Alternative Bases for Modular Forms

• Alternative bases for modular forms are explicit constructions that yield spanning sets with enhanced arithmetic and computational control compared to classical bases.
• They employ canonical row reduction, Zagier duality, and hypergeometric methods to generate triangular q-expansions and dual structures.
• These bases underpin applications ranging from Borcherds products to L-value formulas, offering both theoretical insights and practical computational algorithms.

Alternative bases for modular forms refer to constructions, distinct from the classical Eisenstein/cusp/monomial bases, that yield explicit spanning sets (often with extra structure) for spaces of modular forms, holomorphic or weakly holomorphic, of integral, half-integral, or vector-valued weight over various congruence subgroups. These constructions enable fine control over arithmetic properties, computational complexity, and duality phenomena and connect to deep objects such as Borcherds products, Kac–Moody algebras, and Ramanujan’s theories.

1. Canonical Bases and Row-Reduction

Canonical row-reduced bases are constructed by exploiting the modular forms' Fourier expansions at cusps, performing analogs of Gaussian elimination, so that each basis element has a uniquely specified principal part (e.g., a prescribed pole at infinity), with all holomorphic and cusp-form components canceled. For a Fuchsian group $\Gamma$ commensurable with $\mathrm{SL}_2(\mathbb{Z})$, any finite-dimensional space $M_k^!(\Gamma, v)$ of weight-$k$ weakly holomorphic modular forms can thus be equipped with such bases $\{f_{k,m}\}$ and their duals $\{g_{2-k,n}\}$, with Fourier expansions

$$f_{k,m}(z) = q^{-m} + \sum_{n > -m} a_k(m,n) q^n, \quad g_{2-k,n}(z) = q^{-n} + \sum_{m > -n} b_{2-k}(n,m) q^m,$$

where exponents are chosen to index the basis. This basis construction is systematic for both integral and half-integral weights, as well as for spaces with character or multiplier attached to more general subgroups (Griffin et al., 2020). The key property is the elimination of lower-order $q$-terms in all but the leading (principal part) term, producing a lower- or upper-triangular structure in the $q$-expansion coefficients.

2. ZAGIER Duality and Grid Structures

A central phenomenon in alternative basis construction is Zagier duality: for canonical (row-reduced) bases $\{f_{k,m}\}$ for $M_k^!(\Gamma,v)$ and $\{g_{2-k,n}\}$ for $M_{2-k}^!(\Gamma,v^*)$, the Fourier coefficients satisfy

$a_k(m,n) = -b_{2-k}(n,m)$

outside of a finite exceptional set. This structure assembles into a modular grid, reflected analytically in bivariate generating functions such as

$$\mathcal{G}_k(z,T) = \sum_m f_{k,m}(z) p^m = -\sum_n g_{2-k,n}(T) q^n,$$

where $p = e^{2\pi i T}$, $q = e^{2\pi i z}$. These grids encode the entire duality between bases, admit interpretation via residues at simple poles for $z = \gamma T$, and are stable under Hecke, Fricke, trace, and differentiation operators (Griffin et al., 2020, Zhang, 2013). The generating function can often be written as

$\mathcal{G}_k(z,T) = \frac{K(z,T)}{H(z) - H(T)},$

with $H$ a modular Hauptmodul (weight-0 modular function).

3. Explicit Construction of Alternative Bases in Various Settings

Alternative bases are constructed for broad classes:

• Integral Weight, Level One: For $M_{2k}(\mathrm{SL}_2(\mathbb{Z}))$, Fukuhara introduces the basis $\{G_{2k}\}\cup\{G_{4i} G_{2k-4i}\}$ (or similarly with $G_{4i+2} G_{2k-4i-2}$ when $2k\not\equiv 0 \bmod 4$), offering computational advantages and explicit convolution formulas for Fourier coefficients (Fukuhara, 2010).
• Congruence Subgroups: For genus-zero groups, canonical bases are given by polynomials in Hauptmoduln (e.g., $f^{(p)}_{0,m}(z)=P_{p,m}(\Psi_{(p)}(z))$), yielding q-expansions with a single principal part and recursive structure (Jenkins et al., 2014).
• Weight-Two Generation: When the graded algebra $\bigoplus_k M_k(\Gamma_0(N))$ is generated in weight two (no elliptic fixed points), total bases can be constructed as products of weight-two forms, vastly improving computational efficiency over modular-symbol linear algebra (Lam et al., 2017).
• Strong Modular Unit Construction: Using strong $\eta$-quotients $A_N(\tau)$ (which vanish only at $\infty$ and generate all higher weights), one writes every modular form as a polynomial in $A_N$ times elements from a finite low-weight window, leading to efficient triangular bases (Feauveau, 2018).
• Quaternionic Theta-Series: For newform (especially cuspform) subspaces over arbitrary level, the Jacquet-Langlands correspondence yields an orthogonal basis of theta-series attached to right ideal classes in quaternion orders, giving a concrete alternative to classical modular symbols (Martin, 2018).

4. Special Structures: Vector-Valued, Half-Integral, and Mock Modular Forms

Vector-valued modular forms with respect to the Weil representation admit an explicit correspondence (via discriminant forms) to scalar-valued forms with sign conditions. Canonical bases (sometimes referred to as Miller-like) for such spaces generalize the row-reduction technique, imposing Atkin–Lehner $+$/$-$-conditions and giving rise to bases whose nonzero coefficients enumerate root multiplicities in Kac–Moody type algebras and divisors in Borcherds products (Zhang, 2013).

For half-integral weight, the Kohnen plus space $M_{k+1/2}^+(4)$ enjoys a finite presentation in terms of $\vartheta$ and $F_2$, with efficient canonical bases available via explicit polynomial/convolution formulas (Inam et al., 2020). Alternative bases, such as those constructed via canonical Poincaré series and their harmonic Maass "supplements", realize a mock modular analog of the integral-weight basis phenomenon—these bases satisfy duality via the action of certain differential operators and period pairing, extending the theory to harmonic and mock modular forms with explicit quadratic relations among $L$-values at critical points (Choi et al., 2012).

5. Alternative Modular Bases via Hypergeometric and Theta-Series

Ramanujan's theory and its generalizations, as developed by Berndt, Bhargava, Garvan, Borwein, and others, showcase alternative modular bases constructed from hypergeometric functions and associated theta-series (e.g., $a_0(q) = \sum_{m,n} q^{m^2+mn+n^2}$, $c_0(q)$, $b_0(q)$ of Borwein). These yield modular forms of weight 1 for congruence subgroups ($\Gamma_0(3)$ for cubic signature), with closed forms in terms of elliptic integrals and explicit algebraic modular invariants. The change of basis from Eisenstein series to these "alternative" elements is highly transparent, leading to modular interpretations of special values of $L$-functions and rationality of singular moduli (Bagis, 2010, Bagis, 2015, Arora et al., 11 Jan 2026).

6. Integrality, Congruences, and Computational Properties

Alternative bases often enable precise control over integrality and congruences of Fourier coefficients. In the canonical row-reduced and Miller-like settings, divisibility by high powers of the level is systematically linked to the nature and order of the principal part—this is crucial for explicit $p$-adic applications, congruences between modular forms, and the arithmetic of modular units (Jenkins et al., 2014, Zhang, 2013).

The recursive structure of these bases (e.g., for genus-zero Hauptmodul-based spaces or via strong modular units) yields efficient algorithms for computation and modular form identification at high weight or level. The weight-two-generation and graded algebra paradigms are computationally dominant for moderate levels and high weights (Lam et al., 2017, Feauveau, 2018).

7. Applications and Theoretical Implications

Alternative bases underlie the construction of infinite product expansions (Borcherds products), the explicit realization of generalized Kac–Moody algebras, evaluations of Ramanujan-type $1/\pi$ series via hypergeometric and modular identities, trace formulas (e.g., traces of singular moduli), and the study of zeros and nonvanishing of Fourier coefficients (with implications for Lehmer-type questions and the distribution of zeros in moduli spaces) (Choi et al., 2012, Folsom et al., 2016, Zhang, 2013, Bagis, 2015).

They also provide the infrastructure for explicit L-value formulas for CM modular forms and hypergeometric-type Galois representations, extending the landscape of explicit modular parameterizations with arithmetic significance (Arora et al., 11 Jan 2026).

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Key references: (Fukuhara, 2010, Choi et al., 2012, Zhang, 2013, Jenkins et al., 2014, Bagis, 2015, Folsom et al., 2016, Lam et al., 2017, Martin, 2018, Feauveau, 2018, Inam et al., 2020, Griffin et al., 2020, Bagis, 2010, Arora et al., 11 Jan 2026)