{"id":"795f0890-95b4-493f-8c03-5ee890d16292","arxiv_id":"2505.08660","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For odd square-free levels, the pullback of a Saito-Kurokawa lift decomposes into old and new pieces whose squared norms are exactly the central L-values L(f⊗sym^2 g, 1/2), with all other pieces vanishing.","lead":"This paper gives the complete list of nonzero pieces that remain when a Saito-Kurokawa Siegel modular form is restricted to a diagonal slice, and it identifies the size of each surviving piece with a standard L-function value. It also proposes a conjectural asymptotic for the average mass and verifies the first term over the family.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Old-level pullback formula (1.3), imported from [PV20], is the sole bridge from old-class periods to central L-values; any normalization mismatch in its completed L-function constants propagates through (2.8), (2.10) and the average main term.","rationale":"The central theorem is established by a long chain, but its final equality (2.8) does not prove the old-level central-value formula; it imports (1.3). The paper's own work reduces all old-basis coefficients to the single B_{M_g}-period and decides which vanish. That reduction is substantial and internally coherent, but it cannot fix a normalization error in the input. Thus the mathematical content of Theorem 2.3 is conditional on [PV20]'s formula being exactly as quoted under the paper's normalization. This is precisely the reader's weakest assumption, and I agree with it. The edge-case issue on (2.7) is a genuine statement-level error that should be corrected, but it does not affect the old-class formula itself. The numerical test would also confirm the intended statement. I keep the reader's CONDITIONAL verdict: the claimed spectral decomposition is plausible and well supported, but it should not be ACCEPT until (1.3) is verified under the paper's normalizations and the edge case is amended.","tokens_in":50980,"tokens_out":16566,"duration_ms":163896,"concrete_test":"Compute both sides of (1.3) for a concrete small case, e.g. N=15, N_g=5, M_g=3, and N=21, N_g=7, M_g=3, with k=3. Use explicit Fourier expansions of f∈S_6^new(N), g∈S_4^new(N_g), the SK lift F via the EZI map, and evaluate the Petersson inner product ⟨F^∘,g⊗g|B_3⟩ and Λ(f⊗sym^2 g,1/2) with Dokchitser's gamma-factor normalization. If the right-hand side differs by an M_g-dependent power or a ∏(p+1) factor, adjust (1.3) and re-derive (2.8). Also re-read (2.7) with N_g=N: since 1∉L_f by (2.5), the statement as written predicts vanishing of the new-level coefficient, so the theorem needs the M_g>1 qualifier or L_f must include 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Formula (1.3), taken verbatim from [PV20], is the only place where the paper's old-class periods ⟨F^∘,g⊗g|B_{M_g}⟩ are converted into Λ(f⊗sym^2 g,1/2). The proof of Theorem 2.3(2.8) is: relate every gσ-period to this single B_{M_g}-period (end of §6.6), then invoke (1.3). The paper itself states that its completed L-function has central point at s=1/2 and notes right after (1.3) that the formula differs from [PV19] by a power of M_g, i.e. the normalization of the L-function in (1.3) is not the one used elsewhere. If the true [PV20] constant carries an extra M_g^α, or one of the local Euler factors at p|N_g, then (2.8) is off by that factor for every old class. This propagates unchanged into Theorem 2.6's L^2-mass formula (2.10) and into the average main-term computation in Section 9. Since (1.3) is an input rather than a derived statement, no internal consistency check in the paper can detect such a mismatch. A secondary statement-level bug: Theorem 2.3(2.7) with L_f defined in (2.5) excluding L=1 would force all new-level periods (M_g=1) to vanish; the intended condition is M_g>1 and M_g∉L_f, or L_f should include 1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":51310,"tokens_out":19471,"duration_ms":169086,"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":[{"comment":"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.","section":"Theorem 2.3, Eqs. (2.7)–(2.8), with L_f as in (2.5)"},{"comment":"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.","section":"Section 1, Eq. (1.3); used in the proof of Theorem 2.3 at the end of Section 6.6"}],"minor_comments":[{"comment":"The title contains \"theL2-mass\" without a space, and the key words list \"Saito-Kurokawa lits\", which should be \"lifts\".","section":"Title and abstract"},{"comment":"\"where where v1 = vol.(SL2(Z)\\H)\" contains a duplicated \"where\".","section":"Section 2.3, after Eq. (2.9)"},{"comment":"The text refers to \"Conjecture 2.11\", but the conjecture is numbered 2.7; please correct the cross-reference.","section":"Section 9, first paragraph"},{"comment":"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.","section":"Section 3.1.3"},{"comment":"The entries [DSa] and [DSb] are listed as \"Preprint\" with no arXiv identifier or year; please provide as much information as is available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The statement-level bug in Theorem 2.3(2.7) for Mg=1 is easy to fix but must be corrected before the main theorem can be read as stated. The deeper concern is the dependence on the normalization of the old-level pullback formula (1.3) imported from [PV20]; the authors should be asked to verify the constant explicitly or to prove the formula in their normalization. The rest of the computational core appears sound and well beyond the level of a routine verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper settles Question 1.1 in the affirmative for odd square-free levels. It gives the complete spectral decomposition of the pullback of a Saito-Kurokawa newform onto the diagonal, in an arithmetically orthogonalized basis of oldforms. The vanishing rule (only old classes whose level ratio lies in L_f survive) and the L^2-mass formula (2.10) are new, and they match the level-one result of Liu–Young. The proof is a serious, long computation with coset representatives, Hecke algebra, and explicit Euler products. The central-value identity (2.8) compares a period squared to an independently defined L-function; there are no fitted parameters or circular steps. I believe the main theorems are correct in spirit.\n\nWhere I'd press: the bridge from old-class periods to central L-values is formula (1.3), taken verbatim from Pal–de Vera-Piquero [PV20]. The paper notes the completed L-function is normalized to put the central point at s=1/2 and that this differs from [PV19] by a power of M_g, but it does not prove (1.3) or test its constants. Any normalization mismatch in that input propagates directly into (2.8), (2.10), and the average main-term calculation. The authors are clearly aware of the issue; a referee should ask them to verify (1.3) against a concrete example (small weight, small level, computed numerically) or to derive it from [PV20] with full normalization tracking.\n\nA smaller statement bug: Theorem 2.3(2.7) says the period vanishes whenever M_g is not in L_f, but L_f as defined in (2.5) excludes 1, so as written it would force all new-level periods (M_g=1) to vanish, contradicting (2.8) for M_g in L_f and the newform case of (1.3). The proof (Corollary 5.3) actually requires M_g>1. Fix: either include 1 in L_f or state (2.7) for M_g>1 only.\n\nThe abstract claims the averaged main term matches the CFKRS heuristics; that matching is carried out only for N=p in the appendix, with composite levels deferred. That is an overstatement as written. Minor, but should be qualified.\n\nThis paper deserves peer review. The central result is important and the machinery is substantial. I'd send it to a strong number theory journal and ask the referee to focus on the normalization of (1.3), the M_g=1 edge case, and the scope of the average claim.","headline":"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.","tokens_in":51863,"tokens_out":3833,"would_cite":true,"duration_ms":36710,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F46","11F50","11F67","11F37"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $\\Lambda(f\\otimes \\mathrm{sym}^2 g,1/2)$.","keywords":["Saito-Kurokawa lifts","pullback formulas","central L-values","spectral decomposition","oldforms","square-free level","L2-mass","non-vanishing"],"falsifier":"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.","tokens_in":50775,"feed_emoji":"🧮","tokens_out":25589,"duration_ms":208351,"temperature":0.7,"pith_summary":"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.","feed_headline":"The squares of pullback coefficients are central L-values","feed_subtitle":"For odd square-free levels, the full spectral decomposition of Saito-Kurokawa pullbacks is now explicit.","key_machinery":"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 $w_f(p)=-1$.","core_discovery":"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 $\\pm\\sqrt{\\Lambda(f\\otimes \\mathrm{sym}^2g,1/2)}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the old-class pullback formula (1.3) that converts the period into the central L-value; Theorem 2.3(2.8) inherits its normalization.","marker":"[PV20]"},{"why":"Provides the newform-level central-value formula and the SL(2)-period framework used as input and as the normalization reference for the completed L-function.","marker":"[PV19]"},{"why":"Proves the square-free-level pullback formula for new g, the M_g=1 case of (1.3).","marker":"[Che20]"},{"why":"Gives the full-level pullback formula that the present work extends to every old class.","marker":"[Ich05]"},{"why":"Establishes the EZI lifting and new-and-old theory for square-free levels, including the correspondence f, h, phi, F and the norm relation used in the L2-mass computation.","marker":"[AD24]"},{"why":"Supplies the arithmetically orthogonalized old basis {g_sigma} and its inner-product evaluation.","marker":"[PY19]"},{"why":"Provides the Petersson formula for newforms, the harmonic weights, and the Hecke-eigenvalue identities used throughout the Euler-product computations and the average over f.","marker":"[ILS00]"},{"why":"Gives the Atkin-Lehner eigenvalue w_F(p)=1 for Saito-Kurokawa lifts used in the vanishing proof.","marker":"[Sch07]"},{"why":"Baseline level-one expression for the L2-mass and the moment-heuristic route to its asymptotic, which Theorem 2.6 and Conjecture 2.7 generalize.","marker":"[LY14]"},{"why":"The moment heuristics for families of L-functions from which the conjectural main term in Conjecture 2.7 is derived.","marker":"[Con+05]"}],"fun_headline_variants":["SK pullback squares tied to central L-values","Full decomposition for Saito-Kurokawa pullbacks","Central L-values govern pullback coefficients","Pullback spectral decomposition via L-functions","Non-vanishing of SK pullbacks explained"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SK pullback squares tied to central L-values","Full decomposition for Saito-Kurokawa pullbacks","Central L-values govern pullback coefficients","Pullback spectral decomposition via L-functions","Non-vanishing of SK pullbacks explained"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000262,"raw_usage":{"total_tokens":1721,"prompt_tokens":1195,"completion_tokens":526,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":811,"completion_tokens_details":{"reasoning_tokens":456}},"tokens_in":811,"tokens_out":526,"duration_ms":5014,"temperature":1.0,"reasoning_tokens":456,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:48:49.715609+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}