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arxiv: 1907.01057 · v1 · pith:73NEEKVHnew · submitted 2019-07-01 · 🧮 math.NT

Computing an order complete basis for M^(infty)(N) and Applications

Mark van Hoeij , Cristian-Silviu Radu This is my paper

Pith reviewed 2026-05-25 11:24 UTC · model grok-4.3

classification 🧮 math.NT
keywords modular functionsGamma_0(N)M^infty(N)partition functioncongruencesorder complete basispole orders
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The pith

Given two modular functions for Γ₀(N) with coprime pole orders at infinity, an order-complete basis for all such functions can be computed.

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

The paper supplies a procedure that, starting from any two modular functions f and g for the group Γ₀(N) whose poles lie only at the cusp infinity and whose pole orders are coprime, produces a full ordered basis for the vector space of every modular function with the same restriction on poles. The basis elements satisfy algebraic relations that translate directly into identities among the Fourier coefficients of the partition generating function. One immediate consequence is a pair of explicit identities from which the congruence p(11n+6) ≡ 0 mod 11 follows for the ordinary partition function p(n).

Core claim

If f and g are modular functions for Γ₀(N) with poles only at i∞ and coprime pole orders, then an algorithm produces an order-complete basis for the space M^∞(N) of all modular functions for Γ₀(N) having poles only at i∞; the same construction also yields two new identities that imply the partition congruence p(11n+6) ≡ 0 mod 11.

What carries the argument

The order-complete basis for M^∞(N), a basis ordered by increasing pole orders at infinity that spans every modular function for Γ₀(N) allowed poles solely at the infinite cusp.

If this is right

  • Two explicit new identities are obtained whose coefficients certify the congruence p(11n+6) ≡ 0 mod 11.
  • Every modular function for Γ₀(N) with a pole only at infinity belongs to the span of the computed basis.
  • The same procedure works for arbitrary level N once suitable input functions f and g exist.

Where Pith is reading between the lines

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

  • The same computational pattern may apply to other congruence subgroups once two functions with coprime pole orders are identified.
  • Finding the initial pair f and g for a given N is a prerequisite step whose difficulty is left open by the method.
  • The resulting bases could be used to search for further arithmetic congruences satisfied by coefficients of other modular functions.

Load-bearing premise

Two modular functions f and g for Γ₀(N) with poles only at infinity and coprime pole orders must be supplied as input.

What would settle it

For N=11, execute the algorithm on known input functions f and g, extract the resulting identities, and check whether they certify that p(11n+6) is divisible by 11.

read the original abstract

This paper gives a quick way to construct all modular functions for the group $\Gamma_0(N)$ having only a pole at $\tau = i \infty$. We assume that we are given two modular functions $f,g$ for $\Gamma_0(N)$ with poles only at $i \infty$ and coprime pole orders. As an application we obtain two new identities from which one can derive that $p(11n+6)\equiv 0\pmod{11}$, here $p(n)$ is the usual partition function.

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

2 major / 1 minor

Summary. The paper claims to give a quick construction of an order-complete basis for the space M^∞(Γ₀(N)) of modular functions for Γ₀(N) with poles only at i∞, assuming two input functions f,g ∈ M^∞(Γ₀(N)) with coprime pole orders at infinity are already available. It applies the method to produce two new identities implying the known congruence p(11n+6) ≡ 0 (mod 11).

Significance. If the construction is valid once suitable f and g are supplied, the approach could streamline computation of bases for these spaces and facilitate discovery of arithmetic relations, as illustrated by the partition-function identities. The explicit identities add concrete value even if the underlying congruence is classical.

major comments (2)
  1. [Abstract] Abstract and first paragraph: the central construction is stated to require two input functions f,g with coprime pole orders, yet no general algorithm, recurrence relation, or existence proof is supplied for producing such a pair for arbitrary N; this prerequisite is load-bearing for the title claim of 'computing' an order-complete basis.
  2. [Application] Application section: the derivation of the two new identities for N=11 must explicitly verify that the chosen f and g satisfy the coprime-pole-order hypothesis and include the intermediate steps that convert the basis into the stated partition identities.
minor comments (1)
  1. [Abstract] Notation for the space M^∞(N) should be defined on first use and related explicitly to the standard notation M^∞(Γ₀(N)).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and first paragraph: the central construction is stated to require two input functions f,g with coprime pole orders, yet no general algorithm, recurrence relation, or existence proof is supplied for producing such a pair for arbitrary N; this prerequisite is load-bearing for the title claim of 'computing' an order-complete basis.

    Authors: The abstract and introduction already state explicitly that the construction assumes two such input functions f and g are given. The paper's contribution is the efficient method to obtain the order-complete basis once suitable f and g are available; it does not claim or attempt a general algorithm for producing the inputs themselves for arbitrary N. We will revise the abstract to emphasize this conditional nature more prominently, thereby aligning the presentation with the title. revision: partial

  2. Referee: [Application] Application section: the derivation of the two new identities for N=11 must explicitly verify that the chosen f and g satisfy the coprime-pole-order hypothesis and include the intermediate steps that convert the basis into the stated partition identities.

    Authors: We will revise the application section to include an explicit verification that the pole orders of the chosen f and g for N=11 are coprime, together with the intermediate steps that produce the two new identities and the resulting partition congruence. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is conditional on explicitly stated external inputs f and g

full rationale

The paper states upfront that it assumes two modular functions f and g with coprime pole orders are already given as inputs, then constructs the basis from them. No equations, self-citations, fitted parameters, or reductions of the claimed output back to those inputs by construction are visible in the abstract or described claims. The method is therefore self-contained as a conditional algorithm rather than a closed loop that renames or re-derives its own prerequisites.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no free parameters, axioms, or invented entities are extractable from the provided text.

pith-pipeline@v0.9.0 · 5614 in / 1055 out tokens · 19767 ms · 2026-05-25T11:24:17.560298+00:00 · methodology

discussion (0)

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Reference graph

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