline@v0.9.0 · 5386 in / 1354 out tokens · 29052 ms · 2026-05-15T16:34:22.984947+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

[1]

of cogenus h − 1, we define Hermitian Jacobi forms of cogenus h by φm(τ1, z, w) = X n,r c(f ′; t)e(nτ1 + rt z + rt w). This right hand side converges absolutely, since it is a subseries of the absolutely convergent Fourier expansion of ψm′ by virtue of ψm′(τ′ 1, z′, w ′) = X n′ 22,r ′ 2 φm(τ1, z, w)e ¡ n′ 22τ′ 12 + rt ′ 2z′ 22 + rt ′ 2w ′ 22 ¢ . Further, ...

work page
[2]

Raum where disc(m′) and the theta series are the ones defined in (2.17), and the product on the right hand side is a product of formal series in Ce(τi j)i ,j

θm′,µ′(τ′ 1, z′, w ′), (4.5) –44 – The unitary Kudla conjecture M. Raum where disc(m′) and the theta series are the ones defined in (2.17), and the product on the right hand side is a product of formal series in Ce(τi j)i ,j . The vector-valued formal series with components ˜fµ′ satisfies ¡ ˜fµ′ ¢ µ′ ∈ FS(g −h+1,1) k−h+1 ¡ ρ(g −h+1) ∨ m′ ¢ . Proof. Thro...

work page
[3]

and r ′ 2(µ′) = r ′ 2(µ′

work page
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We first prove the formal decomposition in (4.5)

for their decomposition as in (2.27). We first prove the formal decomposition in (4.5). We set uλ := ³ 1g −h+1 λ 1h−1 ´ ∈ GLg (OF ) for λ ∈ Math−1,g −h+1(OF ). We recall that ψm′(τ1, z, w)e(m′τ′

work page
[5]

Throughout the proof, all references to t implicitly assume this bottom right block

= Pc(f ; t)e(t τ), where the sum runs over t ∈ Mat† g (OF )∨ with bottom right block m′. Throughout the proof, all references to t implicitly assume this bottom right block. We have t[uλ] = ³ 1g −h+1 λt 1h−1 ´ ³ n′ rt ′ r ′ m′ ´ ³ 1g −h+1 λ 1h−1 ´ = ³ n′+ rt ′λ+ λt r ′+ λt m′λ rt ′+ λt m′ r ′+m′λ m′ ´ . For every t, there is a unique λ such that t[uλ] has...

work page
[6]

To simplify the inner sum overλ, we set λ′ = r ′(µ′) + m′λ, which yields m′ −1[λ′] = m′ −1[r ′(µ′)] + r ′(µ′)t λ + λt r ′(µ′) + λt m′λ

= X µ′ X t c(f ; t) X λ e(t[uλ]τ), where the outer sum runs over µ′ as in the statement, the sum over t ∈ Mat† g (OF )∨ is restricted to t with bottom left block r ′(µ′), and the inner sum runs over λ as in the definition of uλ. To simplify the inner sum overλ, we set λ′ = r ′(µ′) + m′λ, which yields m′ −1[λ′] = m′ −1[r ′(µ′)] + r ′(µ′)t λ + λt r ′(µ′) + ...

work page
[7]

–46 – The unitary Kudla conjecture M

= ¡ ψm′(τ′ 1, z′, w ′)e(m′τ′ 2) ¢ |k γ = D ρ(g −h+1) ∨(γ)−1 ˜f (τ′ 1)|k−h+1 γ, θm′(τ′ 1, z′, w ′)e(m′τ′ 2) E . –46 – The unitary Kudla conjecture M. Raum Thus, ˜f (τ′

work page
[8]

= ρ(g −h+1) ∨(γ)−1 ˜f (τ′ 1)|k−h+1 γ, and inspecting the coefficient of e((m11 − m′ −1[r ′ 2(µ′)])τ′

work page
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4.3 The final convergence statement We are now in a position to prove Theorem C, which is a synthesis of Theorem 3.15 with Proposition 4.3 and Proposition 4.5

on both sides establishes the desired transformation behavior of φµ′,m11. 4.3 The final convergence statement We are now in a position to prove Theorem C, which is a synthesis of Theorem 3.15 with Proposition 4.3 and Proposition 4.5. Proof of Theorem C. We show the theorem by induction on the cogenus. Ifh = 1, then Theorem 3.15 in conjunction with Lemma 4...

work page
[10]

Let f be any functional on CH g (Γ\Gr−(LR))

Since Chow groups are not known to be finite dimensional, we have to proceed with care regard- ing the difference between formal series with coefficients in Chow groups and the tensor product of formal series with Chow groups. Let f be any functional on CH g (Γ\Gr−(LR)). It yields a formal Fourier series f (θKudla L (τ)). By Proposition 5.3 it is a formal...

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[1] [1]

of cogenus h − 1, we define Hermitian Jacobi forms of cogenus h by φm(τ1, z, w) = X n,r c(f ′; t)e(nτ1 + rt z + rt w). This right hand side converges absolutely, since it is a subseries of the absolutely convergent Fourier expansion of ψm′ by virtue of ψm′(τ′ 1, z′, w ′) = X n′ 22,r ′ 2 φm(τ1, z, w)e ¡ n′ 22τ′ 12 + rt ′ 2z′ 22 + rt ′ 2w ′ 22 ¢ . Further, ...

work page

[2] [2]

Raum where disc(m′) and the theta series are the ones defined in (2.17), and the product on the right hand side is a product of formal series in Ce(τi j)i ,j

θm′,µ′(τ′ 1, z′, w ′), (4.5) –44 – The unitary Kudla conjecture M. Raum where disc(m′) and the theta series are the ones defined in (2.17), and the product on the right hand side is a product of formal series in Ce(τi j)i ,j . The vector-valued formal series with components ˜fµ′ satisfies ¡ ˜fµ′ ¢ µ′ ∈ FS(g −h+1,1) k−h+1 ¡ ρ(g −h+1) ∨ m′ ¢ . Proof. Thro...

work page

[3] [3]

and r ′ 2(µ′) = r ′ 2(µ′

work page

[4] [4]

We first prove the formal decomposition in (4.5)

for their decomposition as in (2.27). We first prove the formal decomposition in (4.5). We set uλ := ³ 1g −h+1 λ 1h−1 ´ ∈ GLg (OF ) for λ ∈ Math−1,g −h+1(OF ). We recall that ψm′(τ1, z, w)e(m′τ′

work page

[5] [5]

Throughout the proof, all references to t implicitly assume this bottom right block

= Pc(f ; t)e(t τ), where the sum runs over t ∈ Mat† g (OF )∨ with bottom right block m′. Throughout the proof, all references to t implicitly assume this bottom right block. We have t[uλ] = ³ 1g −h+1 λt 1h−1 ´ ³ n′ rt ′ r ′ m′ ´ ³ 1g −h+1 λ 1h−1 ´ = ³ n′+ rt ′λ+ λt r ′+ λt m′λ rt ′+ λt m′ r ′+m′λ m′ ´ . For every t, there is a unique λ such that t[uλ] has...

work page

[6] [6]

To simplify the inner sum overλ, we set λ′ = r ′(µ′) + m′λ, which yields m′ −1[λ′] = m′ −1[r ′(µ′)] + r ′(µ′)t λ + λt r ′(µ′) + λt m′λ

= X µ′ X t c(f ; t) X λ e(t[uλ]τ), where the outer sum runs over µ′ as in the statement, the sum over t ∈ Mat† g (OF )∨ is restricted to t with bottom left block r ′(µ′), and the inner sum runs over λ as in the definition of uλ. To simplify the inner sum overλ, we set λ′ = r ′(µ′) + m′λ, which yields m′ −1[λ′] = m′ −1[r ′(µ′)] + r ′(µ′)t λ + λt r ′(µ′) + ...

work page

[7] [7]

–46 – The unitary Kudla conjecture M

= ¡ ψm′(τ′ 1, z′, w ′)e(m′τ′ 2) ¢ |k γ = D ρ(g −h+1) ∨(γ)−1 ˜f (τ′ 1)|k−h+1 γ, θm′(τ′ 1, z′, w ′)e(m′τ′ 2) E . –46 – The unitary Kudla conjecture M. Raum Thus, ˜f (τ′

work page

[8] [8]

= ρ(g −h+1) ∨(γ)−1 ˜f (τ′ 1)|k−h+1 γ, and inspecting the coefficient of e((m11 − m′ −1[r ′ 2(µ′)])τ′

work page

[9] [9]

4.3 The final convergence statement We are now in a position to prove Theorem C, which is a synthesis of Theorem 3.15 with Proposition 4.3 and Proposition 4.5

on both sides establishes the desired transformation behavior of φµ′,m11. 4.3 The final convergence statement We are now in a position to prove Theorem C, which is a synthesis of Theorem 3.15 with Proposition 4.3 and Proposition 4.5. Proof of Theorem C. We show the theorem by induction on the cogenus. Ifh = 1, then Theorem 3.15 in conjunction with Lemma 4...

work page

[10] [10]

Let f be any functional on CH g (Γ\Gr−(LR))

Since Chow groups are not known to be finite dimensional, we have to proceed with care regard- ing the difference between formal series with coefficients in Chow groups and the tensor product of formal series with Chow groups. Let f be any functional on CH g (Γ\Gr−(LR)). It yields a formal Fourier series f (θKudla L (τ)). By Proposition 5.3 it is a formal...

work page

[11] [11]

H. Aoki. Estimating Siegel modular forms of genus 2 using Jacobi forms. J. Math. Kyoto Univ.40.3 (2000)

work page 2000

[12] [12]

H. Aoki. On the upper bound of the orders of Jacobi forms. Int. J. Math. 33.8 (2022)

work page 2022

[13] [13]

H. Aoki, T . Ibukiyama, and C. Poor. Formal series of Jacobi forms. arXiv:2412.18746. 2024

work page arXiv 2024

[14] [14]

R. E. Borcherds. Automorphic forms with singularities on Grassmannians. Invent. Math. 132.3 (1998)

work page 1998

[15] [15]

R. E. Borcherds. Reflection groups of Lorentzian lattices. Duke Math. J. 104.2 (2000)

work page 2000

[16] [16]

R. E. Borcherds. The Gross-Kohnen-Zagier theorem in higher dimensions. Duke Math. J. 97.2 (1999)

work page 1999

[17] [17]

H. Braun. Hermitian modular functions. Ann. Math. 50.2 (1949)

work page 1949

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H. Braun. Hermitian modular functions III. Ann. Math. (2) 53 (1951)

work page 1951

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H. Braun. Hermitian modular functions. II. Ann. Math. (2) 51 (1950)

work page 1950

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J. H. Bruinier. On the converse theorem for Borcherds products. J. Algebra 397 (2014)

work page 2014

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J. H. Bruinier. Vector valued formal Fourier-Jacobi series. Proc. Amer . Math. Soc.143.2 (2015). –51 – The unitary Kudla conjecture M. Raum

work page 2015

[22] [22]

J. H. Bruinier and M. Westerholt-Raum. Kudla’ s modularity conjecture and formal Fourier-Jacobi series. Forum Math. Pi 3 (2015)

work page 2015

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