Garvan's Formulas: Modular Determinants & k-Ranks

• Garvan's Formulas are two distinct frameworks: one yields determinant identities for powers of the modular discriminant (extended from classical genus-one to genus-two) and the other defines k-rank statistics in partition theory.
• They utilize conformal field theory, Fay's trisecant identities, and classical Ramanujan identities to recast ∆(τ) and related modular objects as determinants with clear analytic implications.
• In partition theory, the k-rank formulas provide precise generating functions and asymptotic estimates (including cases like Dyson's crank), advancing our understanding of partition statistics.

Garvan's formulas, in recent arXiv literature, designate two technically distinct families of number-theoretic identities associated with Garvan: determinant formulas for powers of the modular discriminant and formulas governing Garvan's $k$-rank for integer partitions. In the first setting, the formulas begin with classical genus-one determinant identities for $\Delta(\tau)=\eta(\tau)^{24}$ and are extended by conformal-field-theoretic determinant representations to all powers and to genus two; in the second, they take the form of generating functions, asymptotic formulas, and finite-difference statements for the partition statistic $N_k(m,n)$ and, in the case $k=1$, Dyson's crank distribution (Levin et al., 2 Aug 2025, Zhou, 2018).

1. Classical genus-one determinant identities

In the genus-one setting, the basic objects are the elliptic modulus $\tau\in\mathbb H$, the nome $q=e^{2\pi i\tau}$, the Dedekind eta-function

$\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$

and the modular discriminant

$\Delta(\tau)=\eta(\tau)^{24},$

which is a holomorphic cusp form of weight $12$ for $SL_2(\mathbb Z)$. For even $\Delta(\tau)=\eta(\tau)^{24}$0, the classical Eisenstein series is

$\Delta(\tau)=\eta(\tau)^{24}$1

where $\Delta(\tau)=\eta(\tau)^{24}$2 is the $\Delta(\tau)=\eta(\tau)^{24}$3th Bernoulli number, and $\Delta(\tau)=\eta(\tau)^{24}$4 for odd $\Delta(\tau)=\eta(\tau)^{24}$5. The identities

$\Delta(\tau)=\eta(\tau)^{24}$6

and

$\Delta(\tau)=\eta(\tau)^{24}$7

supply the higher weights used in the determinant formulas. The odd Jacobi theta-function is

$\Delta(\tau)=\eta(\tau)^{24}$8

The two classical determinant formulas recorded in the note are

$\Delta(\tau)=\eta(\tau)^{24}$9

and Garvan's second-power formula

$N_k(m,n)$0

The derivation starts from the classical Ramanujan identities among $N_k(m,n)$1. Since $N_k(m,n)$2 is a weight-$N_k(m,n)$3 cusp form, one writes $N_k(m,n)$4 as a polynomial in the ring $N_k(m,n)$5, or more generally as a determinant of the Hankel matrix $N_k(m,n)$6. Garvan discovered that for $N_k(m,n)$7 this Hankel determinant collapses to the explicit $N_k(m,n)$8 determinant above; later proofs, including Milne's 2001 argument, use identities among Bernoulli numbers and classical dimension arguments for spaces of modular forms (Levin et al., 2 Aug 2025).

2. CFT determinant representations and generalized genus-one formulas

The modern extension of Garvan's determinant formulas proceeds through torus correlation functions in conformal field theory. Let $N_k(m,n)$9 be a $k=1$0 vertex-operator algebra and $k=1$1 a $k=1$2-twisted module. For $k=1$3 and local coordinates $k=1$4, the $k=1$5-point trace is

$k=1$6

where $k=1$7 is the Virasoro zero-mode, $k=1$8 the central charge, and $k=1$9 are twist operators parameterized by real parameters $\tau\in\mathbb H$0 via $\tau\in\mathbb H$1, $\tau\in\mathbb H$2. In the free-fermion model, the partition function has two equivalent expansions: $\tau\in\mathbb H$3 with $\tau\in\mathbb H$4, $\tau\in\mathbb H$5, $\tau\in\mathbb H$6, and simultaneously

$\tau\in\mathbb H$7

Equating these yields the Jacobi triple-product in determinant form,

$\tau\in\mathbb H$8

where the infinite matrix $\tau\in\mathbb H$9 has entries encoding the two-point expansion of the deformed Weierstrass kernel.

The corresponding deformed elliptic function is

$q=e^{2\pi i\tau}$0

with $q=e^{2\pi i\tau}$1 and $q=e^{2\pi i\tau}$2 such that $q=e^{2\pi i\tau}$3. These deformed Eisenstein series enter the matrix elements of $q=e^{2\pi i\tau}$4 and a companion $q=e^{2\pi i\tau}$5, and the genus-one Fay trisecant identity yields closed-form determinant expressions for $q=e^{2\pi i\tau}$6 and its powers.

The genus-one generalization of Garvan's formulas takes two principal forms. First, for $q=e^{2\pi i\tau}$7 and two $q=e^{2\pi i\tau}$8-point sets $q=e^{2\pi i\tau}$9 on the torus, one has determinant formulas for $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$0 in both the twisted and untwisted cases, involving $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$1 or $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$2, theta-function factors, and the product $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$3 of theta-quotients generalizing the prime form. Second, for arbitrary multi-indices $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$4 and multiplicities $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$5 satisfying $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$6, there is a formula

$\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$7

where $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$8 is a block-matrix whose blocks are the higher-order coefficients in the expansion of $\eta(\tau)=q^{1/24}\prod_{n=1}^\infty(1-q^n),$9. The proof uses the full Fay generalized trisecant identity. The resulting framework extends Garvan's classical genus-one determinant formulas for $\Delta(\tau)=\eta(\tau)^{24},$0 to all powers and establishes an explicit dictionary between CFT sewing-determinant methods and determinantal identities among modular forms (Levin et al., 2 Aug 2025).

3. Genus-two counterparts

The genus-two extension is formulated in the self-sewing picture, where a genus-two surface is constructed by sewing two punctured tori with complex parameter $\Delta(\tau)=\eta(\tau)^{24},$1. The fermion partition function then admits two representations. The first is the direct sewing of two torus one-point functions with determinant of a $\Delta(\tau)=\eta(\tau)^{24},$2 block matrix $\Delta(\tau)=\eta(\tau)^{24},$3: $\Delta(\tau)=\eta(\tau)^{24},$4 The second comes from bosonization in terms of the genus-two theta-function divided by $\Delta(\tau)=\eta(\tau)^{24},$5 and a second determinant: $\Delta(\tau)=\eta(\tau)^{24},$6 Equating the two gives a genus-two Jacobi-triple-product analogue and, upon specialization $\Delta(\tau)=\eta(\tau)^{24},$7, a formula for $\Delta(\tau)=\eta(\tau)^{24},$8.

The genus-two analogue of Garvan's determinant formula is stated for fixed $\Delta(\tau)=\eta(\tau)^{24},$9, sewing parameter $12$0, twisting parameters $12$1, $12$2, and points $12$3 on the torus. With the genus-two Szegő kernel $12$4, its $12$5-twisted part $12$6, the finite matrix $12$7, and infinite-dimensional block matrices $12$8, the proposition gives a formal analogue of Garvan's formula expressing $12$9 as a quotient whose numerator contains

$SL_2(\mathbb Z)$0

and whose denominator contains

$SL_2(\mathbb Z)$1

A key identity,

$SL_2(\mathbb Z)$2

is proved via the genus-two Fay trisecant identity. The proof strategy begins with the genus-two partition function in fermionic form and its bosonization, derives a genus-two triple-product analogue by sewing two tori and comparing it to the genus-two theta-function, and then isolates $SL_2(\mathbb Z)$3 by rearranging determinant factors (Levin et al., 2 Aug 2025).

4. Cross-genus structure and analytical consequences

The genus-one and genus-two formulas share a common architecture. In both cases, they arise from equating a fermionic expansion, given as a product over modes, with a bosonic expansion in terms of theta-functions. In both cases, deformed elliptic kernels and Fay's trisecant identities are the main analytic tools. In both cases, the final formulas express powers of $SL_2(\mathbb Z)$4 or $SL_2(\mathbb Z)$5 as determinants of matrices whose entries are deformed Eisenstein-Weierstrass functions.

The genus-two setting introduces substantial additional structure. Sewing two tori brings in the sewing parameter $SL_2(\mathbb Z)$6, the branch-cut index $SL_2(\mathbb Z)$7, and twist parameters $SL_2(\mathbb Z)$8 on each torus. The genus-two kernel $SL_2(\mathbb Z)$9 has a more elaborate expansion in terms of $\Delta(\tau)=\eta(\tau)^{24}$00 and an infinite matrix resolvent $\Delta(\tau)=\eta(\tau)^{24}$01. The determinant formulas combine finite matrices $\Delta(\tau)=\eta(\tau)^{24}$02 with infinite-dimensional sewing matrices $\Delta(\tau)=\eta(\tau)^{24}$03 and $\Delta(\tau)=\eta(\tau)^{24}$04, so block-determinant identities become essential. The note explicitly identifies new phenomena: half-integer powers such as $\Delta(\tau)=\eta(\tau)^{24}$05, multi-component twisting, and explicit dependence on the genus-two period matrix $\Delta(\tau)=\eta(\tau)^{24}$06.

The conclusions of the note state that these results extend Garvan's classical genus-one determinant formulas for $\Delta(\tau)=\eta(\tau)^{24}$07 to all powers and to a new genus-two context. The potential implications listed there are a deeper understanding of Siegel modular forms via vertex-operator algebra techniques, new relations among genus-two analogues of Eisenstein series and theta-constants, and possible arithmetic applications in counting problems on higher-genus curves. The open questions recorded in the same source are extension to arbitrary genus $\Delta(\tau)=\eta(\tau)^{24}$08 by iterated sewing, specialization of the formal series in $\Delta(\tau)=\eta(\tau)^{24}$09 to convergent expansions characterizing the Siegel cusp forms $\Delta(\tau)=\eta(\tau)^{24}$10, and exploration of connections with Krichever-Novikov algebras and higher-genus generalizations of the Jack-Hall-Littlewood identities. This suggests that the determinant side of Garvan's formulas is being reframed as part of a broader higher-genus modular-form and VOA program (Levin et al., 2 Aug 2025).

5. Garvan's $\Delta(\tau)=\eta(\tau)^{24}$11-rank formulas in partition theory

A separate use of Garvan's name occurs in partition theory. For fixed positive integer $\Delta(\tau)=\eta(\tau)^{24}$12, a partition $\Delta(\tau)=\eta(\tau)^{24}$13 of $\Delta(\tau)=\eta(\tau)^{24}$14 has at least $\Delta(\tau)=\eta(\tau)^{24}$15 successive Durfee squares if one can inscribe a square of side $\Delta(\tau)=\eta(\tau)^{24}$16 in the Ferrers diagram of $\Delta(\tau)=\eta(\tau)^{24}$17, then a square of side $\Delta(\tau)=\eta(\tau)^{24}$18, and so on up to side $\Delta(\tau)=\eta(\tau)^{24}$19. The $\Delta(\tau)=\eta(\tau)^{24}$20-rank of such a partition is defined by

$\Delta(\tau)=\eta(\tau)^{24}$21

The counting function is

$\Delta(\tau)=\eta(\tau)^{24}$22

Its two-variable generating function is

$\Delta(\tau)=\eta(\tau)^{24}$23

where

$\Delta(\tau)=\eta(\tau)^{24}$24

A term-by-term comparison gives, for $\Delta(\tau)=\eta(\tau)^{24}$25,

$\Delta(\tau)=\eta(\tau)^{24}$26

with

$\Delta(\tau)=\eta(\tau)^{24}$27

The symmetry relation

$\Delta(\tau)=\eta(\tau)^{24}$28

is also part of the basic formalism.

The proof mechanism recorded in the paper is direct: expand $\Delta(\tau)=\eta(\tau)^{24}$29, multiply by the $\Delta(\tau)=\eta(\tau)^{24}$30-sum, and compare coefficients of $\Delta(\tau)=\eta(\tau)^{24}$31. In this setting, Garvan's formulas are not determinant identities but exact combinatorial generating and coefficient formulas for a family of partition statistics defined through successive Durfee squares (Zhou, 2018).

6. Asymptotics, Dyson's crank, and finite differences

The asymptotic theory begins with

$\Delta(\tau)=\eta(\tau)^{24}$32

For fixed $\Delta(\tau)=\eta(\tau)^{24}$33, the main theorem states first that if

$\Delta(\tau)=\eta(\tau)^{24}$34

then

$\Delta(\tau)=\eta(\tau)^{24}$35

Second, if

$\Delta(\tau)=\eta(\tau)^{24}$36

then

$\Delta(\tau)=\eta(\tau)^{24}$37

for some absolute $\Delta(\tau)=\eta(\tau)^{24}$38. The proof uses the classical Hardy-Ramanujan formula for $\Delta(\tau)=\eta(\tau)^{24}$39, a finite-shift expansion for $\Delta(\tau)=\eta(\tau)^{24}$40 when $\Delta(\tau)=\eta(\tau)^{24}$41, and the fact that the higher-$\Delta(\tau)=\eta(\tau)^{24}$42 terms in the defining sum for $\Delta(\tau)=\eta(\tau)^{24}$43 are exponentially smaller when $\Delta(\tau)=\eta(\tau)^{24}$44.

For $\Delta(\tau)=\eta(\tau)^{24}$45, one has $\Delta(\tau)=\eta(\tau)^{24}$46, Dyson's crank. Theorem 3.1 gives the exact threshold for Dyson's asymptotic formula: it holds if and only if

$\Delta(\tau)=\eta(\tau)^{24}$47

and the paper also records a uniform estimate for $\Delta(\tau)=\eta(\tau)^{24}$48. The proof combines the general asymptotic for $\Delta(\tau)=\eta(\tau)^{24}$49 with the explicit difference

$\Delta(\tau)=\eta(\tau)^{24}$50

and compares $\Delta(\tau)=\eta(\tau)^{24}$51 with $\Delta(\tau)=\eta(\tau)^{24}$52. In the formulation given there, the condition $\Delta(\tau)=\eta(\tau)^{24}$53 is what ensures that the main $\Delta(\tau)=\eta(\tau)^{24}$54-factor is in the correct range and that the error terms from both the tail of the defining series and the Hardy-Ramanujan expansions are simultaneously $\Delta(\tau)=\eta(\tau)^{24}$55.

The same paper studies backward finite differences

$\Delta(\tau)=\eta(\tau)^{24}$56

For fixed integers $\Delta(\tau)=\eta(\tau)^{24}$57 and $\Delta(\tau)=\eta(\tau)^{24}$58, Corollary 4.1 states that there exists an absolute $\Delta(\tau)=\eta(\tau)^{24}$59, in fact

$\Delta(\tau)=\eta(\tau)^{24}$60

such that if

$\Delta(\tau)=\eta(\tau)^{24}$61

then one obtains asymptotic formulas for $\Delta(\tau)=\eta(\tau)^{24}$62. The proof uses Theorem 2.1 together with the classical Odlyzko formula for finite differences of $\Delta(\tau)=\eta(\tau)^{24}$63, and then matches its exponential factor with the Hardy-Ramanujan form of $\Delta(\tau)=\eta(\tau)^{24}$64. A plausible implication is that Garvan's $\Delta(\tau)=\eta(\tau)^{24}$65-rank formulas delimit several asymptotic regimes: an exact large-$\Delta(\tau)=\eta(\tau)^{24}$66 regime, a $\Delta(\tau)=\eta(\tau)^{24}$67 regime controlled by $\Delta(\tau)=\eta(\tau)^{24}$68, the Dyson crank regime with threshold $\Delta(\tau)=\eta(\tau)^{24}$69, and a finite-difference regime beginning at order $\Delta(\tau)=\eta(\tau)^{24}$70 (Zhou, 2018).