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Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves the complete spectral decomposition of the diagonal pullback of a Saito-Kurokawa lift of odd square-free level: each coefficient is zero or its square is a scalar multiple of $\Lambda(f\otimes \mathrm{sym}^2 g,1/2)$.

desk verdict Genuinely new spectral decomposition for pullbacks of SK lifts, but the old-level normalization is imported from [PV20] and the abstract oversells the average—worth a careful referee. read the letter →

arxiv 2505.08660 v2 pith:JLEYKXXZ submitted 2025-05-13 math.NT

Pramath Anamby , Soumya Das This is my paper · ORCID
classification math.NT MSC 11F4611F5011F6711F37
keywords Saito-KurokawaliftspullbackformulascentralL-valuesspectraldecompositionoldformssquare-freelevelL2-massnon-vanishing
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

This paper studies what happens when a Saito-Kurokawa lift $F$ of odd square-free level $N$ is pulled back to the diagonal of the Siegel upper half-plane. In the natural arithmetic basis of the oldspace—forms of lower level raised to level $N$ and then orthogonalized—every coefficient $\langle F^\circ,g_\sigma\otimes g_\sigma\rangle$ is either forced to vanish (exactly when the elliptic newform $f$ whose lift is $F$ has Atkin-Lehner eigenvalue $-1$ at some prime dividing the old level) or its square is an explicit constant times the central value $\Lambda(f\otimes \mathrm{sym}^2 g,1/2)$. Because these basis elements span the full space $S_{k+1}(N)$, this is the complete spectral decomposition, up to the signs of the square roots. The paper then turns the decomposition into a formula for the $L^2$-mass of the pullback as a weighted average of those central $L$-values, derives a conjectural asymptotic for that mass from standard moment heuristics, verifies the main term on average over $f$, and proves several non-vanishing results, including all weight-2 cases.

What carries the argument

The load-bearing object is the arithmetically orthogonalized old basis $g_\sigma=\sum_{d|M_g}\sigma(d)\,g|W_N(d)$, indexed by characters $\sigma$ of $(\mathbb{Z}/2\mathbb{Z})^{\omega(M_g)}$; it turns the oldspace of $S_{k+1}(N)$ into an orthogonal family of Hecke-eigenforms, one family per underlying newform $g$. Two mechanisms carry the argument. The first is an explicit set of coset representatives $\mathcal{C}(N,M)$ of $\Gamma_0^{(2)}(N)\backslash \Gamma_0^{(2)}(M)$ chosen so that the pullback map and the relevant Hecke operators commute at the level of cosets; this lets the authors unfold $\langle F^\circ,g\otimes g|B_M\rangle$ and, after a long calculation in the $\mathrm{GL}(2)$ Hecke algebra, reduce it to the Jacobi-form period $\langle \phi^\circ,g|W_N(M_g)\rangle$ with all intervening Euler products collapsing to elementary factors (Theorem 6.14). The second is the vanishing mechanism: for a newform $F$, the trace identity $F|\mathrm{Tr}^{(2)}(N,N/p)=0$ is unfolded with these cosets to derive $(1+p^{1-k}\lambda_F(p)w_F(N))\langle F^\circ,g\otimes g|V\rangle=0$, and since Saito-Kurokawa lifts have $w_F(p)=1$, this forces the coefficient to vanish whenever $w_f(p)=-1$.

What would settle it

Take $N=15$, $k=3$, choose $f\in S^{\mathrm{new}}_6(15)$ and $g\in S^{\mathrm{new}}_4(3)$ with $w_f(5)=+1$ so that $M_g=5\in L_f$, compute the diagonal restriction $F^\circ$ from the Fourier expansion of $F$ (via the half-integral-weight form $h$), and evaluate both sides of (2.8). The equality must hold with the exact constants displayed, including $M_g^{7/4}=5^{7/4}$; any residual power of $5$ or a missing power of $2$ would falsify the normalization of the imported formula.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.3. Let $N$ be odd and square-free, let $f\in S^{\mathrm{new}}_{2k}(N)$, and let $F=F_f$ be its Saito-Kurokawa lift. For a newform $g\in S^{\mathrm{new}}_{k+1}(N_g)$ with $N_g|N$, put $M_g=N/N_g$; for each character $\sigma$ of $(\mathbb{Z}/2\mathbb{Z})^{\omega(M_g)}$ the element $g_\sigma=\sum_{d|M_g}\sigma(d)\,g|W_N(d)$ belongs to an orthogonal basis of $S_{k+1}(N)$. The theorem states that $\langle F^\circ,g_\sigma\otimes g_{\sigma'}\rangle=0$ for $\sigma\ne\sigma'$; that $\langle F^\circ,g_\sigma\otimes g_\sigma\rangle=0$ whenever $M_g\notin L_f$, the set of divisors on which $f$ has Atkin-Lehner eigenvalue $+1$; and that when $M_g\in L_f$, $$\Lambda(f\otimes \mathrm{sym}^2 g,\tfrac12)=$2^{{k+1-\omega(M_g)}}$ $M_g^{{7/4}}$ $N^{{-1}}$ \prod_{p|N_g}(p+1)^2 \frac{\langle f,f\rangle}{\langle h,h\rangle} \frac{|\langle F^\circ,g_\$\sigma$\otimes g_\$\sigma$\rangle|^2}{\langle g_\$\sigma$,g_\$\sigma$\$rangle^{2}$},$$ where $h$ is the half-integral-weight form corresponding to $f$. Since the $g_\sigma$, together with the newforms of level $N$, form an orthogonal basis of $S_{k+1}(N)$, this gives the full spectral decomposition of $F^\circ$ up to the signs $\pm\sqrt{\Lambda(f\otimes \mathrm{sym}^2g,1/2)}$.

Load-bearing premise

The load-bearing input is the old-level pullback formula imported from the literature and used in exactly the normalized form stated as (1.3); any error in its constants—in particular the power of the old level $M_g$—propagates directly into the spectral decomposition and everything built on it.

Editorial extensions

If this is right

  • The $L^2$-mass $\mathcal{N}(F_f)$ becomes a finite weighted average of central values $L(f\otimes \mathrm{sym}^2 g,1/2)$, so non-negativity of those central values directly bounds the pullback norm from below.
  • In weight 2 every pullback is nonvanishing; hence every $f\in S^{\mathrm{new}}_2(N)$ has at least one newform $g$ of level dividing $N$ with $L(f\otimes \mathrm{sym}^2 g,1/2)\neq 0$, and if all Atkin-Lehner eigenvalues of $f$ are $-1$, $g$ can be chosen of full level $N$.
  • Nonvanishing of $F^\circ_f$ is equivalent to the existence of an even Fourier coefficient of $H=h\cdot\vartheta_0$, and only the finite range $m\le K$ needs checking; this gives a finite algorithm for deciding whether a given pullback vanishes.
  • For prime level $p$ and odd $k>2$, a proportion $>1/7$ of newforms $f$ have nonvanishing pullbacks, hence nonvanishing of at least one central value in the average.
  • The average over $f$ of $\mathcal{N}(F_f)$ has the same main term as the conjectural asymptotic, giving the first averaged confirmation of the $L^2$-mass conjecture in the level aspect.

Reading between the lines

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

  • The signs of the individual coefficients are left undetermined, so the decomposition fixes the Petersson norm of $F^\circ$ but not the function itself; a natural extension is to determine the signs by matching the Fourier-Jacobi expansion near the cusps, upgrading (2.8) to an exact identity.
  • Because the vanishing condition depends only on the Atkin-Lehner signs of $f$ and on $M_g$, a testable prediction is that the same dichotomy (all coefficients zero unless $M_g\in L_f$) persists in neighbouring settings—non-square-free levels, or pullbacks of Hermitian Maass lifts—where the analogous old-level formula is not yet available.
  • Theorem 2.9 reduces nonvanishing to a finite Fourier-coefficient check; scanning small square-free $N$ and odd $k>2$ with this criterion would give a quantitative census of how often pullbacks vanish, which the paper does not attempt.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the diagonal restriction (pullback) F^∘ of a Saito-Kurokawa newform F of odd square-free level N, where F corresponds to an elliptic newform f∈S_new_{2k}(N). Introducing an arithmetically orthogonalized basis of oldforms g_σ indexed by characters of (Z/2Z)^{ω(M_g)}, the authors prove (Theorem 2.3) that the periods ⟨F^∘,g_σ⊗g_{σ'}⟩ vanish for σ≠σ', vanish whenever the level ratio M_g lies outside a set L_f of divisors determined by the Atkin-Lehner eigenvalues of f, and otherwise satisfy an exact formula for |⟨F^∘,g_σ⊗g_σ⟩|²/⟨g_σ,g_σ⟩² in terms of the central L-value Λ(f⊗sym²g,1/2). This gives the full spectral decomposition of F^∘ up to signs. The authors then derive an L²-mass formula (Theorem 2.6), state a CFKRS-based conjecture for its asymptotic (Conjecture 2.7), and in the appendix compute the average over f of the main term, matching the conjecture. The paper also contains several non-vanishing results, including a positive-proportion result for pullbacks at prime level and a complete weight-2 non-vanishing theorem.

Significance. If the normalization issues identified below are resolved, this is a substantial contribution: it answers Question 1.1 completely for Saito-Kurokawa lifts of square-free level, providing the first full spectral decomposition of the pullback in the level aspect. The core computations in Sections 5 and 6 are detailed and apparently original, with explicit Euler products and extensive Hecke-algebra manipulations; Section 4 gives a self-contained construction of coset representatives likely to be useful independently. The paper contains no fitted parameters, no circularity, and several falsifiable predictions. The derived L²-mass formula and the average matching with CFKRS heuristics are of independent analytic interest, and the non-vanishing applications (weight 2 all cases; positive proportion for prime level) are valuable.

major comments (2)
  1. [Theorem 2.3, Eqs. (2.7)–(2.8), with L_f as in (2.5)] As written, the theorem is false for Mg=1. Since (2.5) defines L_f to exclude L=1, the condition "Mg∉L_f" is satisfied for every newform g of level N, so (2.7) would force ⟨F^∘,g⊗g⟩=0 for all such g. This contradicts formula (1.3) with Mg=1 (the Chen/PV19 result), and would make the later spectral decomposition, Theorem 2.6, and Corollary 2.8 inconsistent. The proof via Corollary 5.3 only establishes vanishing when there exists p|Mg with wf(p)=-1, i.e., for Mg>1 and Mg∉L_f. Please correct by either including 1 in L_f (the vacuous condition "wf(p)=+1 for all p|L" is satisfied for L=1) or by writing (2.7) with "Mg>1 and Mg∉L_f" and (2.8) with "Mg∈L_f∪{1}", and adjusting the surrounding text and the basis statement accordingly.
  2. [Section 1, Eq. (1.3); used in the proof of Theorem 2.3 at the end of Section 6.6] The central-value formula (2.8) for old classes is ultimately an immediate consequence of the imported formula (1.3), which is attributed to [PV20] with only the note that the completed L-function has been renormalized to have central point s=1/2. Any mismatch in the constant of (1.3) — for example an extra power of Mg or a missing local Euler factor at primes p|Ng — would propagate unchanged through (2.8), Theorem 2.6(2.10), and the average main-term computation in Section 9. Since the paper's own completed L-function (Section 3.4) has normalization N^{3s/2}Ng^{s/2} and the authors note in Remark 3.2 that their normalization differs from [PV19]/[PV20] by powers of Mg, the paper should either prove (1.3) in the present normalization or give a precise verification of the translation, including the exact powers of N, Ng, and Mg and the local factors at p|Ng. Without this, the central claim of the paper is conditional on an unverified external constant.
minor comments (5)
  1. [Title and abstract] The title contains "theL2-mass" without a space, and the key words list "Saito-Kurokawa lits", which should be "lifts".
  2. [Section 2.3, after Eq. (2.9)] "where where v1 = vol.(SL2(Z)\H)" contains a duplicated "where".
  3. [Section 9, first paragraph] The text refers to "Conjecture 2.11", but the conjecture is numbered 2.7; please correct the cross-reference.
  4. [Section 3.1.3] The notation W_N(d) is used for both the degree-1 and degree-2 Atkin-Lehner operators, and the displayed identity "W_N(d) = W_N(d) × W_N(d)" is confusing; consider using distinct notation for the two settings.
  5. [References] The entries [DSa] and [DSb] are listed as "Preprint" with no arXiv identifier or year; please provide as much information as is available.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral decomposition is obtained by applying the external pullback formula (1.3) to an explicit orthogonal basis; the central L-values are independent inputs, not fitted outputs.

full rationale

The central formula (2.8) is a substitution of the known pullback formula (1.3) (Ichino full-level, Chen/PV19 new-level, PV20 old-level) into the algebraic identity relating the basis periods <F^o,g_sigma x g_sigma> to <F^o,g x g|B_{M_g}>. Neither side of (1.3) is defined in terms of the other, no parameter is fitted to data, and no subset of L-values is used to predict itself; the paper explicitly flags the normalization dependence of (1.3). The self-citations to [AD24] supply auxiliary equivalences (EZI/SK coincidence and norm relations) and are not the derivation of the central value formula; a mismatch there would be an input error, not a circular reduction. The conjecture/average comparison is a consistency check between CFKRS heuristics applied to (2.10) and the same expression, not an independent derivation from the target. A statement-level issue exists: L_f in (2.5) excludes L=1, so (2.7) as written would force all new-level periods (M_g=1) to vanish, while Section 7 treats L=1 separately; this is a correctness bug, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants or new physical or arithmetic entities. Its free-parameter count is zero: the Euler products, Hecke eigenvalues, and central L-values are all already determined by f and g. The axioms are either standard background in the modular forms program, the fixed level restriction to odd square-free N, the imported [PV20] formula, or the explicitly heuristic CFKRS recipe. No invented entities appear.

assumptions (6)
  • standard math Multiplicity-one for GL(2): off-diagonal periods ⟨F^∘,g'⊗g''⟩ vanish unless g' and g'' come from the same underlying newform (Lemma 3.3).
    Basis for the diagonal support of the spectral expansion and for reducing the decomposition to gσ⊗gσ terms.
  • domain assumption N is odd and square-free throughout, with all forms of trivial nebentypus and Hecke level N.
    The main theorems are stated only in this level aspect; the proofs use square-freeness for the divisor group structure and Atkin-Lehner decompositions.
  • domain assumption The old-level central-value formula (1.3) from [PV20] is accepted in the paper's normalization, with completed L-function central point at s=1/2.
    All old-class squared periods are converted to Λ(f⊗sym^2 g, 1/2) through this identity; the paper does not reprove it and only notes a normalization difference from [PV19].
  • standard math Functional equation and root number for L(f⊗sym^2 g, s): the sign is +1 for the central values appearing in the mass formula.
    Used to assert non-negativity of central values via Lapid's theorem and to set up the L2-mass formula in Section 3.4.
  • domain assumption The CFKRS (Conrey, Iwaniec, Keating) moment recipe is applied in Section 7.2 to conjecture the asymptotic for sums of L(f⊗sym^2 g, 1/2).
    Conjecture 2.7 is conditional on this heuristic; the paper labels it as conjectural and checks only a diagonal main term.
  • standard math Published results on spaces with eta multipliers, including Arakawa-Böcherer and dimension formulas, are used for the non-vanishing proportions.
    Theorems 8.2 and 8.3 rely on these external results, which are not re-proved in the paper.

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Pith. "Pith review of Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass." pith.science (2026). https://pith.science/paper/JLEYKXXZ

@misc{pith2026250508660,
  author       = {Pith},
  title        = {Pith review of: Pullbacks of Saito-Kurokawa lifts of square-free levels, their non-vanishing and the $L^2$-mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLEYKXXZ}},
  note         = {Machine review of arXiv:2505.08660}
}
abstract

We obtain the full spectral decomposition of the pullback of a Saito-Kurokawa (SK) newform $F$ of odd, square-free level; and show that the projections onto the elements $\mathbf g \otimes \mathbf g$ of an arithmetically orthogonalized old-basis are either zero or whose squares are given by the certain $\mathrm{GL}(3)\times \mathrm{GL}(2)$ central $L$-values $L(f\otimes \mathrm{sym}^2 g, \frac{1}{2})$, where $F$ is the lift of the $\mathrm{GL}(2)$ newform $f$ and $g$ is the newform underlying $\mathbf g$. Based on this, we work out a conjectural formula for the $L^2$-mass of the pullback of $F$ via the CFKRS heuristics, which becomes a weighted average (over $g$) of the central $L$-values. We show that on average over $f$, the main term predicted by the above heuristics matches with the actual main term. We also provide several results and sufficient conditions that ensure the non-vanishing of the pullbacks.

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Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages

  1. [1]

    Vanishing of certain spaces of elliptic modular forms and some applications

    [AB03] T. Arakawa and S. Böcherer. “Vanishing of certain spaces of elliptic modular forms and some applications”. In:Journal für reine und angweandte Mathematik599 (2003), pp. 25–51 (cit. on pp. 7, 34, 35, 37, 39). [AB99] T. Arakawa and S. Böcherer. “A note on the restriction map for Jacobi forms”. In: Abh. Math. Sem. Univ. Hamburg69 (1999), pp. 309–317 (...

    work page arXiv 2003

  2. [46]

    Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

    Prog.. Math., Birkhäuser Boston. 1984, pp. 114–137 (cit. on p. 2). [GGP11] W. T. Gan, B. H. Gross, and D. Prasad. “Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups”. In: Astérisque (2011) (cit. on p. 1). [GGS09] W. T. Gan, N. Gurevich, and G. Savin. “Restrictions of Saito-K...

    work page 2011

  3. [51]

    Statistique de la variable aléatoireL(sym2f, 1)

    Cambridge University Press Cambridge, 1977 (cit. on p. 35). [Roy01] E. Royer. “Statistique de la variable aléatoireL(sym2f, 1)”. In:Mathematische Annalen 321 (2001), pp. 667–687 (cit. on p. 42). [Sch05] R. Schmidt. “Iwahori-spherical representations ofGSp(4) and Siegel modular forms of degree 2 with square-free level”. In:Journal of the Mathematical Socie...

    arXiv 2001

  4. [53]

    Low lying zeros of families ofL-functions

    American Mathematical Soc., 2021 (cit. on p. 41). [ILS00] H. Iwaniec, W. Luo, and P. Sarnak. “Low lying zeros of families ofL-functions”. In: Publications Mathématiques de l’IHÉS91 (2000), pp. 55–131 (cit. on pp. 10, 24, 31, 32). [Kle04] M. Klein. “Verschwindungssätze für Hermitesche Modulformen sowie Siegelsche Mod- ulformen zu den KongruenzuntergruppenΓ...

    work page 2000

  5. [55]

    Pullbacks of Eisenstein series; applications

    Springer (cit. on pp. 2, 9, 34). [Gar84] P. B. Garrett. “Pullbacks of Eisenstein series; applications”. In:Automorphic forms of several variables (Katata, 1983). Vol

    work page 1983

  6. [228]

    Nonvanishing of Siegel Poincaré series

    Springer, 2005 (cit. on p. 19). [DS12] S. Das and J. Sengupta. “Nonvanishing of Siegel Poincaré series”. In:Mathematische Zeitschrift 272 (2012), pp. 869–883 (cit. on p. 38). [EZ] M. Eichler and D. Zagier.The theory of Jacobi forms. Vol

    work page 2012

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