pith. sign in

arxiv: 2605.17972 · v1 · pith:PGGTPCSEnew · submitted 2026-05-18 · 🧮 math.NT · math.AG· math.GR

Decay of weighted cusp counts for congruence subgroups of SL₂ over number fields

Shengyuan Zhao This is my paper

Pith reviewed 2026-05-20 00:54 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.GR
keywords congruence subgroupscusp countsSL_2number fieldsweighted countslevel decayfinite rings
0
0 comments X

The pith

The ratio of weighted cusp counts to subgroup index for SL_2 over number fields is bounded by a negative power of the level norm.

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

This paper shows that for congruence subgroups of SL_2 that are commensurable with the full modular group over any number field, the total weighted cusp count divided by the index of the subgroup decays as the norm of the congruence level raised to a negative power. A sympathetic reader would care because this bound implies that cusp contributions are subleading in many formulas from topology, arithmetic, and representation theory as the level gets larger. The result extends a known theorem for the field of rational numbers to general number fields. The argument works by localizing the counting problem at each prime and reducing it to a finite group counting task over non-reduced local rings.

Core claim

The paper establishes that if Gamma is a congruence subgroup commensurable with SL_2 over a number field, then the weighted cusp count of Gamma divided by the index of Gamma in SL_2 is bounded above by a constant times the norm of the level of Gamma raised to a negative exponent.

What carries the argument

Localization of the cusp counting problem to finite quotients over non-reduced principal local rings, where subgroup counting is performed using methods reminiscent of additive combinatorics.

If this is right

  • Cusp terms in topological, arithmetic and representation-theoretical formulas are subleading as the congruence level increases.
  • The result generalizes the Cox-Parry theorem from the rational numbers to arbitrary number fields.
  • The bound supports heuristics about the relative size of cusp contributions in various contexts.

Where Pith is reading between the lines

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

  • If the decay rate can be made effective, it could lead to explicit error terms in applications involving these groups.
  • Similar localization and counting techniques might extend to cusp counts for other algebraic groups or higher rank cases.
  • The approach could connect to problems in additive combinatorics over finite rings more broadly.

Load-bearing premise

The global cusp counting problem can be localized at a prime and reduced to a counting problem for subgroups of SL_2 over finite non-reduced principal local rings that is solvable by an analysis reminiscent of additive combinatorics.

What would settle it

An explicit computation of the weighted cusp count and the index for a family of congruence subgroups with levels of arbitrarily large norm, verifying whether the ratio goes to zero polynomially fast.

read the original abstract

For congruence subgroups commensurable with $\operatorname{SL}_2$ over number fields, we study cusp counts with certain multiplicities. We prove that the ratio of the total weighted cusp count to the group index is bounded by a negative power of the norm of the congruence level. This generalizes a theorem of Cox--Parry over $\mathbb Q$, and supports the heuristic that cusp terms occurring in topological, arithmetic and representation-theoretical formulas are subleading. The proof proceeds by localizing at a prime and reducing the problem to finite quotients, where it becomes a counting problem for finite groups. The main technical part is a counting problem for subgroups of $\operatorname{SL}_2$ over finite non-reduced principal local rings, proved by an analysis reminiscent of additive combinatorics.

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

1 major / 1 minor

Summary. The paper proves a decay estimate for weighted cusp counts associated to congruence subgroups of SL_2 over number fields. It shows that the ratio of the total weighted cusp count to the index of the subgroup is bounded by a negative power of the norm of the congruence level. The proof reduces the global problem to local finite counting problems for subgroups of SL_2 over non-reduced local rings via localization at primes, with the local count handled by additive-combinatorics style arguments. This generalizes the Cox--Parry theorem over the rationals.

Significance. This result is of interest as it provides quantitative support for the idea that cusp contributions are subleading in topological, arithmetic, and representation-theoretic formulas. The localization technique and the analysis over non-reduced rings constitute a technical contribution that may apply to similar problems in the arithmetic of algebraic groups. The reduction to an independent finite counting problem is a positive feature.

major comments (1)
  1. [Localization and reduction step] The reduction from the global weighted cusp count to the local subgroup counting problem over finite rings must be shown to preserve the decay rate. Specifically, it is necessary to confirm that the weights do not contain non-local factors (such as class group orders or regulators) that could grow with the level and negate the local O(N(p^k)^{-δ}) bound. This is load-bearing for the central claim.
minor comments (1)
  1. [Abstract] The phrase 'bounded by a negative power' in the abstract could be strengthened by indicating the dependence of the exponent on the number field or the group.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the interest of the result and the technical contributions of the localization technique. We address the major comment below.

read point-by-point responses

  1. Referee: The reduction from the global weighted cusp count to the local subgroup counting problem over finite rings must be shown to preserve the decay rate. Specifically, it is necessary to confirm that the weights do not contain non-local factors (such as class group orders or regulators) that could grow with the level and negate the local O(N(p^k)^{-δ}) bound. This is load-bearing for the central claim.

    Authors: The weighted cusp counts in the manuscript are defined using multiplicities arising from local data at the primes dividing the congruence level (specifically, orders of stabilizers in the local quotients). Because the base number field is fixed, any global arithmetic invariants of the field, such as its class number or regulator, are independent of the level and appear as multiplicative constants in both the weighted cusp count and the group index. These constants therefore cancel in the ratio and cannot affect the decay rate. The proof reduces the global ratio to a product of local ratios via the localization map at each prime (detailed in the reduction step preceding the main theorem). Each local ratio is then bounded by O(N(p^k)^{-δ}) using the finite counting argument over the non-reduced local rings. Consequently the global bound inherits the same power decay. We are prepared to insert an explicit paragraph after the definition of the weights to record this cancellation if the referee considers the current presentation insufficiently explicit. revision: partial

Circularity Check

0 steps flagged

Reduction to independent finite subgroup counting over local rings is self-contained

full rationale

The derivation localizes the weighted cusp count at each prime, reduces to a finite counting problem for subgroups of SL_2(R) where R is a finite non-reduced principal local ring, and solves the latter via additive-combinatorics-style enumeration. No equations or definitions in the abstract or described proof chain make the target decay bound equivalent to the input data by construction; the finite counting step is presented as an independent technical result. The global-to-local passage is claimed to preserve the weighted ratio without introducing growing multiplicative factors from class groups or regulators, and no self-citation chain or fitted parameter is invoked to force the O(N(p^k)^{-δ}) bound. This is the most common honest non-finding for a paper whose central claim rests on an explicit reduction to a separately solved finite problem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from algebraic number theory and group theory; no free parameters, new axioms beyond domain standards, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of congruence subgroups, their indices, and commensurability with SL_2
    Used to relate weighted cusp counts to the group index.
  • domain assumption Existence of localization at primes and reduction to finite quotients for arithmetic groups
    Central step that converts the infinite-group problem into finite counting.

pith-pipeline@v0.9.0 · 5661 in / 1248 out tokens · 57812 ms · 2026-05-20T00:54:59.284961+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

32 extracted references · 32 canonical work pages

  1. [1]

    Semin., Honolulu/Hawaii 1987, Contemp

    Eiichi Abe,Normal subgroups of Chevalley groups over commutative rings, Al- gebraicK-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 1-17 (1989)., 1989

    work page 1987

  2. [2]

    Second Series 185(2017), no

    Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, and Iddo Samet,On the growth ofL 2-invariants for sequences of lattices in Lie groups, Annals of Mathematics. Second Series 185(2017), no. 3, 711–790

    work page 2017

  3. [3]

    Second Series84(1966), 442–528

    jun.Baily, W.L.andArmandBorel,Compactification of arithmetic quotients of bounded symmetric domains, Annals of Mathematics. Second Series84(1966), 442–528. DECAY FOR CUSPS OF CONGRUENCE SUBGROUPS 91

    work page 1966

  4. [4]

    Baker and Alan W

    Mark D. Baker and Alan W. Reid,Principal congruence link complements, Annales de la Faculté des sciences de Toulouse: Mathématiques23(2014), no. 5, 1063–1092

    work page 2014

  5. [5]

    Hyman Bass,AlgebraicK-theory, Mathematics Lecture Note Series, The Ben- jamin/Cummings Publishing Company, Reading, MA, 1968

    work page 1968

  6. [6]

    Hyman Bass, John Milnor, and Jean-Pierre Serre,Solution of the congruence subgroup problem for SLn (n≥3) and Sp2n (n≥2), Publications Mathéma- tiques de l’IHÉS33(1967), 59–137

    work page 1967

  7. [7]

    Second Se- ries172(2010), 2197–2221

    Mikhail Belolipetsky, Tsachik Gelander, Alexander Lubotzky, and Aner Shalev, Counting arithmetic lattices and surfaces, Annals of Mathematics. Second Se- ries172(2010), 2197–2221

    work page 2010

  8. [8]

    Armand Borel,Density and maximality of arithmetic subgroups, Journal für die Reine und Angewandte Mathematik224(1966), 78–89

    work page 1966

  9. [9]

    Classe di Scienze8(1981), no

    ,Commensurability classes and volumes of hyperbolic 3-manifolds, An- nali della Scuola Normale Superiore di Pisa. Classe di Scienze8(1981), no. 1, 1–33

    work page 1981

  10. [10]

    Publications Mathématiques69(1989), 119–171

    Armand Borel and Gopal Prasad,Finiteness theorems for discrete subgroups of bounded covolume in semisimple groups, Institut des Hautes Études Scien- tifiques. Publications Mathématiques69(1989), 119–171

    work page 1989

  11. [11]

    Armand Borel and Jean-Pierre Serre,Corners and arithmetic groups, Com- mentarii Mathematici Helvetici48(1973), 436–491

    work page 1973

  12. [12]

    Second Series170(2009), no

    Frank Calegari and Matthew Emerton,Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms, Annals of Math- ematics. Second Series170(2009), no. 3, 1437–1446

    work page 2009

  13. [13]

    Cox and Walter R

    David A. Cox and Walter R. Parry,Genera of congruence subgroups inQ- quaternion algebras, Journal für die Reine und Angewandte Mathematik351 (1984), 66–112

    work page 1984

  14. [14]

    Cremona and M

    John E. Cremona and M. T. Aranés,Congruence subgroups, cusps and Manin symbols over number fields, Computations with Modular Forms (Gebhard Böckle and Gabor Wiese, eds.), Contributions in Mathematical and Computa- tional Sciences, vol. 6, Springer, Cham, 2014, pp. 109–127

    work page 2014

  15. [15]

    228, Springer, New York, 2005

    Fred Diamond and Jerry Shurman,A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer, New York, 2005

    work page 2005

  16. [16]

    II: Application to the limit multiplicity problem, Mathematische Zeitschrift289(2018), no

    Tobias Finis and Erez Lapid,An approximation principle for congruence subgroups. II: Application to the limit multiplicity problem, Mathematische Zeitschrift289(2018), no. 3-4, 1357–1380

    work page 2018

  17. [17]

    Summer School on Group Representations of the Bolyai János Math

    Günter Harder,On the cohomology of SL(2,O), Lie Groups and Their Repre- sentations (Proc. Summer School on Group Representations of the Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 139–150

    work page 1971

  18. [18]

    2e Série19(1973), 183–281

    Friedrich Hirzebruch,Hilbert modular surfaces, L’Enseignement Mathématique. 2e Série19(1973), 183–281

    work page 1973

  19. [19]

    Friedrich Hirzebruch and Antonius Van de Ven,Hilbert modular surfaces and the classification of algebraic surfaces, Inventiones Mathematicae23(1974), 1–29

    work page 1974

  20. [20]

    92 SHENGYUAN ZHAO

    Friedrich Hirzebruch and Don Zagier,Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Inventiones Mathematicae 36(1976), 57–113. 92 SHENGYUAN ZHAO

    work page 1976

  21. [21]

    17, Springer- Verlag, Berlin, 1991

    Grigori˘ ı Aleksandrovich Margulis,Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 17, Springer- Verlag, Berlin, 1991

    work page 1991

  22. [22]

    Third Series105(2012), no

    Alexander Mason and Andreas Schweizer,The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain, Proceedings of the London Mathematical Society. Third Series105(2012), no. 2, 311–341

    work page 2012

  23. [23]

    3, 841–881

    Jasmin Matz,Limit multiplicities for SL 2(OF )in SL 2(Rr1⊕CR2), Groups, Geometry, and Dynamics13(2019), no. 3, 841–881

    work page 2019

  24. [24]

    1, 237–260

    Mehmet Haluk Sengün and Seyfi Türkelli,Lower bounds on the dimension of the cohomology of Bianchi groups via Sczech cocycles, Journal de Théorie des Nombres de Bordeaux28(2016), no. 1, 237–260

    work page 2016

  25. [25]

    3, 489–527

    Jean-Pierre Serre,Le problème des groupes de congruence pour SL2, Annals of Mathematics92(1970), no. 3, 489–527

    work page 1970

  26. [26]

    247, Springer-Verlag, Berlin, 1982

    Michio Suzuki,Group theory I, Grundlehren der mathematischen Wis- senschaften, vol. 247, Springer-Verlag, Berlin, 1982

    work page 1982

  27. [27]

    Vu,Additive combinatorics, Cambridge Studies in Ad- vanced Mathematics, vol

    Terence Tao and Van H. Vu,Additive combinatorics, Cambridge Studies in Ad- vanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006

    work page 2006

  28. [28]

    Gerard van der Geer,Minimal models for Hilbert modular surfaces of principal congruence subgroups, Topology18(1979), 29–39

    work page 1979

  29. [29]

    ,Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge, vol. 16, Springer-Verlag, Berlin/Heidelberg, 1988

    work page 1988

  30. [30]

    Vaserstein,On normal subgroups of Chevalley groups over commu- tative rings, Tôhoku Mathematical Journal

    Leonid N. Vaserstein,On normal subgroups of Chevalley groups over commu- tative rings, Tôhoku Mathematical Journal. Second Series38(1986), no. 1-2, 219–230

    work page 1986

  31. [31]

    800, Springer, Cham, 1980

    Marie-France Vignéras,Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Cham, 1980

    work page 1980

  32. [32]

    288, Springer, 2021

    John Voight,Quaternion algebras, Graduate Texts in Mathematics, vol. 288, Springer, 2021. Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118, route de Narbonne, F-31062 Toulouse, France Email address:shengyuan.zhao@math.univ-toulouse.fr

    work page 2021