Congruences for traces of singular moduli and Hurwitz - Kronecker class numbers

1. Problem Statement

The paper addresses the study of congruences for traces of singular moduli, denoted as $B(D, d)$. These integers represent the traces of values of the modular $j$-invariant at imaginary quadratic arguments (singular moduli).

• Context: Introduced by Zagier, these traces are simultaneously values of modular functions and Fourier coefficients of weight $3/2$ modular forms.
• Existing Knowledge: Previous research (e.g., by Ahlgren, Ono, Edixhoven) established congruences of the form $B(D, p^{2n}d) \equiv 0 \pmod{p^n}$ under specific splitting conditions. Other results (Bruinier-Ono) linked these traces to Hurwitz-Kronecker class numbers $H(d)$ modulo small primes.
• The Gap: While specific congruences were known for small primes (like $p=2, 3, 5, 7$) or specific discriminants, a unified framework explaining the rate of convergence of these congruences and their relationship to class numbers for general primes (specifically $p=11$) was lacking. The paper aims to provide a general mechanism to derive these congruences and quantify the modulus of convergence.

2. Methodology

The author employs a sophisticated approach combining modular forms of half-integral weight, $p$-adic analysis, and the theory of overconvergent modular forms.

• Generating Functions: The traces $B(D, d)$ are treated as Fourier coefficients of weight $3/2$ modular forms $g_D$ on $\Gamma_0(4)$ satisfying Kohnen's plus-space condition.
• The $U$-Operator: The core mechanism involves the repeated action of the $U_p$ operator (which maps $\sum a(n)q^n$ to $\sum a(pn)q^n$) on these forms. The paper investigates the $p$-adic limits of sequences like $g_D | U^{2n}$.
• Modified Eisenstein Series ($\tilde{H}$): The author constructs a specific weight $3/2$ holomorphic Eisenstein series $\tilde{H}$ on $\Gamma_0(4p)$. This series is a modification of Zagier's non-holomorphic Eisenstein series, designed to be a modular form with specific eigen-properties under the Hecke operators and the $U$-operator.
• Overconvergent Forms: The proofs rely heavily on the theory of overconvergent half-integral weight modular forms (developed by Ramsey). By viewing these forms in a Banach space of overconvergent forms, the author utilizes the spectral properties of the $U_p$ operator (specifically, the existence of a discrete spectrum and the projection onto zero-slope subspaces) to prove the existence of $p$-adic limits.
• Shimura Correspondence: For the specific application to elliptic curves, the paper uses the Shimura correspondence to relate weight $3/2$ forms to weight $2$ cusp forms (associated with elliptic curves like $X_0(11)$).

3. Key Contributions and Results

A. General Theorem on $p$-adic Limits (Theorem 2)

The paper establishes a general result regarding the $p$-adic limit of the sequence $g_D | U^{2n}$.

• Let $D = p^{2t}D_0$ with $p^2 \nmid D_0$ and $\epsilon = -(\frac{D_0}{p})$.
• The limit $\tilde{g}_D = \lim_{n \to \infty} (-1)^n \epsilon^{(\epsilon+1)/2} g_D | U^{2n}$ exists.
• Classification of the Limit:
  • If $\epsilon = 0$, the limit is 0.
  • If $D$ is not a perfect square and $\epsilon \neq 0$, the limit lies in the subspace $S_\epsilon$ (cusp forms with specific character properties).
  • If $D$ is a perfect square, the limit is related to the modified Eisenstein series $\tilde{H}$ and a cusp form.

B. Quantification of Convergence Rates (Theorem 3)

The paper quantifies the "unexpectedly fast" convergence of these congruences for small primes ($p \in \{3, 5, 7, 11, 13\}$).

• It defines a parameter $s$ (the smallest non-zero slope in the spectrum of the $U_p$ operator) for each prime.
• Result: The $p$-adic valuation of the difference between the truncated series and the limit grows linearly with $n$:
  $$\text{ord}_p( (g_D - \tilde{g}_D) | U^{2n} ) = O(s \cdot n)$$
• For example, for $p=3$, $s=3$, implying a convergence rate of $3^{3n}$, which explains numerical observations that were previously unexplained.

C. Application to Elliptic Curves and Class Numbers (Theorem 1)

A major application is derived for the elliptic curve $E \cong X_0(11)$ and the prime $p=11$.

• Condition: Let $-d$ be a fundamental discriminant with $(\frac{-d}{11}) = -1$.
• Equivalence: The vanishing of the central $L$-value $L(E_{-d}, 1) = 0$ is equivalent to a specific relation between the trace of singular moduli and the Hurwitz-Kronecker class number $H(d)$.
• Formula:
  $$H(d) = -\frac{5}{24} \lim_{n \to \infty} B(11^{2n}d)$$
  (where the limit is $11$-adic).
• Congruence: This yields the explicit congruence:
  $$\text{If } L(E_{-d}, 1) = 0, \text{ then } H(d) \equiv 6 B(d) \pmod{11}$$
  This provides a concrete explanation for Mazur's earlier qualitative results regarding the finiteness of the Mordell-Weil group.

4. Significance

• Unification: The paper unifies previously disparate congruence results (those of Ahlgren-Ono and Bruinier-Ono) under a single framework involving the action of the $U$-operator on weight $3/2$ forms.
• Explanation of Phenomena: It provides a theoretical explanation for why congruences for small primes often hold modulo higher powers of $p$ than expected (e.g., $p^{sn}$ instead of $p^n$), linking this to the spectral slopes of the $U$-operator on overconvergent forms.
• New Tools: By utilizing overconvergent modular forms and the specific construction of $\tilde{H}$, the author offers a method to derive congruences for primes where previous geometric or combinatorial methods failed to generalize.
• Arithmetic Geometry: The results deepen the understanding of the relationship between the arithmetic of elliptic curves (specifically the vanishing of $L$-values) and the arithmetic of quadratic fields (class numbers), extending the scope of the Birch and Swinnerton-Dyer conjecture's implications.

In summary, Guerzhoy's work transforms the study of singular moduli traces from a collection of isolated numerical coincidences into a systematic theory based on the spectral properties of half-integral weight modular forms in the $p$-adic setting.