Miller Basis in Modular Forms

• Miller basis is a canonical basis for the space Mₖ of holomorphic modular forms, defined by prescribing the initial Fourier expansion at the cusp infinity.
• It is constructed using modular invariants such as Δ, Eisenstein series Eₖ′, and the j-invariant to generate Faber polynomials that simplify the study of zeros and asymptotic behavior.
• The framework has been employed to derive effective bounds and asymptotic results on the distribution of zeros on the standard fundamental domain and the unit arc.

The Miller basis is a canonical basis of the space $M_k(SL_2(\mathbb{Z}))$ of holomorphic modular forms of even weight $k$, defined by prescribing the initial segment of the Fourier expansion at the cusp $\infty$. If $k=12\ell+k'$ with $k' \in \{0,4,6,8,10,14\}$, the basis consists of the unique forms $f_{k,m}$ or $g_{k,m}$, for $0 \le m \le \ell$, satisfying

$$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$

Equivalently, $\operatorname{ord}_\infty(f_{k,m})=m$, and the range of $k$0 matches $k$1. In recent work the Miller basis has become a central framework for studying Faber polynomials, zero loci on the boundary of the standard fundamental domain, and asymptotic zero distributions depending on the ratio $k$2 (Zilka, 7 Oct 2025, Raveh, 2024).

1. Definition and ambient modular-form setting

Let $k$3 denote the space of holomorphic modular forms of even weight $k$4 for the full modular group. Writing

$k$5

the Miller basis of $k$6 is the family

$k$7

characterized by

$k$8

For each $k$9 there is a unique modular form with that $\infty$0-expansion, and these forms constitute a basis of $\infty$1 (Zilka, 7 Oct 2025).

The same basis is written in the literature as $\infty$2, with

$\infty$3

and the two descriptions coincide at the level of definition: the basis is adapted to the cusp $\infty$4, normalized by the coefficient of $\infty$5, and indexed by the vanishing order at infinity (Raveh, 2024). The subfamily with $\infty$6 lies in $\infty$7, so the Miller basis contains a distinguished basis of cusp forms. In the notation of the cusp-form paper, $\infty$8 are precisely the cusp forms of the Miller basis (Raveh, 2024).

This basis is “natural” and “canonical” in the sense used in the modular-form literature. It is adapted simultaneously to weight and to the truncated Fourier expansion, and in one formulation the Fourier coefficient matrix up to order $\infty$9 is in reduced row echelon form (Raveh, 2024).

2. Construction through $k=12\ell+k'$0, $k=12\ell+k'$1, $k=12\ell+k'$2, and Faber polynomials

The structural representation of the Miller basis uses the modular discriminant

$k=12\ell+k'$3

the Eisenstein series $k=12\ell+k'$4 of weight $k=12\ell+k'$5, and the classical $k=12\ell+k'$6-invariant

$k=12\ell+k'$7

Every modular form $k=12\ell+k'$8 with $k=12\ell+k'$9 can be written uniquely as

$k' \in \{0,4,6,8,10,14\}$0

where $k' \in \{0,4,6,8,10,14\}$1 is a polynomial in one variable. For a Miller basis element, the associated polynomial is the Faber polynomial $k' \in \{0,4,6,8,10,14\}$2, characterized by

$k' \in \{0,4,6,8,10,14\}$3

with $k' \in \{0,4,6,8,10,14\}$4 (Zilka, 7 Oct 2025).

A more algorithmic construction begins with the seed forms

$k' \in \{0,4,6,8,10,14\}$5

Since $k' \in \{0,4,6,8,10,14\}$6 and $k' \in \{0,4,6,8,10,14\}$7, one has

$k' \in \{0,4,6,8,10,14\}$8

Applying Gaussian elimination to the Fourier coefficient matrix of $k' \in \{0,4,6,8,10,14\}$9 up to order $f_{k,m}$0 produces the Miller basis in reduced row echelon form. The resulting Faber polynomial $f_{k,m}$1 has integer coefficients (Raveh, 2024).

The Faber-polynomial description converts zero problems for modular forms into zero problems for ordinary polynomials. If $f_{k,m}$2 are the zeros of $f_{k,m}$3, counted with multiplicity, then

$f_{k,m}$4

because $f_{k,m}$5 is nonvanishing on $f_{k,m}$6 (Zilka, 7 Oct 2025). On the arc

$f_{k,m}$7

one has

$f_{k,m}$8

so zeros of $f_{k,m}$9 on $g_{k,m}$0 correspond exactly to zeros of $g_{k,m}$1 in $g_{k,m}$2 (Zilka, 7 Oct 2025).

3. Zeros on the standard arc and effective large-weight results

The zero geometry of the Miller basis is studied in the standard fundamental domain

$g_{k,m}$3

whose circular boundary arc is

$g_{k,m}$4

For a Miller cusp form $g_{k,m}$5, the finite zeros in $g_{k,m}$6 are the zeros away from the cusp $g_{k,m}$7. Since $g_{k,m}$8, the form has exactly $g_{k,m}$9 finite zeros in the fundamental domain, counted with multiplicity (Raveh, 2024).

For fixed $0 \le m \le \ell$0, there exist constants $0 \le m \le \ell$1 such that if

$0 \le m \le \ell$2

then every finite zero of $0 \le m \le \ell$3 lies on the arc $0 \le m \le \ell$4. Moreover, as $0 \le m \le \ell$5, these zeros become uniformly distributed on $0 \le m \le \ell$6 (Raveh, 2024). The same paper makes the bound explicit: $0 \le m \le \ell$7 This yields an effective linear bound in $0 \le m \le \ell$8 for the weight (Raveh, 2024).

A special case is markedly stronger. For every $0 \le m \le \ell$9, equivalently for every even weight $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$0, all finite zeros of

$$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$1

lie on $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$2. Thus the first cusp form in the Miller basis exhibits the arc-zero phenomenon in every admissible weight, not merely asymptotically (Raveh, 2024).

The proof strategy follows the Rankin–Swinnerton-Dyer and Duke–Jenkins method. On the arc, the normalized function

$$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$3

is real-valued, and for $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$4 sufficiently large relative to $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$5 it satisfies

$$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$6

The sign changes of the cosine term then force at least $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$7 zeros on the arc, and since $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$8 is the total number of finite zeros, all finite zeros must lie there (Raveh, 2024).

This large-weight restoration of arc concentration coexists with explicit counterexamples when $$f_{k,m}(\tau)=q^m+O\bigl(q^{\ell+1}\bigr), \qquad q=e^{2\pi i\tau}.$$9 is not fixed. Duke–Jenkins observed that for $\operatorname{ord}_\infty(f_{k,m})=m$0 the statement “all zeros on $\operatorname{ord}_\infty(f_{k,m})=m$1” fails in general, and $\operatorname{ord}_\infty(f_{k,m})=m$2 is given as a counterexample with zeros off the arc (Raveh, 2024).

4. Faber zeros, linear moments, and limiting distributions

A second line of work studies the zeros $\operatorname{ord}_\infty(f_{k,m})=m$3 of the Faber polynomial $\operatorname{ord}_\infty(f_{k,m})=m$4. For $\operatorname{ord}_\infty(f_{k,m})=m$5, the $\operatorname{ord}_\infty(f_{k,m})=m$6-th power sum and normalized moment are

$\operatorname{ord}_\infty(f_{k,m})=m$7

The fundamental structural result is that these power sums are linear functions of the weight and the index. For each integer $\operatorname{ord}_\infty(f_{k,m})=m$8 there exist constants $\operatorname{ord}_\infty(f_{k,m})=m$9 and a function $k$00, depending only on $k$01, such that for all $k$02 with $k$03 and $k$04,

$k$05

Moreover,

$k$06

$k$07 is the coefficient of $k$08 in the $k$09-expansion of $k$10, and

$k$11

The same analysis yields a nonrecursive explicit formula for the coefficients of $k$12 via Newton’s identities (Zilka, 7 Oct 2025).

Because $k$13, positivity of even moments constrains whether all zeros can remain on the arc. Using $k$14, the paper proves explicit upper ranges in which at least one zero leaves the arc. If $k$15, then: $k$16 or

$k$17

implies that at least one zero of $k$18 is not on $k$19. More asymptotically, if $k$20, $k$21, and $k$22 is large enough, then at least one zero is not on $k$23 (Zilka, 7 Oct 2025).

When $k$24 with $k$25, the limiting moments depend only on $k$26: $k$27 Under the hypothesis $k$28 and that all zeros lie on the arc, the limiting distribution of zeros along the arc is explicitly non-uniform. For

$k$29

the limiting probability that a zero lies between $k$30 and $k$31 is

$k$32

At $k$33, this reduces to the uniform distribution on $k$34; for $k$35, the density is shifted by the $k$36 term and therefore depends nontrivially on $k$37 (Zilka, 7 Oct 2025).

5. Logarithmic Szegő curves, phase transitions, and algebraic zeros

The parameter

$k$38

organizes the large-weight variation of zeros across the Miller basis. In this regime the central geometric object is the logarithmic analogue of the Szegő curve

$k$39

The paper defines

$k$40

and then

$k$41

with sign $k$42 determined by the sign of the weight. Equivalently, $k$43 is the curve of points $k$44 such that

$k$45

As $k$46,

$k$47

(Chiriac et al., 10 May 2026).

For the holomorphic Miller basis, the first main zero-counting result gives a lower bound for the proportion of non-elliptic zeros on the unit arc $k$48: $k$49 where $k$50. In particular, if $k$51 and $k$52 is sufficiently large, then all zeros lie on $k$53; if $k$54, then at least one non-elliptic zero lies on $k$55 (Chiriac et al., 10 May 2026).

In the opposite regime, when

$k$56

for some fixed $k$57, the zeros asymptotically approach $k$58. This is the regime where $k$59 is asymptotically close to $k$60, and the Faber polynomials behave like truncated exponentials. The paper accordingly posits that for all $k$61, the zeros asymptotically approach the upper hull of the union of the unit arc and the logarithmic Szegő curve (Chiriac et al., 10 May 2026).

The same work formulates conjectural threshold values

$k$62

with the conjectural picture that for $k$63 all zeros are on the unit arc, while for $k$64 no zeros remain on the arc. The proven thresholds $k$65 and $k$66 are described as numerically very close to these conjectural values (Chiriac et al., 10 May 2026).

Algebraic zeros form a separate arithmetic phenomenon. Up to $k$67, the classification result is that $k$68 has a non-elliptic algebraic zero if and only if $k$69, and the corresponding CM points are

$k$70

with discriminants $k$71. For $k$72, the paper proves a weaker density statement: up to height $k$73, all but possibly $k$74 weights yield no non-elliptic algebraic zero in that range (Chiriac et al., 10 May 2026).

6. Terminological extensions and non-modular usage

Although the expression “Miller basis” is concrete in the modular-form setting, it is not fully standardized across the literature. In Rhoades’ paper on symmetric-group character embeddings, the phrase does not actually appear, but the paper’s content makes it natural to interpret a “Miller basis” as the family of embedded class functions

$k$75

for fixed $k$76, or equivalently their images under the Frobenius characteristic map. Here $k$77 is obtained by multiplying each part of $k$78 by $k$79 and then repeating each part $k$80 times, and

$k$81

Under the Frobenius characteristic, the relevant plethystic identity is

$k$82

and Theorem 1 realizes $k$83 as the character of an explicitly described $k$84-module obtained by restricting an induction product of $k$85 copies of $k$86 (Rhoades, 2022).

This distinct usage is representation-theoretic rather than modular. It is connected to Miller’s embedding and congruence phenomena for symmetric-group characters, whereas the modular-form Miller basis is the canonical $k$87-expansion-adapted basis of $k$88. The two usages share the idea of a distinguished family indexed by combinatorial data, but they arise in different parts of mathematics and are not interchangeable (Rhoades, 2022).