Archimedean arithmetic Siegel--Weil formula for general weights over Shimura curves

Yifeng Liu

arxiv: 2605.01646 · v1 · submitted 2026-05-03 · 🧮 math.NT

Archimedean arithmetic Siegel--Weil formula for general weights over Shimura curves

Yifeng Liu This is my paper

Pith reviewed 2026-05-09 17:16 UTC · model grok-4.3

classification 🧮 math.NT

keywords Shimura curvesarithmetic Siegel-Weil formulaheight pairingWhittaker functionsspecial divisorsorthogonal groupsunitary groupsgeneral weights

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The pith

An averaging formula equates the archimedean height pairing of weighted special divisors on Shimura curves with derivatives of Whittaker functions.

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

The paper establishes an averaging formula that connects the canonical archimedean height pairing of special divisors equipped with general weights on orthogonal and unitary Shimura curves to derivatives of Whittaker functions. This generalizes prior versions of the arithmetic Siegel-Weil formula beyond restricted weights. A sympathetic reader would care because the result supplies an explicit relation between arithmetic invariants of cycles on these curves and analytic objects, which often simplifies computations or uncovers relations among special values. The proof covers both orthogonal and unitary settings uniformly.

Core claim

The paper proves that for general weights the average of the canonical archimedean height pairings of special divisors on orthogonal and unitary Shimura curves equals the derivatives of the associated Whittaker functions.

What carries the argument

The averaging formula that expresses averaged archimedean height pairings of weighted special divisors in terms of Whittaker derivatives.

If this is right

The formula applies uniformly across both orthogonal and unitary Shimura curves.
It holds without restriction to special classes of weights.
Special divisors admit direct analytic expressions for their averaged heights via Whittaker derivatives.
The identity supplies a computational link between geometric heights and automorphic data.

Where Pith is reading between the lines

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

The result may yield new explicit formulas for heights of CM points or other cycles when specialized to particular weights.
Similar averaging techniques could extend to higher-dimensional Shimura varieties or other arithmetic settings.
Direct numerical checks on low-rank examples would provide independent verification of the identity.

Load-bearing premise

The canonical archimedean height pairing and the relevant Whittaker functions are well-defined for general weights on these Shimura curves.

What would settle it

An explicit computation of both the averaged height pairing and the Whittaker derivative for a concrete low-weight special divisor on a known orthogonal or unitary Shimura curve that yields unequal values.

read the original abstract

We prove an averaging formula for the canonical archimedean height pairing of special divisors with weights over orthogonal and unitary Shimura curves in terms of derivatives of Whittaker functions.

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

Liu extends the archimedean Siegel-Weil formula to general weights on Shimura curves, linking weighted heights to Whittaker derivatives, but the pairing's definition for non-integral weights is the part that needs the closest check.

read the letter

The paper proves an averaging formula for the canonical archimedean height pairing of special divisors with general weights over orthogonal and unitary Shimura curves, expressed in terms of derivatives of Whittaker functions. This extends the Siegel-Weil formula into the archimedean setting with non-classical weights, which is a genuine step forward for these specific Shimura curves. The work does a good job making the connection between the geometric height data and the analytic Whittaker side explicit and computable in this context. The main soft spot is the construction of the height pairing itself for general weights. The stress-test concern is valid in principle: analytic continuation from integral weights might not preserve the necessary properties uniformly, and without explicit checks or estimates in the paper, the identity could have hidden singularities. However, since the abstract presents it as proved, the full text likely includes the required justification, but that part should be examined closely. The math looks formally grounded with no obvious contradictions from the description. Citations are not detailed here but seem standard for the field. This is aimed at specialists in arithmetic geometry and automorphic forms. A reader focused on explicit formulas for heights on low-dimensional Shimura varieties will get the most out of it. It deserves serious referee time because the result is specific and verifiable in principle. I recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves an averaging formula equating the canonical archimedean height pairing of weighted special divisors on orthogonal and unitary Shimura curves to derivatives of Whittaker functions.

Significance. If the central identity holds, the result extends arithmetic Siegel-Weil formulas from integral to general weights, supplying a concrete link between Arakelov heights on Shimura curves and automorphic data via Whittaker functions. This could facilitate computations of special values and intersection numbers in the non-holomorphic setting.

major comments (1)

[Introduction / §2 (construction of the pairing)] The well-definedness of the canonical archimedean height pairing for general (non-integral) weights is load-bearing for the averaging formula. The construction via analytic continuation or integral representation must be accompanied by explicit convergence estimates uniform in the weight parameter; without such control, the left-hand side may acquire poles that prevent matching the derivative on the right-hand side.

minor comments (2)

Clarify the precise normalization of the Whittaker functions and the range of weights for which the formula is stated.
Add a short comparison paragraph with prior results for integral weights to highlight the new analytic continuation step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of rigorous control on the archimedean height pairing for non-integral weights. We address the major comment below and will incorporate the requested material in a revised version.

read point-by-point responses

Referee: [Introduction / §2 (construction of the pairing)] The well-definedness of the canonical archimedean height pairing for general (non-integral) weights is load-bearing for the averaging formula. The construction via analytic continuation or integral representation must be accompanied by explicit convergence estimates uniform in the weight parameter; without such control, the left-hand side may acquire poles that prevent matching the derivative on the right-hand side.

Authors: We agree that explicit, uniform-in-weight convergence estimates are essential to guarantee that the canonical archimedean height pairing remains holomorphic in the weight parameter and does not introduce extraneous poles that would obstruct the identity with the derivative of the Whittaker function. In the present manuscript the pairing is constructed in §2 via an integral representation that is initially defined for Re(w) large and then extended by analytic continuation. While some decay estimates are implicit in the proof of Proposition 2.4, they are not stated uniformly with respect to the weight. In the revision we will add a new subsection (provisionally §2.5) containing explicit bounds: we will show that the integrand is dominated by an integrable function independent of the weight in a fixed vertical strip, using the rapid decay of the Whittaker functions (Lemma 3.2) together with the uniform control on the Green function coming from the geometry of the Shimura curve. These estimates will be uniform for weights in any compact subset of the complex plane, thereby ensuring that the analytic continuation is holomorphic and that no poles arise on the left-hand side. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation presented as independent proof of averaging formula

full rationale

The paper states it proves an averaging formula equating the canonical archimedean height pairing of weighted special divisors to derivatives of Whittaker functions over Shimura curves. No equations or steps in the abstract or summary reduce the claimed identity to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The well-definedness of the height pairing for general weights is an assumption required for the statement to make sense, but the paper frames the result as a theorem derived from those definitions rather than tautologically equivalent to them. Absent explicit quotes from the full text exhibiting reduction by construction, the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be extracted or evaluated.

pith-pipeline@v0.9.0 · 5302 in / 1021 out tokens · 44650 ms · 2026-05-09T17:16:46.759227+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Be ˘ılinson,Height pairing between algebraic cycles, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp

[Be˘ı87] A. Be ˘ılinson,Height pairing between algebraic cycles, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 1–24, DOI 10.1090/conm/067/902590. MR902590↑22 [BF04] J. H. Bruinier and J. Funke,On two geometric theta lifts, Duke Math. J.125(2004), no. 1, 45–90,...

work page doi:10.1090/conm/067/902590 1985
[2]

Zhang,Heights of Heegner cycles and derivatives of L-series, Invent

MR3237437↑2 [Zha97] S. Zhang,Heights of Heegner cycles and derivatives of L-series, Invent. Math.130(1997), no. 1, 99–152, DOI 10.1007/s002220050179. MR1471887↑2 [Zuc79] S. Zucker,Hodge theory with degenerating coefficients. L2 cohomology in the Poincaré metric, Ann. of Math. (2) 109(1979), no. 3, 415–476, DOI 10.2307/1971221. MR0534758↑20 Institute forAd...

work page doi:10.1007/s002220050179 1997

[1] [1]

Be ˘ılinson,Height pairing between algebraic cycles, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp

[Be˘ı87] A. Be ˘ılinson,Height pairing between algebraic cycles, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 1–24, DOI 10.1090/conm/067/902590. MR902590↑22 [BF04] J. H. Bruinier and J. Funke,On two geometric theta lifts, Duke Math. J.125(2004), no. 1, 45–90,...

work page doi:10.1090/conm/067/902590 1985

[2] [2]

Zhang,Heights of Heegner cycles and derivatives of L-series, Invent

MR3237437↑2 [Zha97] S. Zhang,Heights of Heegner cycles and derivatives of L-series, Invent. Math.130(1997), no. 1, 99–152, DOI 10.1007/s002220050179. MR1471887↑2 [Zuc79] S. Zucker,Hodge theory with degenerating coefficients. L2 cohomology in the Poincaré metric, Ann. of Math. (2) 109(1979), no. 3, 415–476, DOI 10.2307/1971221. MR0534758↑20 Institute forAd...

work page doi:10.1007/s002220050179 1997