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A new symmetric spectral reciprocity formula yields the uniform subconvexity bound L(1/2, π) ≪ C(π)^{1/4 − 1/120 + ε} for every unitary cuspidal GL(2) form over a number field.

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2026-07-11 18:51 UTC pith:VTA6V6XG

load-bearing objection Solid new 4=2+2 reciprocity and a claimed 1/120 uniform saving over number fields, but the square-level dual conductors are not forced for every π, so the final exponent has a gap. the 2 major comments →

arxiv 2607.04476 v1 pith:VTA6V6XG submitted 2026-07-05 math.NT

Symmetric Spectral Reciprocity for GL(2) and Uniform Subconvexity

Liyang Yang This is my paper
classification math.NT MSC 11F6611F7211M4111F70
keywords subconvexityspectral reciprocityGL(2) L-functionsEisenstein seriesrelative trace formulaamplificationArtin L-functionsclass groups
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper regularizes the formally divergent Petersson identity for products of Eisenstein series on GL(2) and extracts from it an explicit symmetric spectral reciprocity that relates two twisted fourth moments of GL(2) L-functions. The symmetry comes from the rank decomposition 4 = 2 + 2 and produces a non-negative spectral side that is compatible with amplification. Amplification forces the dual family to have square-level conductors; that arithmetic rigidity lets the author replace ordinary test vectors in the relative trace formula by special matrix coefficients, improving hybrid bounds for twisted L-functions. Combining the reciprocity identity with these refined hybrid bounds produces a uniform power saving of 1/120 over the convexity bound for every unitary cuspidal representation of GL(2) over an arbitrary number field. The same saving applies to certain Artin L-functions attached to class groups and improves known estimates for the shortest ideal of a given character or cyclic subgroup.

Core claim

After analytic regularization of the Petersson norm identity for four Eisenstein series, the resulting symmetric spectral reciprocity formula (Theorem A) together with the square-level phenomenon of amplification yields the uniform subconvexity estimate L(1/2, π) ≪ C(π)^{1/4 − 1/120 + ε} for every unitary cuspidal automorphic representation π of GL(2) over a number field F.

What carries the argument

Symmetric spectral reciprocity (Theorem A): an explicit identity equating two meromorphically continued Rankin–Selberg kernels of Eisenstein series, with all residual and geometric correction terms written as explicit period integrals of Hecke L-functions.

Load-bearing premise

The extra saving from 1/128 to 1/120 requires that amplification always produces dual forms whose finite conductors are products of two distinct squares; without that square-level rigidity the special test vectors used in the relative trace formula are unavailable.

What would settle it

If a numerical check for a family of GL(2) forms of large square-free level over Q finds that the fourth-moment dual side after amplification is not supported on square-level conductors, or if the resulting hybrid bound for L(1/2, σ imes ω) fails to improve on the earlier 1/128 exponent, the 1/120 claim collapses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every unitary cuspidal GL(2) L-function over any number field satisfies a uniform power saving of 1/120 over convexity, including derivatives and t-aspect twists.
  • Artin L-functions attached to characters of Picard groups of quadratic orders improve from the previous 1/1889 saving to the same 1/120 exponent.
  • The shortest ideal realizing a non-trivial class-group character is O(D^{1/2 − 1/60 + ε}), improving earlier ineffective bounds.
  • Hybrid subconvexity for L(1/2, σ imes ω) sharpens whenever the level of σ is a product of two distinct squares.

Where Pith is reading between the lines

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

  • The same Rankin–Selberg-kernel regularization should apply to other formally divergent multi-Eisenstein identities, potentially yielding symmetric reciprocity for higher-rank or higher-moment problems.
  • If a fifth-moment estimate of comparable quality becomes available, the paper’s own amplification machinery would immediately improve the fixed-central-character exponent beyond 1/24.
  • The square-level phenomenon may appear in other amplification settings (GL(2) imes GL(2) convolutions, cubic moments) and could be exploited systematically whenever dual conductors factor as products of squares.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs an analytic regularization of the formal Petersson identity for products of GL(2) Eisenstein series over a number field via Rankin–Selberg kernels, yielding an explicit symmetric spectral reciprocity (Theorem A) that equates twisted fourth moments of GL(2) L-functions up to twenty-four explicit correction terms. After specializing the automorphic data and amplifying, the dual side is controlled by hybrid subconvexity for L(1/2, σ × ω). The author identifies a “square-level phenomenon” and proves refined hybrid bounds (Theorems 10.7–10.8) for dual forms of conductor p1²p2², which are then optimized to obtain the uniform subconvexity L(1/2, π) ≪ C(π)^{1/4−1/120+ε} (Corollary 1.4), together with hybrid bounds for Rankin–Selberg L-functions, Artin L-functions, and applications to Picard-group arithmetic and fourth-moment estimates.

Significance. If correct, the work supplies a new, representation-theoretic realization of the 4=2+2 decomposition that preserves non-negativity of the original spectral side and works uniformly over number fields and arbitrary central characters. The regularization via Rankin–Selberg kernels is conceptually clean and of independent interest for higher-moment problems. Even the weaker uniform exponent 1/128 already improves on the best previously available bounds over general number fields (Wu) and removes technical restrictions (square-free level, primitive nebentypus) present in Blomer–Khan. The moment estimates (Theorems E–F) and the Picard-group applications are useful by-products. The claimed extra saving to 1/120 would be a striking numerical coincidence with the classical Duke–Friedlander–Iwaniec threshold and would strengthen several arithmetic corollaries.

major comments (2)
  1. §1.8 and §10.4–§10.7: the claim that amplification “necessarily” produces dual conductors of the form p1²p2² is false. The amplifier expansion (10.10) contains cross terms with n = p1^{ℓ1}p2^{ℓ2} where ℓi ∈ {1,2}; whenever both ℓi = 1 one obtains square-free n = p1p2. For such n the dual family F0(n,1) has square-free level, so the special test vectors of Theorems 10.7–10.8 (matrix coefficients of depth-zero supercuspidals or unipotent translations) are unavailable. The optimization that upgrades Theorem H (1/128) to Corollary 1.4 (1/120) therefore applies only to a proper subset of the amplifier terms. The paper must either restrict the amplifier to primes with ℓp = 2 (and prove that the resulting thinner set still amplifies), perform a case-by-case analysis of the mixed terms, or retract the uniform 1/120 claim.
  2. §10.7.1, display (10.37) and the subsequent optimization: the refined hybrid bound (10.35) derived from Theorem 10.7 is inserted indiscriminately into the cross-term contribution of (10.10). Because that contribution includes non-square n, the insertion is unjustified. The resulting range distinctions that produce the three pieces of Theorem B (and hence the 1/120 of Corollary 1.4) rest on an invalid majorization.
minor comments (4)
  1. The phrase “squarefree square” in §1.8 is ambiguous; replace by “square of a square-free ideal” or “product of two distinct prime squares”.
  2. Many estimates carry 1000ε, 50ε, etc. While harmless, a uniform convention (e.g., “≪ε”) would improve readability.
  3. The dependence of all implied constants on the fixed smooth cut-offs αv (v|∞) should be stated once in the notation section rather than repeated in every proposition.
  4. Cross-references to the author’s earlier works [Yan26b] and [HY26] are frequent; a short “black-box” summary of the precise statements imported from those papers would help the reader.

Circularity Check

1 steps flagged

Minor load-bearing self-citation of the author's prior RTF paper for hybrid bounds; the new reciprocity (Theorem A) and square-level refinements are independently derived and not forced by construction.

specific steps
  1. self citation load bearing [§1.8 and Theorems 10.7–10.8 (§10.6)]
    "By applying [Yan26b, Theorem 1.6 or Corollary 1.9], one obtains hybrid estimates for L(1/2, σ×ω), which in turn yield L(1/2, π)≪C(π)^{1/4−1/128+ε}... This additional structure enables us to construct refined test functions within the amplified relative trace formula developed in [Yan26b]... Theorems 10.7 and 10.8"

    The passage from the weaker uniform exponent 1/128 (Theorem H, using only the prior hybrid of [Yan26b]) to the claimed 1/120 (Corollary 1.4 / Theorem B) is obtained by refining the same author's prior RTF with special test vectors available only for square-level dual conductors. The prior RTF is not re-derived; it is imported as black-box machinery. This is a mild self-citation dependency for the numerical improvement, but the new reciprocity identity and the square-level analysis itself remain independent content.

full rationale

The paper's core derivation of the symmetric spectral reciprocity (Theorem A, via Rankin–Selberg kernel regularization in §§2–3, meromorphic continuation, and geometric/spectral expansions) is self-contained and does not reduce to its inputs by definition or fit. Amplification (§10.4) and moment estimates (Theorems E/F) follow from it by standard contour shifts and local integral bounds. The improvement from the 1/128 of Theorem H to the 1/120 of Corollary 1.4 does rely on refined hybrid bounds (Theorems 10.7–10.8) that invoke the relative trace formula of the author's prior work [Yan26b]; this is a normal self-citation of independent prior machinery, not a circular reduction (no uniqueness theorem forbidding alternatives, no fitted parameters renamed as predictions, no ansatz smuggled as first principles). The square-level claim is an arithmetic observation about dual conductors after amplification, not a definitional tautology. No free parameters are fitted to data; the exponent emerges from optimizing amplification length L against the hybrid estimates. Score 2 reflects only the non-load-bearing character of the self-citation chain for the final numerical saving.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 2 invented entities

The paper works entirely inside the standard adelic theory of automorphic forms on GL(2). The only free parameters are the usual ε-powers and the Ramanujan exponent ϑ ≤ 7/64, both of which are classical. No new physical or arithmetic entities are postulated; the Rankin–Selberg kernel is a re-packaging of existing period integrals.

free parameters (2)
  • Ramanujan exponent ϑ = ≤ 7/64
    Appears in all hybrid and uniform bounds; taken as any number ≤ 7/64 (Kim–Sarnak / Blomer–Brumley). The final 1/120 is independent of the precise value of ϑ once ϑ is small enough.
  • ε-powers in all estimates = arbitrary positive
    Standard analytic-number-theory convention; every bound is of the form X^ε. Not fitted to data.
axioms (4)
  • standard math Standard Rankin–Selberg theory and meromorphic continuation of Eisenstein series on GL(2)
    Used throughout §2–§3 to define the kernels and their spectral expansions.
  • domain assumption Relative-trace-formula machinery and hybrid bounds of the author’s earlier paper [Yan26b]
    The refined hybrid bounds of Theorems 10.7–10.8 are obtained by modifying the test vectors inside the RTF of [Yan26b]; the base RTF is taken as given.
  • standard math Zero-free regions for Hecke L-functions (classical)
    Used in §3.3 to control contour shifts for the continuous spectrum.
  • standard math Katz’s estimate and quasi-orthogonality of trace functions
    Invoked in the proof of Theorem 10.7 to bound the regular orbital integrals at depth-zero places.
invented entities (2)
  • Rankin–Selberg kernel Eis(G_i h_j) independent evidence
    purpose: Provides a convergent automorphic kernel whose Petersson product regularizes the divergent product of Eisenstein series.
    Defined in §2.2 as the sum of the generic part of one Eisenstein series against a Godement section of another; it is a re-arrangement of existing period integrals rather than a new physical object.
  • Square-level phenomenon after amplification independent evidence
    purpose: Forces dual conductors to be of the form p1^{2}p2^{2}, enabling special test vectors.
    An arithmetic observation about the support of the amplified dual family; not a new entity but a structural feature exploited for the first time in this way.

pith-pipeline@v1.1.0-grok45 · 86075 in / 3327 out tokens · 30249 ms · 2026-07-11T18:51:32.529352+00:00 · methodology

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read the original abstract

We construct a new analytic regularization of the Petersson norm identity for Eisenstein series on $\mathrm{GL}_2$ over a number field $F$, and derive from it an explicit symmetric spectral reciprocity formula for twisted fourth moments of $\mathrm{GL}_2$ $L$-functions, reflecting the intrinsic rank decomposition $4=2+2$. Independently, we identify a square-level phenomenon arising from amplification, whereby the dual spectral family acquires square-level conductors. This additional arithmetic rigidity permits a refined analysis within the relative trace formula and leads to refined hybrid subconvexity bounds for twisted $L$-functions. As a consequence, we obtain new uniform subconvexity bounds for $\mathrm{GL}_2/F$; in particular, \begin{align*} L(1/2,\pi)\ll C(\pi)^{\frac14-\frac{1}{120}+\varepsilon} \end{align*} for every unitary cuspidal representation $\pi$ of $\mathrm{GL}_2/F$. We also obtain refined bounds for certain Artin $L$-functions and applications to class group arithmetic.

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Reference graph

Works this paper leans on

52 extracted references · 19 canonical work pages

  1. [1]

    Blomer, Valentin and Brumley, Farrell , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.4007/annals.2011.174.1.18 , URL =

    doi:10.4007/annals.2011.174.1.18 2011

  2. [2]

    The local Langlands conjecture for

    Bushnell, Colin J and Henniart, Guy , year=. The local Langlands conjecture for

  3. [3]

    Blomer, Valentin and Harcos, Gergely and Michel, Philippe , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.ansens.2007.05.003 , URL =

    doi:10.1016/j.ansens.2007.05.003 2007

  4. [4]

    , TITLE =

    Blomer, Valentin and Jana, Subhajit and Nelson, Paul D. , TITLE =. Duke Math. J. , FJOURNAL =. 2023 , NUMBER =. doi:10.1215/00127094-2022-0058 , URL =

    work page doi:10.1215/00127094-2022-0058 2023

  5. [5]

    Duke Math

    Blomer, Valentin and Khan, Rizwanur , TITLE =. Duke Math. J. , FJOURNAL =. 2019 , NUMBER =. doi:10.1215/00127094-2018-0060 , URL =

    doi:10.1215/00127094-2018-0060 2019

  6. [6]

    Blomer, Valentin and Khan, Rizwanur , TITLE =. J. Funct. Anal. , FJOURNAL =. 2019 , NUMBER =. doi:10.1016/j.jfa.2018.11.009 , URL =

    work page doi:10.1016/j.jfa.2018.11.009 2019

  7. [7]

    Blomer, Valentin and Mili\'cevi\'c, Djordje , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00039-015-0318-7 , URL =

    work page doi:10.1007/s00039-015-0318-7 2015

  8. [8]

    and Str\"ombergsson, Andreas and Then, Holger , TITLE =

    Booker, Andrew R. and Str\"ombergsson, Andreas and Then, Holger , TITLE =. LMS J. Comput. Math. , FJOURNAL =. 2013 , PAGES =. doi:10.1112/S1461157013000028 , URL =

    work page doi:10.1112/s1461157013000028 2013

  9. [9]

    , TITLE =

    Cogdell, James W. , TITLE =. Automorphic forms and applications , SERIES =. 2007 , ISBN =. doi:10.1090/pcms/012/04 , URL =

    work page doi:10.1090/pcms/012/04 2007

  10. [10]

    , TITLE =

    Cogdell, James W. , TITLE =. Indian J. Pure Appl. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s13226-014-0081-8 , URL =

    work page doi:10.1007/s13226-014-0081-8 2014

  11. [11]

    Conrey, J. B. and Iwaniec, H. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2000 , NUMBER =. doi:10.2307/121132 , URL =

    doi:10.2307/121132 2000

  12. [12]

    and Friedlander, J

    Duke, W. and Friedlander, J. B. and Iwaniec, H. , TITLE =. Invent. Math. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF01231759 , URL =

    work page doi:10.1007/bf01231759 1994

  13. [13]

    Inventiones mathematicae , volume=

    The subconvexity problem for Artin L--functions , author=. Inventiones mathematicae , volume=. 2002 , publisher=

    2002

  14. [14]

    Algebra Number Theory , FJOURNAL =

    Diamantis, Nikolaos and O'Sullivan, Cormac , TITLE =. Algebra Number Theory , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/ant.2013.7.1883 , URL =

    work page doi:10.2140/ant.2013.7.1883 2013

  15. [15]

    Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.4007/annals.2011.173.2.5 , URL =

    work page doi:10.4007/annals.2011.173.2.5 2011

  16. [16]

    Colloquium de Giorgi 2013 and 2014 , pages=

    Trace functions over finite fields and their applications , author=. Colloquium de Giorgi 2013 and 2014 , pages=. 2014 , organization=

    2013

  17. [17]

    Forms of

    Gelbart, SS and Jacquet, H , journal=. Forms of

  18. [18]

    Gradshteyn, I. S. and Ryzhik, I. M. , TITLE =. 2015 , PAGES =

    2015

  19. [19]

    Annals of Mathematics , pages=

    Coefficients of Maass forms and the Siegel zero , author=. Annals of Mathematics , pages=. 1994 , publisher=

    1994

  20. [20]

    Harcos, Gergely and Michel, Philippe , TITLE =. Invent. Math. , FJOURNAL =. 2006 , NUMBER =. doi:10.1007/s00222-005-0468-6 , URL =

    doi:10.1007/s00222-005-0468-6 2006

  21. [21]

    Humphries, Peter and Yang, Liyang , journal=

  22. [22]

    Iwaniec, Henryk , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1992 , PAGES =. doi:10.1515/crll.1992.428.139 , URL =

    work page doi:10.1515/crll.1992.428.139 1992

  23. [23]

    2004 , PAGES =

    Iwaniec, Henryk and Kowalski, Emmanuel , TITLE =. 2004 , PAGES =. doi:10.1090/coll/053 , URL =

    doi:10.1090/coll/053 2004

  24. [24]

    Number theory (

    Jutila, Matti , TITLE =. Number theory (. 2001 , ISBN =

    2001

  25. [25]

    and Zagier, D

    Jacquet, H. and Zagier, D. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1987 , NUMBER =. doi:10.2307/2000586 , URL =

    doi:10.2307/2000586 1987

  26. [26]

    Journal of the American Mathematical Society , volume=

    An estimate for character sums , author=. Journal of the American Mathematical Society , volume=. 1989 , publisher=

    1989

  27. [27]

    Khan, Rizwanur , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1017/s0305004118000944 , URL =

    work page doi:10.1017/s0305004118000944 2020

  28. [28]

    Knapp, A. W. , TITLE =. Motives (. 1994 , ISBN =. doi:10.1090/pspum/055.2/1265560 , URL =

    work page doi:10.1090/pspum/055.2/1265560 1994

  29. [29]

    Kuznetsov, N. V. , TITLE =. Number theory and related topics (. 1989 , ISBN =

    1989

  30. [30]

    Refined estimates towards the Ramanujan and Selberg conjectures , author=. J. Amer. Math. Soc , volume=

  31. [31]

    K ral, Eren Mehmet and Young, Matthew , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2021 , NUMBER =. doi:10.4171/jems/1011 , URL =

    work page doi:10.4171/jems/1011 2021

  32. [32]

    and Michel, P

    Kowalski, E. and Michel, P. and VanderKam, J. , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002220000086 , URL =

    work page doi:10.1007/s002220000086 2000

  33. [33]

    , TITLE =

    Michel, P. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2004 , NUMBER =. doi:10.4007/annals.2004.160.185 , URL =

    work page doi:10.4007/annals.2004.160.185 2004

  34. [34]

    Automorphic forms and applications , volume=

    Analytic Number Theory and Families of , author=. Automorphic forms and applications , volume=. 2007 , publisher=

    2007

  35. [35]

    Miyauchi, Michitaka , TITLE =. J. Math. Soc. Japan , FJOURNAL =. 2014 , NUMBER =. doi:10.2969/jmsj/06610017 , URL =

    work page doi:10.2969/jmsj/06610017 2014

  36. [36]

    , TITLE =

    Motohashi, Y. , TITLE =. Proceedings of the. 2003 , MRCLASS =

    2003

  37. [37]

    1966 , PAGES =

    Magnus, Wilhelm and Oberhettinger, Fritz and Soni, Raj Pal , TITLE =. 1966 , PAGES =

    1966

  38. [38]

    Michel, Philippe and Venkatesh, Akshay , TITLE =. Publ. Math. Inst. Hautes \'. 2010 , PAGES =. doi:10.1007/s10240-010-0025-8 , URL =

    doi:10.1007/s10240-010-0025-8 2010

  39. [39]

    Eisenstein series and the cubic moment for

    Nelson, Paul D , journal=. Eisenstein series and the cubic moment for

  40. [40]

    Schmidt, Ralf , TITLE =. J. Ramanujan Math. Soc. , FJOURNAL =. 2002 , NUMBER =

    2002

  41. [41]

    Forum Math

    Templier, Nicolas , TITLE =. Forum Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1515/FORM.2011.041 , URL =

    work page doi:10.1515/form.2011.041 2011

  42. [42]

    Wu, Han , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00039-014-0277-4 , URL =

    work page doi:10.1007/s00039-014-0277-4 2014

  43. [43]

    Lagarias, J. C. and Montgomery, H. L. and Odlyzko, A. M. , TITLE =. Invent. Math. , FJOURNAL =. 1979 , NUMBER =. doi:10.1007/BF01390234 , URL =

    doi:10.1007/bf01390234 1979

  44. [44]

    Wu, Han , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/tran/8750 , URL =

    doi:10.1090/tran/8750 2022

  45. [45]

    Mathematika , FJOURNAL =

    Wu, Han , TITLE =. Mathematika , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/mtk.12147 , URL =

    work page doi:10.1112/mtk.12147 2022

  46. [46]

    Yang, Liyang , journal=. A

  47. [47]

    Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin--Selberg

    Yang, Liyang , journal=. Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin--Selberg

  48. [48]

    Relative Trace Formula and Twisted

    Yang, Liyang , journal=. Relative Trace Formula and Twisted

  49. [49]

    Spectral Reciprocity: A

    Yang, Liyang , journal=. Spectral Reciprocity: A

  50. [50]

    Yang, Liyang , title =

  51. [51]

    2020 , eprint=

    Periods and Reciprocity II , author=. 2020 , eprint=

    2020

  52. [52]

    Zagier, Don , TITLE =. J. Fac. Sci. Univ. Tokyo Sect. IA Math. , FJOURNAL =. 1981 , NUMBER =

    1981