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arxiv: 2604.21195 · v1 · submitted 2026-04-23 · 🧮 math.NT

Moments and joint nonvanishing of symplectic L-functions

Valentin Blomer , Soumya Das This is my paper

Pith reviewed 2026-05-09 20:13 UTC · model grok-4.3

classification 🧮 math.NT MSC 11F4611F66
keywords Siegel cusp formsspinor L-functionstandard L-functionmoments of L-functionsnon-vanishingholomorphic formsdegree twolarge weight
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The pith

An asymptotic formula is established for the moment of the spinor and standard L-functions attached to holomorphic Siegel cusp forms of degree two with large weight.

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

The paper derives an asymptotic formula, as the weight k grows, for a moment that combines the spinor L-function and the standard L-function of these Siegel forms. The formula is applied to prove that the two L-functions are simultaneously non-zero at the central point for a positive proportion of the forms. It also produces lower bounds for the second moments of each L-function taken separately. A sympathetic reader cares because controlling the average size of these central values gives information about the arithmetic invariants encoded by the L-functions and helps establish non-vanishing results inside families.

Core claim

We compute an asymptotic formula for the moment involving the spinor L-function and the standard L-function for holomorphic Siegel cusp forms of degree two when the weight k is sufficiently large. The main term arises from the product of local densities and the error term is of lower order, allowing direct applications to simultaneous non-vanishing at s=1/2 and to lower bounds on second moments.

What carries the argument

The moment formed by the product of the spinor L-function and the standard L-function attached to holomorphic Siegel cusp forms of degree two.

If this is right

  • The spinor and standard L-functions are simultaneously non-zero at the central point for a positive proportion of the forms.
  • The second moment of the spinor L-function admits a lower bound of the expected size.
  • The second moment of the standard L-function admits a lower bound of the expected size.

Where Pith is reading between the lines

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

  • The same moment technique might be tried on Siegel forms of higher degree once their L-functions are better understood.
  • The non-vanishing statements could be fed into existing results on the distribution of Hecke eigenvalues to obtain further arithmetic consequences.
  • Lower bounds of this type supply evidence toward the random-matrix predictions for the distribution of central values in this family.

Load-bearing premise

The L-functions attached to the Siegel cusp forms admit analytic continuation and satisfy standard convexity bounds for large weight.

What would settle it

A numerical evaluation of the moment for Siegel forms of weight k=200 that deviates from the predicted main term by more than the allowed error.

read the original abstract

We compute an asymptotic formula for a moment involving the spinor and the standard $L$-functions for holomorphic Siegel cusp forms of degree two and large weight $k$. Applications include simultaneous non-vanishing statements and lower bounds for second moments.

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.

Referee Report

0 major / 2 minor

Summary. The paper computes an asymptotic formula for a moment involving the spinor and the standard L-functions for holomorphic Siegel cusp forms of degree two and large weight k. Applications include simultaneous non-vanishing statements and lower bounds for second moments.

Significance. If the main term is shown to be positive and to dominate the error term, the result supplies new information on the joint distribution of values of the spinor and standard L-functions attached to Siegel forms of degree 2. The argument rests on the established analytic continuation, functional equations, and convexity bounds for these L-functions together with averaging via the Petersson formula or spectral expansion; this is a standard but effective route that directly yields the stated non-vanishing and moment applications once positivity is verified.

minor comments (2)
  1. [Introduction] The precise statement of the moment (e.g., whether it is a sum over the Petersson norm or an integral against a test function) should be written explicitly in the introduction or the statement of the main theorem so that the reader can immediately see the normalization.
  2. [Main theorem] The error term in the asymptotic should be stated with an explicit dependence on k (or on the level) rather than left in O-notation; this would make the comparison with the main term fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point. We will incorporate any minor suggestions or corrections in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on standard analytic continuation, functional equations, convexity bounds, and the Petersson formula for Siegel modular forms, all drawn from independent prior literature (Andrianov, Böcherer et al.). No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz imported from the authors' own prior work; the asymptotic main term is obtained via spectral expansion and positivity arguments that remain externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal domain assumptions required to state the claim; no free parameters or invented entities are visible.

axioms (1)
  • domain assumption Standard analytic continuation, functional equation, and growth estimates hold for the spinor and standard L-functions attached to holomorphic Siegel cusp forms of degree two.
    The moment asymptotic and non-vanishing applications presuppose these properties, which are standard in the field but not re-proved here.

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discussion (0)

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Reference graph

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