Special Values at CM Points

• Special values at CM points are special invariants that arise at complex multiplication points, generating abelian extensions and underpinning explicit class field theory.
• They enable explicit transcendence results by identifying the rare cases where modular parameters and their invariants are both algebraic, as seen in classical j-invariant results.
• Their study extends to higher-dimensional abelian varieties, where computational algorithms and Galois actions illuminate cryptographic applications and further arithmetic geometry insights.

Special values at CM (complex multiplication) points play a central role in transcendence theory, algebraic geometry, arithmetic geometry, and the explicit class field theory of modular and automorphic forms. CM points are special because they parametrize abelian varieties (e.g., elliptic curves, abelian surfaces, or Drinfeld modules in positive characteristic) with extra endomorphisms, leading to dramatic algebraic and arithmetic behavior of automorphic invariants. The structure of special values at CM points—such as modular invariants, modular forms, L-functions, or periods—illuminates deep connections between transcendence, Galois theory, and the geometry of Shimura varieties.

1. Classical Transcendence Theorems and CM Points

For the elliptic modular j-invariant, two classical transcendence results illustrate the exceptional nature of CM points:

• Schneider's Theorem (1934): For τ in the Poincaré upper half-plane ℍ,

$$\operatorname{trdeg}_\mathbb{Q}(\tau, j(\tau)) \geq 1$$

with equality (i.e., both algebraic) if and only if τ is quadratic imaginary—a CM point. Thus, CM points are the only source of algebraic dependence for the pair $(\tau, j(\tau))$ (Gómez, 13 May 2025).

• Stéphanois' Theorem (ca. 1933–38): For any nonzero $q$ with $|q|<1$,

$\operatorname{trdeg}_\mathbb{Q}(q, j(q)) \geq 1$

meaning $j(q)$ and $q$ are never both algebraic. In contrast to Schneider's theorem, CM points do not provide an exception to the Fourier-based transcendence statement.

This distinction underscores the exceptional transcendental and algebraic structure of automorphic invariants at CM points: they are the only loci where the “function and value” perspective (parameter and modular invariant) are both algebraic.

2. Algebraicity and Factorization of Modular Invariants at CM Points

At CM points, modular functions and their special values generate maximal abelian extensions (class fields) of imaginary quadratic fields. This underpins the Hilbert class field theory and the theory of singular moduli:

• j-Invariant: For a CM point $\tau$ of discriminant $D<0$, $j(\tau)$ is an algebraic integer generating the ring class field $H$ of $\mathbb{Q}(\sqrt{D})$. The conjugates of $j(\tau)$ over $\mathbb{Q}$ are given by $j(\omega_Q)$, where $Q$ ranges over reduced quadratic forms of discriminant $D$. This is algorithmically accessible via the extended form class group and Shimura reciprocity (Koo et al., 17 Jun 2025).
• Other Invariants: For classical functions such as the $\lambda$-invariant, explicit Borcherds product realizations and big CM value formulas afford norm and unit results at CM points. For example, for $\tau \in \mathrm{CM}(d,2)$,
  • $\lambda(\tau)$ is an algebraic integer.
  • $\lambda(\tau)$ or $1-\lambda(\tau)$ may be a unit, depending on $d \bmod 8$.
  • Differences $\lambda(\tau_1)-\lambda(\tau_2)$ have explicit norm factorizations in ray class fields (Yang et al., 2018).
• Higher Green Functions: The Gross–Zagier conjecture and subsequent work show that special values of Green functions at pairs of CM points (even for higher Green functions) are always logarithms of algebraic numbers in suitable class fields (Viazovska, 2011, Li, 2021).

3. Special Values at CM Points in Higher Dimensions

When generalizing to genus 2 (principally polarized abelian surfaces), the behavior diverges sharply from the genus 1 case:

• Igusa Invariants: The natural analogues of $j$ are the absolute Igusa invariants $j_1, j_2, j_3$ (ratios of Clebsch–Igusa invariants). Each has a multi-variable Fourier expansion in $(q_1, q_2, q_3)$. The naive expectation by analogy with Stéphanois is

$$\operatorname{trdeg}_\mathbb{Q} (q_1, q_2, q_3, j_1(q), j_2(q), j_3(q)) \geq 3.$$

However, for CM points, positive-dimensional exceptional subvarieties appear (Gómez, 13 May 2025): - Humbert Surfaces: Classical linear relations (Humbert singular relations) correspond to special divisors (Humbert surfaces). Every CM principally polarized abelian surface lies on such a Humbert surface, leading to loci where the transcendence degree in the expected set drops below 3. - Conditional on Schanuel's Conjecture: For a simple CM surface (quartic CM field) or non-simple (product of two nonisogenous CM elliptic curves), the minimal transcendence degree $\delta_m = 2$; for decomposable isotypic products $E^2$, $\delta_m = 1$. Unconditionally, $\delta_m \leq 2$ (Gómez, 13 May 2025).

4. Algebraic Independence and Galois Structure of Special Values

A broad theme is the algebraic independence of special values of modular, automorphic, or Drinfeld modular forms at CM points:

• Drinfeld Modular Forms (characteristic $p$): For nearly holomorphic Drinfeld modular forms, the special values at CM points with non-isomorphic endomorphism rings are algebraically independent over $\overline{K}$ (Chen et al., 2023). For higher-rank meromorphic Drinfeld modular forms, similar algebraic independence statements hold for ratios of periods at distinct CM points, generalizing earlier results of Chang (Chen et al., 4 Dec 2025).
• Motivic Galois Perspective: The transcendence degree computations exploit motivic Galois group calculations (Papanikolas’s theory), showing independence of periods and quasi-periods for CM Drinfeld modules of linearly disjoint endomorphism fields (Chen et al., 4 Dec 2025).
• Shimura Varieties and Theta Lifts: For orthogonal or unitary Shimura varieties, values of automorphic Green functions at CM cycles or points are logarithms of algebraic numbers in class fields. The difference of such values at two CM points can be explicitly written (for suitable functions) in terms of periods and Galois conjugates, with actions computable via explicit algorithms (Li, 2021, Koo et al., 17 Jun 2025).
• Fourier Coefficients of Harmonic Maass Forms: CM values of Borcherds products and their connection with the holomorphic parts of weight 1 harmonic weak Maass forms encode factorization data and Galois actions at CM points (Ehlen, 2012, Ehlen, 2017).

5. Explicit Special Value Formulas and Applications

Several families of special values at CM points admit explicit, closed-form formulas:

• Eta-Quotients and Ramanujan Gamma Values: Evaluations of Goswami–Sun $q$-series at CM values $q = e^{2\pi i \tau}$, with $\tau$ CM, are algebraic multiples of powers of Gamma values (e.g., $\Gamma(1/4)$), via Chowla–Selberg formula and the modular properties of eta-functions (Dawsey et al., 2018).
• Hypergeometric and Modular Functions: On Shimura curves, values of {}$_2F_1$ hypergeometric functions at points associated with CM discriminants are explicit rational multiples of CM periods (Chowla–Selberg periods), yielding closed-form values in terms of fundamental periods of the underlying curves (Yang, 2015).
• Green Functions and Borcherds Products: For higher genus or orthogonal type Shimura varieties, the values of Green functions or Borcherds products at CM points contribute directly to arithmetic intersection theory, special cycles, and explicit class field theory. The CM values can be linked to periods or cycles, and their factorization determined (Li, 2018, Yu, 2017).
• Arithmetic Units and Field Generators: CM values of modular invariants (e.g., $\lambda$ or $j$) generate unit groups in abelian extensions and allow for explicit construction of class fields (Yang et al., 2018).

6. Practical Algorithms and Computational Aspects

The explicit determination of special values and their Galois conjugates at CM points is algorithmically tractable:

Object	Galois Orbits/Conjugates	Computational Method
$j(\tau)$, classical	Roots of reduced quadratic forms	Form class groups, explicit transforms
Modular functions (gen.)	Conjugates via form class group action	Extended form class group, Shimura reciprocity
Drinfeld invariants	Cyclotomic/ray-class extensions	Class field theory, periods, t-motives

Key references for these computational techniques include (Koo et al., 17 Jun 2025) and the classical theory exposed in singular moduli calculations.

7. Broader Arithmetic and Geometric Context

Special values at CM points unify aspects of transcendence theory, the arithmetic of modular forms, and explicit class field theory:

• From Number Fields to Function Fields: Many results have analogues in positive characteristic, notably for Drinfeld modules and modular forms (Chen et al., 2023, Chen et al., 4 Dec 2025).
• Link to Motivic Theory: Period relations and motivic Galois groups provide a conceptual explanation for the independence results.
• Implications for Cryptography: In higher genus, special CM values of modular invariants (e.g., Rosenhain invariants) are central to constructing cryptographically secure genus-2 curves, with formulas controlling denominators and field of definition (Yu, 2017).
• Nonabelian and $p$-adic Extensions: In the setting of $L$-values, special values at CM points appear in congruence formulas underpinning nonabelian Iwasawa theory, with explicit Eisenstein series and $p$-adic measure interpretations (Bouganis, 2010).

• (Gómez, 13 May 2025) D. Gomez, On the CM exception to a generalization of the Stéphanois theorem
• (Koo et al., 17 Jun 2025) A simplified algorithmic realization of Galois actions on special values of modular functions
• (Chen et al., 2023) Nearly holomorphic Drinfeld modular forms and their special values at CM points
• (Chen et al., 4 Dec 2025) On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points
• (Yang et al., 2018) The lambda invariants at CM points
• (Viazovska, 2011) CM Values of Higher Green's Functions
• (Li, 2021) Algebraicity of higher Green functions at a CM point
• (Dawsey et al., 2018) CM Evaluations of the Goswami-Sun Series
• (Yang, 2015) Special values of hypergeometric functions and periods of CM elliptic curves
• (Li, 2018) Average CM-values of Higher Green's Function and Factorization
• (Yu, 2017) CM Values of Green Functions Associated to Special Cycles on Shimura Varieties with Applications to Siegel 3-Fold $X_2(2)$
• (Bouganis, 2010) Non-abelian congruences between special values of $L$-functions of elliptic curves; the CM case
• (Larson et al., 2011) Integrality Properties of the CM-values of Certain Weak Maass Forms
• (Ehlen, 2012, Ehlen, 2017): Borcherds products and harmonic weak Maass forms at CM points

These works collectively establish a rich tapestry of explicit formulas, independence results, and algorithmic tools cementing the exceptional role of special values at CM points in arithmetic geometry and transcendence.