[Paper Review] Zeta functions of Ramanujan graphs and modular forms

Kennichi Sugiyama|arXiv (Cornell University)|May 10, 2019

Graph theory and applications3 citations

TL;DR

This paper establishes a deep connection between Ihara's zeta function of Ramanujan graphs and the Hasse-Weil zeta function of modular curves $X_0(N)$ over finite fields. By constructing a $(p+1)$-regular Ramanujan graph $G_N(p)$ from Brandt matrices associated to supersingular elliptic curves, the authors show that the zeta functions are reciprocal and derive a congruence for Fourier coefficients of weight 2 Hecke eigenforms, linking graph complexity to modular forms via the class number formula analogues.

ABSTRACT

We will investigate the relationship between Ihara's zeta functions of Ramanujan graphs and Hasse-Weil's congruent zeta functions of modular curves. As an application we will describe the limit value of Hasse-Weil's congruent zeta functions in terms of the corresponding Ramanujan graphs. Moreover we will show a congruence relation of the Fourier coefficients of a normalized Hecke eigenform of weight 2.

Motivation & Objective

• To explore the relationship between Ihara's zeta function of Ramanujan graphs and Hasse-Weil zeta functions of modular curves over finite fields.
• To construct a family of Ramanujan graphs $G_N(p)$ from Brandt matrices of supersingular elliptic curves when $N \equiv 1 \pmod{12}$.
• To establish a reciprocal relation between the zeta functions of $G_N(p)$ and $X_0(N)_\mathbb{F_p}$, generalizing class number formula analogues.
• To derive a congruence relation for the Fourier coefficients of normalized Hecke eigenforms of weight 2 on $\Gamma_0(N)$.

Proposed method

• Construct a graph $G_N(p)$ using the adjacency matrix derived from the Brandt matrix $B(p)$ of Hecke operator $T_p$ acting on supersingular elliptic curves over $\overline{\mathbb{F}}_N$.
• Prove that $G_N(p)$ is a connected $(p+1)$-regular Ramanujan graph when $N \equiv 1 \pmod{12}$, using spectral bounds from the Eichler-Shimura relation and Weil conjectures.
• Use the Grothendieck-Lefschetz trace formula to express the Hasse-Weil zeta function of $X_0(N)_\mathbb{F_p}$ in terms of $l$-adic étale cohomology and Frobenius action.
• Establish reciprocity between Ihara's zeta function $Z(G_N(p);t)$ and Hasse-Weil zeta function $W(X_0(N)_\mathbb{F_p};t)$ via determinant identities involving eigenvalues of $T_p$.
• Relate the complexity $\tau(G_N(p))$ of the graph to the limit of the Hasse-Weil zeta function at $t=1$, yielding a formula analogous to the class number formula.
• Derive a congruence for the sum $\mu_N(p) = \sum_{i=1}^{n-1} a_p(f_i)$, showing it is divisible by $n = (N-1)/12$ for primes $p$ with $p+1 \equiv 0 \pmod{n}$.

Experimental results

Research questions

• RQ1 How are Ihara's zeta function of Ramanujan graphs and Hasse-Weil zeta functions of modular curves related over finite fields?
• RQ2 Can the zeta function of a Ramanujan graph be used to express or recover the Hasse-Weil zeta function of a modular curve $X_0(N)$?
• RQ3 What is the arithmetic significance of the complexity $\tau(G_N(p))$ of the constructed Ramanujan graph $G_N(p)$?
• RQ4 Does the sum of Fourier coefficients $a_p(f_i)$ of weight 2 Hecke eigenforms on $\Gamma_0(N)$ satisfy a congruence modulo $n = (N-1)/12$?
• RQ5 Can the spectral properties of the graph adjacency matrix $B(p)$ be used to derive arithmetic invariants of modular forms?

Key findings

• The Hasse-Weil zeta function $W(X_0(N)_\mathbb{F_p};t)$ and Ihara's zeta function $Z(G_N(p);t)$ are reciprocal: $W \cdot Z = \frac{1}{(1-t)^2(1-pt)^2(1-t^2)^{n(p-1)/2}}$.
• The limit $\lim_{t\to 1} (t-1)W(X_0(N)_\mathbb{F_p};t)$ equals $\frac{n \tau(G_N(p))}{p-1}$, linking graph complexity to zeta function residue.
• The graph $G_N(p)$ is a connected $(p+1)$-regular Ramanujan graph that is not bipartite, with eigenvalues bounded by $|\lambda| \leq 2\sqrt{p}$.
• The Fourier coefficients $a_p(f_i)$ of normalized Hecke eigenforms satisfy $\sum_{i=1}^{n-1} a_p(f_i) \equiv 0 \pmod{n}$ for primes $p$ with $p+1 \equiv 0 \pmod{n}$.
• The complexity $\tau(G_N(p))$ satisfies the bounds $\frac{(√p-1)^{2(n-1)}}{n} \leq \tau(G_N(p)) \leq \frac{(√p+1)^{2(n-1)}}{n}$.
• Numerical tables confirm that $\mu_N(p) = \sum a_p(f_i)$ is divisible by $n$ for $N=37, 61, 73$, with explicit computations showing $\mu_N(p) \in n\mathbb{Z}$.