On the Argument of the Lerch, Chowla--Selberg Formula and CM Values of $\eta(\tau)$

Technical Summary: On the Argument of the Lerch, Chowla–Selberg Formula and CM Values of $\eta(\tau)$

Problem Statement
The paper addresses the explicit evaluation of the Dedekind eta function $\eta(\tau)$ at Complex Multiplication (CM) points. While the classical Lerch–Chowla–Selberg formula provides an explicit expression for the product of the moduli $|\eta(\tau)|$ over a set of CM points of a given discriminant $D$, it does not determine the argument (phase) of the individual values or the product of the values themselves without absolute values. The author seeks to provide a completely explicit formula for the product of $\eta(\tau)^4$ (normalized by the Chowla–Selberg gamma quotient) without absolute values, thereby isolating the argument of these values. Furthermore, the paper investigates the algebraic nature of the individual CM values of $\eta(\tau)$ and formulates precise conjectures regarding their minimal polynomials and degrees.

Methodology
The approach combines modular form theory, the arithmetic of binary quadratic forms, and explicit computation.

1. Modular Invariance and Representatives: The author defines a product $P(D)$ over a standard set of representatives $F(D)$ of the class group of discriminant $D$. Since the term $\Im(\tau)|\eta(\tau)|^4$ is $SL_2(\mathbb{Z})$-invariant, the modulus of $P(D)$ is known via the Chowla–Selberg formula. The challenge lies in the non-invariant term $\eta(\tau)^4$.
2. Boundary Analysis: The proof of the main theorem relies on analyzing the boundary cases of the fundamental domain where the argument of $\eta(\tau)$ is non-trivial. Specifically, the author examines quadratic forms $(a, b, c)$ where $a=c$ or $a=b$. By utilizing the transformation properties of $\eta(\tau)$ (specifically $\eta(-1/\tau)$ and $\eta(\tau+1)$), the author relates the arguments of these boundary forms to specific roots of unity and algebraic factors involving $\sqrt{D}$.
3. Arithmetic Conditions: The paper establishes precise arithmetic conditions on the divisors of $|D|$ (denoted as the set $U(D)$) that determine which boundary cases contribute to the argument. This involves detailed analysis of the 2-adic valuation of $D$ and congruence conditions modulo 16 and 32.
4. Algebraic Recognition: To study individual values, the author defines an algebraic number $\alpha(\tau)$ such that $\eta(\tau)^4$ is proportional to $\alpha(\tau)$ times the Chowla–Selberg constant. Using high-precision numerical computation and algebraic recognition algorithms (specifically algdep in Pari/GP), the author computes $\alpha(\tau)$ for a wide range of discriminants to identify patterns in their minimal polynomials and degrees.

Key Contributions and Results

1. Explicit Formula for the Argument (Theorem 2.1):
   The paper derives a closed-form expression for the product $P(D) = \prod_{\tau \in F(D)} \eta(\tau)^4 / |\eta(\tau)|^4$. The result is given as:
   $$P(D) = e^{-(i\pi/6)(|U^-(D)| - 3|U^+(D)|)} \prod_{d \in U^+(D)} \frac{d - \sqrt{D}}{d + \sqrt{D}}$$
   where $U^+(D)$ and $U^-(D)$ are specific subsets of divisors of $|D|$ determined by the congruence class of $D$ modulo 16 or 32.

   • Corollary 2.2 provides simplified results for specific cases: if $D \equiv 0 \pmod 8$, $P(D) = 1$; if $D = -p$ ($p \equiv 3 \pmod 4$) or $D = -4p$ ($p \equiv 1 \pmod 4$), $P(D) = e^{-2i\pi/12}$.

2. Algebraicity of Individual Values (Theorem 3.2):
   The paper proves that for any CM point $\tau$, the quantity $\eta(\tau)^4$ is equal to an algebraic number $\alpha(\tau)$ multiplied by the known Chowla–Selberg constant and geometric factors. Consequently, the ratio $\eta(\tau)/|\eta(\tau)|$ is an algebraic number. This is extended to general modular forms $f$ with algebraic Fourier coefficients (Proposition 3.3).

3. Conjectures on Minimal Polynomials (Conjecture 4.1):
   Based on extensive numerical data, the author proposes four conjectures regarding the algebraic number $\alpha(\tau)$ (and a normalized version $\beta(\tau)$):

   • Integrality: $a \cdot \alpha(\tau)$ and $a/\alpha(\tau)$ are algebraic integers.
   • Norm: The norm of $\alpha(\tau)$ is $\pm 1$ (likely always 1).
   • Polynomial Structure: The minimal polynomial of $a \cdot \beta(\tau)$ has a specific form $B(x^{jm})$, where the degree depends on the class number $h(D)$ and the 2-adic valuation of $h(D)$.
   • Field of Definition: $\alpha(\tau)^{\text{lcm}(6, h(D))}$ belongs to the ring class field of the quadratic order of discriminant $D$.

4. Empirical Data:
   The paper includes a large table (spanning discriminants from $-3$ to $-180$) listing the degrees of $\alpha(\tau)$, $\beta(\tau)$, and their powers. The data supports the conjecture that the degree of these algebraic numbers grows roughly proportional to $h(D)^2$, contrasting with the linear growth of degrees for modular functions like $j(\tau)$.

Significance and Claims
The author claims the primary significance of this work is the provision of a "completely explicit" formula for the argument of the Lerch–Chowla–Selberg product, removing the ambiguity of absolute values. This allows for the precise determination of the phase of $\eta(\tau)$ at CM points.

The paper modestly frames the conjectures as emerging from "a reasonable amount of data" rather than rigorous proof, noting that they "perhaps not difficult to prove using known results on CM theory." The significance of the conjectures lies in their structural description of the algebraic nature of $\eta(\tau)$ values, suggesting a deeper connection between the degrees of these values and the square of the class number, and providing an algorithmic pathway to compute explicit expressions for these values. The work serves as a computational and theoretical bridge between the classical Chowla–Selberg formula and the explicit arithmetic of individual eta values.