Explicit Hypergeometric Modularity Method

• EHMM is a framework that explicitly transforms hypergeometric series, periods, and sums into modular and automorphic structures using transformation identities, congruences, and finite-field methods.
• It spans various settings—from rigid Calabi–Yau families to q-hypergeometric and differential equations—yielding explicit constructions of Hecke eigenforms, eta-quotients, and quantum modular forms.
• The method enables practical results such as deriving Hecke eigenvalue formulas, linking Frobenius periods with critical L-values, and constructing degree-4 Galois representations.

Explicit Hypergeometric Modularity Method (EHMM) denotes a family of explicit procedures that connect hypergeometric objects to modular or automorphic structures. In a central formulation, for certain length-$3$ and length-$4$ hypergeometric data, the method produces an explicit modular function $t(q)$, a differential form whose specialization is a weight-$n$ holomorphic cusp form, and—under Hecke-eigen and integrality hypotheses—a Hecke eigenform $f_\HD^\sharp$ whose Deligne–Galois representation matches Katz’s hypergeometric representation up to twist (Allen et al., 2024). In the rigid Calabi–Yau setting, the method identifies Hecke eigenvalues from $p$-adic truncations of hypergeometric periods and relates the remaining Frobenius-basis solutions at $z=1$ to critical $L$-values of the same modular form (Zudilin, 2018). The same label is also used for explicit q-hypergeometric constructions of Maass cusp forms and quantum modular forms, for modularity criteria for hypergeometric-type differential equations, for modular generating series of D4–D2–D0 invariants, and for hypergeometric decompositions of Delsarte K3 pencils (Bringmann et al., 2016, Saber et al., 2021, Alexandrov et al., 2023, Davis et al., 20 Aug 2025).

1. Scope, terminology, and recurring structure

The literature uses the name EHMM for several explicit pipelines rather than for a single universally fixed formalism. Across these variants, the input is a hypergeometric series, period, differential equation, finite-field character sum, or q-hypergeometric series; the output is a modular form, Maass form, quantum modular form, Galois representation, or modular generating series. A recurring feature is that the modular object is not inferred abstractly but built through explicit formulas, congruences, transformation identities, or Frobenius-trace calculations (Allen et al., 2024, Zudilin, 2018).

Setting	Hypergeometric input	Modular or automorphic output
Rigid Calabi–Yau families	${}_4F_3$ periods and Frobenius basis	Weight-$4$ modular forms and critical $4$0-values
Length-$4$1/$4$2 data	Hypergeometric datum $4$3, Euler integral, $4$4	Weight-$4$5 Hecke eigenforms and Katz–Deligne matching
q-hypergeometric Habiro-ring series	$4$6	Maass cusp forms and weight-$4$7 quantum modular forms
Hypergeometric-type ODEs	Schwarzian form of $4$8	Modular solutions on $4$9, $t(q)$0
D4–D2–D0 theory	Hypergeometric Calabi–Yau geometry and wall-crossing data	Vector-valued modular forms of weight $t(q)$1
Delsarte K3 pencils	Finite-field hypergeometric sums and Picard–Fuchs operators	Hypergeometric $t(q)$2-series and Dedekind zeta factors

In the Allen–Grove–Long–Tu formulation, EHMM is presented as a three-stage process: express finite-sum and truncated hypergeometric values in terms of an auxiliary differential form; specialize that form via a known modular Hauptmodul to obtain a candidate modular form; then use commutative formal-group recurrences together with Gross–Koblitz and Dwork supercongruences to match Hecke eigenvalues with hypergeometric character sums (Allen et al., 2024). In the earlier rigid Calabi–Yau formulation, the two principal steps are instead $t(q)$3-adic extraction of Hecke eigenvalues and analytic comparison of Frobenius-basis periods with critical $t(q)$4-values (Zudilin, 2018). This suggests that “EHMM” functions as a shared editorial label for a cluster of explicit hypergeometric-to-modular constructions.

2. Rigid Calabi–Yau formulation

In the rigid Calabi–Yau threefold setting, EHMM begins with the unique holomorphic period

$t(q)$5

and in the fourteen hypergeometric cases one has

$t(q)$6

with $t(q)$7 (Zudilin, 2018). More generally one considers

$t(q)$8

viewed as the generator of the Picard–Fuchs solution space at $t(q)$9.

The associated hypergeometric differential equation is

$n$0

Near $n$1, the standard Frobenius basis is obtained from the bilateral sum

$n$2

expanded as

$n$3

Then $n$4 form a basis of local solutions at $n$5 (Zudilin, 2018).

The arithmetic step uses the $n$6-term truncation

$n$7

For good primes $n$8, one checks numerically and then proves by finite-field hypergeometric methods that

$n$9

where $f_\HD^\sharp$0 is the $f_\HD^\sharp$1-th Fourier coefficient of a weight-$f_\HD^\sharp$2 modular form, and $f_\HD^\sharp$3 is chosen so that Weil’s bound

$f_\HD^\sharp$4

forces the smallest-absolute-value residue of $f_\HD^\sharp$5 to equal the true Hecke eigenvalue. In the weight-$f_\HD^\sharp$6 case one takes $f_\HD^\sharp$7 and $f_\HD^\sharp$8; for weight $f_\HD^\sharp$9, $p$0 and $p$1; for weight $p$2, $p$3 and $p$4 (Zudilin, 2018).

The analytic step evaluates the other Frobenius-basis elements at $p$5. In weight $p$6, the conjectural pattern is

$p$7

For the $p$8 case, one additionally has

$p$9

yielding

$z=1$0

In the worked example, the truncations

$z=1$1

recover the coefficients of

$z=1$2

for all primes $z=1$3, while $z=1$4, $z=1$5, and $z=1$6 numerically recover $z=1$7, $z=1$8, and $z=1$9 up to rational factors (Zudilin, 2018).

3. Length-$L$0 and length-$L$1 hypergeometric data

In the 2024 formalization, EHMM is designed for hypergeometric data

$L$2

with

$L$3

The stated goal is to build a weight-$L$4 Hecke eigenform whose Deligne–Galois representation occurs inside the Katz hypergeometric representation attached to the datum (Allen et al., 2024).

The first explicit output is a modular function $L$5, obtained through Ramanujan’s alternative-base theory. The second is the differential form

$L$6

whose specialization $L$7 is a weight-$L$8 holomorphic cusp form $L$9. Under suitable Hecke-eigen and integrality hypotheses on ${}_4F_3$0 and on the finite-field hypergeometric sums attached to ${}_4F_3$1, one constructs

${}_4F_3$2

a genuine Hecke eigenform of weight ${}_4F_3$3 whose Galois representation matches, up to twist, the Katz representation (Allen et al., 2024).

The algorithm is given in five steps. Step 1 uses an Euler integral to reduce length ${}_4F_3$4 to length ${}_4F_3$5: ${}_4F_3$6 Step 2 chooses a Hauptmodul ${}_4F_3$7 on the genus-${}_4F_3$8 triangle group attached to ${}_4F_3$9. Step 3 forms $4$0 and verifies holomorphy and Hecke-eigen behavior. Step 4 compares Hecke eigenvalues with

$4$1

using commutative formal-group congruences mod $4$2 and Gross–Koblitz plus Dwork unit-root supercongruences mod $4$3. Step 5 solves for the sign or twist character and, when necessary, takes a Hecke-orbit linear combination (Allen et al., 2024).

The main theorem asserts that, under the stated hypotheses, $4$4 is a Hecke eigenform of weight $4$5, level dividing an explicit divisor of $4$6, and for $4$7, $4$8,

$4$9

The Galois-theoretic form is

$4$00

A companion supercongruence theorem states that for ordinary $4$01 with $4$02,

$4$03

where $4$04 is the Dwork unit root of the local Euler factor (Allen et al., 2024).

4. Explicit realizations by eta-quotients, theta functions, and higher Galois constructions

Several later papers turn EHMM into a mechanism for constructing explicit eta-quotient families. In the length-$4$05 family $4$06, one defines

$4$07

a weight-$4$08 eta-quotient whose Fourier coefficients satisfy Hecke recursion at primes $4$09; Hecke-orbit combinations then produce true eigenforms in $4$10 (Allen et al., 2024). The same circle of ideas is used in the second paper of the series to derive weight-$4$11 modular forms labeled $4$12-functions, and then, through Whipple and McCarthy well-poised formulae, to construct degree-$4$13 Galois representations that extend to $4$14 and whose $4$15-functions coincide with those of automorphic forms (Allen et al., 2024).

The cubic-theta variant introduces the family

$4$16

with

$4$17

Using the Borwein cubic theta functions $4$18, $4$19, and $4$20, together with

$4$21

the Euler-integral realization becomes

$4$22

This construction is then used to prove five Dawsey–McCarthy modularity conjectures of the form

$4$23

for $4$24, and to derive special $4$25-value formulas and clique-count formulas for generalized Paley graphs (Grove, 26 Jul 2025).

The 2026 extension constructs two additional families. For

$4$26

one obtains

$4$27

for $4$28. For

$4$29

one obtains

$4$30

for $4$31. EHMM then yields explicit formulas expressing the Fourier coefficients of the resulting weight-$4$32 and weight-$4$33 eigenforms in terms of finite-field period functions, together with identities involving special values of the finite-field Appell series $4$34 and $4$35 (Maity et al., 3 Apr 2026).

A notable feature of these papers is that the modular form is not merely detected after the fact. It is written down explicitly as an eta-quotient or theta-quotient, and the hypergeometric side then supplies the precise Hecke-eigenvalue formula, the Galois interpretation, or the special-$4$36-value identity.

5. Other constructions carrying the same label

In the q-hypergeometric Maass-form setting, EHMM is a four-step procedure: construct a q-hypergeometric series in the Habiro ring from special polynomials; rewrite it by Bailey-pair manipulations as an indefinite-theta-type sum; apply Zwegers’ theory of indefinite theta functions to produce a Maass waveform of eigenvalue $4$37; and use Habiro-ring convergence at roots of unity to prove quantum modularity and cuspidality (Bringmann et al., 2016). The resulting series $4$38 are weight-$4$39 quantum modular forms on a congruence subgroup, and each is the holomorphic part of a Maass cusp form $4$40.

In the differential-equation setting, EHMM refers to the passage from

$4$41

to the normal form

$4$42

then to the Schwarzian equation

$4$43

The modularity criterion is exact: the Schwarzian equation admits a modular-function solution if and only if $4$44 with $4$45 and $4$46; equivalently,

$4$47

When this occurs, the invariance group is $4$48, and two linearly independent modular solutions are recovered from a Hauptmodul $4$49 by

$4$50

This use of the term emphasizes explicit modular-function uniformization rather than finite-field trace matching (Saber et al., 2021).

In quantum geometry, EHMM is a three-step modular bootstrap for Abelian D4–D2–D0 invariants on one-parameter Calabi–Yau threefolds of hypergeometric type. The geometry begins with the hypergeometric Picard–Fuchs equation

$4$51

and the direct integration of the holomorphic anomaly equations. An exact PT$4$52DT wall-crossing formula is then used to extract polar coefficients of

$4$53

and modularity predicts that $4$54 is a vector-valued modular form of weight $4$55. Matching the polar part fixes the generating series and yields infinitely many DT, PT, and GV invariants (Alexandrov et al., 2023).

For Delsarte K3 pencils, EHMM is presented as an arithmetic-geometric pipeline: finite-field point counts are written explicitly as polynomials in $4$56 plus hypergeometric sums $4$57; the Picard–Fuchs operator is identified as a $4$58-type operator with the same parameters; the incomplete Hasse–Weil $4$59-function is factored into hypergeometric $4$60-series and Dedekind zeta factors; and the Shioda–Katz–Kloosterman decomposition identifies the corresponding hypergeometric motives geometrically (Davis et al., 20 Aug 2025).

6. Hypotheses, limitations, and mathematical significance

The formal length-$4$61/$4$62 method imposes several explicit constraints. It requires $4$63; the modular function $4$64 must have integral Fourier expansion and must lead to a bona fide Hecke eigenform; and the datum must satisfy certain Galois conditions on its conjugates so that the appropriate Hecke orbit is preserved. The papers state that this happens for exactly the classical triangle groups $4$65 with $4$66 in length $4$67 and their extensions in length $4$68 (Allen et al., 2024). The rigid Calabi–Yau formulation similarly assumes good primes and relies on Weil’s bound to lift a congruence class modulo $4$69 to the actual Hecke eigenvalue (Zudilin, 2018). In the differential-equation setting, modularity occurs only for denominators $4$70 (Saber et al., 2021).

At the same time, the method is notable for the explicitness of its outputs. In the rigid Calabi–Yau setting it provides “an explicit, completely elementary algorithm” to compute Hecke eigenvalues $4$71 via hypergeometric truncations and Weil’s bound, and a bridge between Frobenius-basis periods and critical $4$72-values (Zudilin, 2018). In the Allen–Grove–Long–Tu program, it gives an explicit modular function, an explicit cusp form, and, after comparison with finite-field sums and supercongruences, an explicit Hecke eigenform whose Galois representation matches the Katz hypergeometric representation up to twist (Allen et al., 2024). In later extensions it produces explicit eta-quotients, degree-$4$73 Galois representations extending to $4$74, and identities relating Fourier coefficients to finite-field Appell series (Allen et al., 2024, Maity et al., 3 Apr 2026).

A recurring misconception is to treat EHMM as synonymous with a single theorem about modularity of hypergeometric motives. The literature does not support that simplification. The same name is attached to several procedures with different inputs, different target categories, and different proof technologies: Frobenius bases and supercongruences, Katz–Deligne representation matching, Bailey pairs and indefinite theta functions, Schwarzian uniformization, wall-crossing and vector-valued modularity, and hypergeometric decomposition of $4$75-functions. A plausible implication is that EHMM is best understood as a methodological family centered on explicit hypergeometric data and explicit modular realization, rather than as one canonical algorithm.