pith. sign in

arxiv: 2204.06976 · v4 · pith:FCIGHR4Mnew · submitted 2022-04-14 · 🧮 math.NT · math.AG

Arithmetic level raising for certain quaternionic unitary Shimura variety

Haining Wang This is my paper

Pith reviewed 2026-05-24 12:27 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords arithmetic level raisingquaternionic unitary Shimura varietysymplectic group of degree fourramified characteristicssupersingular locusSiegel threefoldBeilinson-Bloch-Kato conjectureGan-Gross-Prasad conjecture
0
0 comments X

The pith

An arithmetic level raising theorem holds for the symplectic group of degree four in the ramified case.

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

The paper proves an arithmetic level raising theorem for the symplectic group of degree four when the prime is ramified. This result functions as a key step toward the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives attached to orthogonal groups under the Gan-Gross-Prasad framework. It also supplies an analogue of Ihara's lemma or the Tate conjecture for the special fibers of the relevant Shimura varieties at ramified characteristics. The argument depends on a sufficiently detailed description of the supersingular locus of a quaternionic unitary Shimura variety that is closely related to the classical Siegel threefold.

Core claim

We prove an arithmetic level raising theorem for the symplectic group of degree four in the ramified case. This theorem is a key step towards the Beilinson-Bloch-Kato conjecture for certain Rankin-Selberg motives associated to orthogonal groups within the framework of the Gan-Gross-Prasad conjecture, and can be viewed as an analogue of the Ihara's lemma or the Tate conjecture for special fibers of Shimura varieties at ramified characteristics.

What carries the argument

The description of the supersingular locus of the quaternionic unitary Shimura variety, which carries the level-raising argument at ramified characteristics.

Load-bearing premise

The description of the supersingular locus of the quaternionic unitary Shimura variety is sufficiently detailed and accurate to carry the level-raising argument at ramified characteristics.

What would settle it

A concrete counterexample in which the level raising map fails to be surjective because the supersingular locus at a ramified prime has a different structure or cardinality than the one used in the proof.

read the original abstract

In this article we prove an arithmetic level raising theorem for the symplectic group of degree four in the ramified case. This result is a key step towards the Beilinson-Bloch-Kato conjecture for certain Rankin-Selberg motives associated to orthogonal groups within the framework of the Gan-Gross-Prasad conjecture. The theorem itself can be also viewed as an analogue of the Ihara's lemma or the Tate conjecture for special fibers of Shimura varieties at ramified characteristics. The proof relies heavily on the description of the supersingular locus of certain quaternionic unitary Shimura variety which is closely related to the classical Siegel threefold.

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 / 1 minor

Summary. The manuscript proves an arithmetic level raising theorem for the symplectic group of degree four in the ramified case. This is presented as a key step toward the Beilinson-Bloch-Kato conjecture for certain Rankin-Selberg motives associated to orthogonal groups in the Gan-Gross-Prasad framework. The result is framed as an analogue of Ihara's lemma or the Tate conjecture for special fibers of Shimura varieties at ramified characteristics, with the proof relying on a detailed description of the supersingular locus of a quaternionic unitary Shimura variety closely related to the classical Siegel threefold.

Significance. If correct, the theorem supplies a missing arithmetic ingredient for special-value conjectures on motives attached to orthogonal groups via GGP, extending level-raising techniques to ramified primes. The explicit use of the supersingular locus description, if sufficiently detailed and accurate as claimed, constitutes a concrete technical contribution that could be reused in related settings.

minor comments (1)
  1. The abstract refers to both the 'symplectic group of degree four' and a 'quaternionic unitary Shimura variety'; a brief clarification of the precise group and the relation to the Siegel threefold would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 1 unresolved

We thank the referee for the summary recognizing the arithmetic level raising theorem as a step toward the Beilinson-Bloch-Kato conjecture in the Gan-Gross-Prasad setting, and for noting its analogy to Ihara's lemma at ramified primes. The report provides no enumerated major comments despite the 'uncertain' recommendation, so we respond to the overall assessment below.

standing simulated objections not resolved
  • The basis for the 'uncertain' recommendation is not articulated in the report, preventing a targeted defense or revision on specific technical points such as the supersingular locus description.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract states that the proof relies on a description of the supersingular locus of the quaternionic unitary Shimura variety. No equations, definitions, or steps are exhibited in the supplied material that reduce the level-raising theorem to a fitted parameter, self-definition, or self-citation chain by construction. The central existence claim is presented as resting on an external description of the locus rather than on quantities derived from the same data within this paper. This is the most common honest finding when no load-bearing reduction is visible from the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the result is stated to rely on a prior description of the supersingular locus whose status is unknown.

pith-pipeline@v0.9.0 · 5621 in / 1154 out tokens · 29208 ms · 2026-05-24T12:27:54.583838+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 1 internal anchor

  1. [1]

    Arthur, A stable trace formula

    J. Arthur, A stable trace formula. I. General expansions , J. Inst. Math. Jussieu 1 (2002), no. 2 , 175--277

    work page 2002

  2. [2]

    Arthur, A stable trace formula .II

    J. Arthur, A stable trace formula .II. Global descent , Invent. Math. 143 (2001), no. 1 , 157--220

    work page 2001

  3. [3]

    Arthur, A stable trace formula .III

    J. Arthur, A stable trace formula .III. Proof of the main theorem , Ann. of Math. (2) 158 (2003), 769--837

    work page 2003

  4. [4]

    Arthur, Automorphic representations of (4) in Contributions to automorphic forms, geometry, and number theory , Johns Hopkins Univ

    J. Arthur, Automorphic representations of (4) in Contributions to automorphic forms, geometry, and number theory , Johns Hopkins Univ. Press , Baltimore, MD , (2004), 65--81

    work page 2004

  5. [5]

    Bertolini and H

    M. Bertolini and H. Darmon, Iwasawa's main conjecture for elliptic curves over anticyclotomic Z_p -extensions , Ann. of Math. (2) 162 , (2005), 1--64

    work page 2005

  6. [6]

    Brooks and R

    R. Brooks and R. Schmidt, Local newforms for GSp(4) , Lecture Notes in Mathematics , 1918 (2007), Springer, Berlin , viii+307

    work page 1918

  7. [7]

    Clozel, Ribet's theorem for U(3) , Amer

    L. Clozel, Ribet's theorem for U(3) , Amer. J. Math. , 122 (2000), 1265--1287

    work page 2000

  8. [8]

    Clozel, M

    L. Clozel, M. Harris and R. Taylor Automorphy for some l -adic lifts of automorphic mod l Galois representations , Publ. Math. Inst. Hautes \' E tudes Sci. 63 (1991), 281--332

    work page 1991

  9. [9]

    Chan and W-T

    P-S. Chan and W-T. Gan, The local Langlands conjecture for GSp(4) III: Stability and twisted endoscopy , J. Number Theory 146 (2015), 69--133

    work page 2015

  10. [10]

    Caraiani and P

    A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties , Ann. of Math. (2) , 186 (2017), no. 3 , 649--766

    work page 2017

  11. [11]

    Caraiani and P

    A. Caraiani and P. Scholze, On the generic part of the cohomology of non-compact unitary Shimura varieties , arXiv:1909.01898v1 , preprint

    work page arXiv 1909

  12. [12]

    Deligne and G

    P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1 , 103--161

    work page 1976

  13. [13]

    Gee, Automorphic lifts of prescribed types , Math

    T. Gee, Automorphic lifts of prescribed types , Math. Ann. 350 (1) , (2011), 107--144

    work page 2011

  14. [14]

    Gross, On the Satake isomorphism , Galois representations in arithmetic algebraic geometry (Durham, 1996) , London Math

    B. Gross, On the Satake isomorphism , Galois representations in arithmetic algebraic geometry (Durham, 1996) , London Math. Soc. Lecture Note Ser. Vol. 254 , Cambridge Univ. Press, Cambridge , 1998

    work page 1996

  15. [15]

    Hales, The fundamental lemma for Sp(4) , Proc

    T. Hales, The fundamental lemma for Sp(4) , Proc. American. Math. Soc. 125 (1) (1997), 301--308

    work page 1997

  16. [16]

    Ichino and T

    A. Ichino and T. Ikeda, On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross--Prasad Conjecture , Geom. Funct. Anal. 19 (2010), 1378--1425

    work page 2010

  17. [17]

    Illusie, Sur la Formule de Picard-Lefschetz , Algebraic Geometry 2000, Azumino , Adv

    L. Illusie, Sur la Formule de Picard-Lefschetz , Algebraic Geometry 2000, Azumino , Adv. Stud. Pure Math. 36 , Math. Soc. Japan, Tokyo (2002) , 249--268

    work page 2000

  18. [18]

    Koshikawa, Vanishing theorems for the mod p cohomology of some simple Shimura varieties , arXiv:1910.03147 , preprint, (2019)

    T. Koshikawa, Vanishing theorems for the mod p cohomology of some simple Shimura varieties , arXiv:1910.03147 , preprint, (2019)

    work page arXiv 1910

  19. [19]

    Koshikawa, On the generic part of the cohomology of local and global Shimura varieties , arXiv:2106.10602 , preprint, (2021)

    T. Koshikawa, On the generic part of the cohomology of local and global Shimura varieties , arXiv:2106.10602 , preprint, (2021)

    work page arXiv 2021

  20. [20]

    Kottwitz, Sign changes in harmonic analysis on reductive groups , Trans

    R. Kottwitz, Sign changes in harmonic analysis on reductive groups , Trans. Amer. Math. Soc. 278 (1983), 289--297

    work page 1983

  21. [21]

    Kudla and M

    S. Kudla and M. Rapoport, Cycles on Siegel threefolds and derivatives of E isenstein series , Ann. Sci. \' E cole Norm. Sup. (4) 33 (2000), no. 5 , 695--756

    work page 2000

  22. [22]

    Laumon, Sur la cohomologie \`a supports compacts des vari\' e t\' e s de Shimura pour GSp (4)_ Q , Compositio Math

    G. Laumon, Sur la cohomologie \`a supports compacts des vari\' e t\' e s de Shimura pour GSp (4)_ Q , Compositio Math. 105 (1997), no. 3 , 267--359

    work page 1997

  23. [23]

    Laumon, Fonctions z\^ e tas des vari\' e t\' e s de Siegel de dimension trois , Formes automorphes

    G. Laumon, Fonctions z\^ e tas des vari\' e t\' e s de Siegel de dimension trois , Formes automorphes. II. Le cas du groupe GSp(4) Ast\' e risque. 302 (2005), 1--66

    work page 2005

  24. [24]

    Liu, Bounding cubic-triple product S elmer groups of elliptic curves , J

    Y. Liu, Bounding cubic-triple product S elmer groups of elliptic curves , J. Eur. Math. Soc. (JEMS). 21 (2017) no.5 1411--1508

    work page 2017

  25. [25]

    Lan and J

    K-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL -type S himura varieties , Adv. Math. 242 (2013), 228--286

    work page 2013

  26. [26]

    Lan and B

    K-W. Lan and B. Stroh, Nearby cycles of automorphic \' e tale sheaves , Compos. Math. 154 (2018), no. 1 , 80--119

    work page 2018

  27. [27]

    Lan and B

    K-W. Lan and B. Stroh, Nearby cycles of automorphic \' e tale sheaves II , Cohomology of arithmetic groups, Springer Proc. Math. Stat. 245 (2018), Springer, Cham, 83--106

    work page 2018

  28. [28]

    Y. Liu, Y. Tian, L. Xiao, W. Zhang and X. Zhu, On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives , arXiv:1912.11942 , online first article in Invent. Math. (2022)

    work page arXiv 1912

  29. [29]

    Y. Liu, Y. Tian, L. Xiao, W. Zhang and X. Zhu, Deformation of rigid conjugate self-dual Galois representations , arXiv:2108.06998 , preprint,(2021)

    work page arXiv 2021

  30. [30]

    Mokrane and J

    A. Mokrane and J. Tilouine, Cohomology of Siegel varieties with p-adic integral coefficients and applications , Cohomology of Siegel varieties , Asterisque no. 280 (2002), 1--95

    work page 2002

  31. [31]

    Mok, Galois representations attached to automorphic forms on GL _2 over CM fields , Compos

    C. Mok, Galois representations attached to automorphic forms on GL _2 over CM fields , Compos. Math. 150 (2014), no. 4 , 523--567

    work page 2014

  32. [32]

    Nekov \'a r , On the parity of ranks of S elmer groups

    J. Nekov \'a r , On the parity of ranks of S elmer groups. III . , Doc. Math. , 12:243--274, 2007

    work page 2007

  33. [33]

    Oki, On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2 , Manuscripta Math

    Y. Oki, On supersingular loci of Shimura varieties for quaternionic unitary groups of degree 2 , Manuscripta Math. 167 (2022), 263--343

    work page 2022

  34. [34]

    Ribet, Congruence relations between modular forms , Proceedings of the International Congress of Mathematicians , Vol

    K. Ribet, Congruence relations between modular forms , Proceedings of the International Congress of Mathematicians , Vol. 1, 2 (Warsaw, 1983) , 503--514, PWN, Warsaw , 1984

    work page 1983

  35. [35]

    Shelstad, Characters and inner forms of a quasi-split group over , Compositio

    D. Shelstad, Characters and inner forms of a quasi-split group over , Compositio. Math. 39 (1979), 11--45

    work page 1979

  36. [36]

    Rosner and R

    M. Rosner and R. Weissauer, Global liftings between inner forms GSp4 , arXiv:2103.14715 , preprint, 2021

    work page arXiv 2021

  37. [37]

    Rapoport and T

    M. Rapoport and T. Zink, Period spaces for p -divisible groups , Annals of Mathematics Studies 141 , Princeton University Press, Princeton, NJ (1996), xxii+324

    work page 1996

  38. [38]

    Santos, Upcoming thesis at LSNTG

    M. Santos, Upcoming thesis at LSNTG

    work page

  39. [39]

    Deligne and N

    P. Deligne and N. Katz, Groupes de monodromie en g\'eom\'etrie alg\'ebrique. II , S \'e minaire de G \'e om \'e trie Alg \'e brique du Bois-Marie 1967--1969 (SGA 7, II ) , Lect. Notes Math. Vol. 340 , Springer-Verlag, Berlin , 1973

    work page 1967

  40. [40]

    Schmidt, Iwahori-spherical representations of GSp(4) and Siegel modular forms of degree 2 with square-free level , J

    R. Schmidt, Iwahori-spherical representations of GSp(4) and Siegel modular forms of degree 2 with square-free level , J. Math. Soc. Japan , 57 (2005), no.1 , 259--293

    work page 2005

  41. [41]

    Schmidt, Packet structure and paramodular forms , Trans

    R. Schmidt, Packet structure and paramodular forms , Trans. Amer. Math. Soc. , 370 (2018) no.5 , 3085--3112

    work page 2018

  42. [42]

    Sally and M

    P. Sally and M. Tadi\' c , Induced representations and classifications for GSp (2,F) and Sp (2,F) , M\' e m. Soc. Math. France (N.S.) 52 (1993), 75--133

    work page 1993

  43. [43]

    Sorensen, Level-raising for Saito--Kurokawa forms , Compositio Math

    C. Sorensen, Level-raising for Saito--Kurokawa forms , Compositio Math. 145 (2010), no. 4 , 915--953

    work page 2010

  44. [44]

    Sorensen, Galois representations attached to Hilbert-Siegel modular forms , Doc

    C. Sorensen, Galois representations attached to Hilbert-Siegel modular forms , Doc. Math. 15 (2010), 623--670

    work page 2010

  45. [45]

    Taylor, Galois representations associated to Siegel modular forms of low weight , Duke Math

    R. Taylor, Galois representations associated to Siegel modular forms of low weight , Duke Math. J. 63 (1991), 281--332

    work page 1991

  46. [46]

    Taylor, On the l -adic cohomology of Siegel threefolds , Invent

    R. Taylor, On the l -adic cohomology of Siegel threefolds , Invent. Math. 114 (1993), 289--310

    work page 1993

  47. [47]

    Wang, On the Bruhat--Tits stratification of a quaternionic unitary Shimura variety , Math

    H. Wang, On the Bruhat--Tits stratification of a quaternionic unitary Shimura variety , Math. Ann. 376 (2020), 1107--1144

    work page 2020

  48. [48]

    Waldspurger, Le lemme fondamental implique le transfert , Compositio

    J. Waldspurger, Le lemme fondamental implique le transfert , Compositio. Math. 105 (1997), 153--236

    work page 1997

  49. [49]

    Wang, On quaternionic unitary Rapoport--Zink spaces with parahoric level structures , arXiv:1909.12263 , online first article in IMRN (2020)

    H. Wang, On quaternionic unitary Rapoport--Zink spaces with parahoric level structures , arXiv:1909.12263 , online first article in IMRN (2020)

    work page arXiv 1909

  50. [50]

    Wang, Arithmetic level raising on triple product of Shimura curves and Gross-Schoen Diagonal cycles I: Ramified case , Preprint, arXiv:2004.00555

    H. Wang, Arithmetic level raising on triple product of Shimura curves and Gross--Schoen Diagonal cycles I: Ramified case , arXiv:2004.00555 , to appear in Algebra Number Theory

    work page arXiv 2004

  51. [51]

    Wang, Deformation of rigid Galois representations and cohomology of certain quaternionic unitary Shimura varieties , preprint

    H. Wang, Deformation of rigid Galois representations and cohomology of certain quaternionic unitary Shimura varieties , preprint

    work page

  52. [52]

    Wang, Level lowering on Siegel modular threefold of paramodular level , preprint

    H. Wang, Level lowering on Siegel modular threefold of paramodular level , preprint

    work page

  53. [53]

    Weissauer, Four dimensional Galois representations , Formes automorphes

    R. Weissauer, Four dimensional Galois representations , Formes automorphes. II. Le cas du groupe GSp(4) , Ast\' e risque , 302 , 67--150

    work page

  54. [54]

    Cycles on Shimura varieties via geometric Satake

    L. Xiao and X. Zhu, Cycles on Shimura varieties via geometric Satake , arXiv:1707.05700 , preprint

    work page internal anchor Pith review Pith/arXiv arXiv