[Paper Review] Uniformization, Unipotent Flows and the Riemann Hypothesis

Sergio L. Cacciatori, Matteo A. Cardella|arXiv (Cornell University)|Feb 6, 2011

Algebraic Geometry and Number Theory39 references3 citations

TL;DR

This paper establishes a deep connection between unipotent flows on the moduli space of genus $g$ principally polarized abelian varieties and the Riemann Hypothesis. By analyzing the asymptotic behavior of $\Gamma_g$-automorphic forms averaged along unipotent directions using Iwasawa coordinates and the Rankin-Selberg method, it proves that the error term in equidistribution is controlled by the supremum of the real parts of the non-trivial zeros of the Riemann zeta function, thus linking ergodic theory to number theory via the Riemann Hypothesis.

ABSTRACT

We prove equidistribution of certain multidimensional unipotent flows in the moduli space of genus $g$ principally polarized abelian varieties (ppav). This is done by studying asymptotics of $\pmbΓ_{g} \sim Sp(2g,\mathbb{Z})$-automorphic forms averaged along unipotent flows, toward the codimension-one component of the boundary of the ppav moduli space. We prove a link between the error estimate and the Riemann hypothesis. Further, we prove $\pmbΓ_{g - r}$ modularity of the function obtained by iterating the unipotent average process $r$ times. This shows uniformization of modular integrals of automorphic functions via unipotent flows.

Motivation & Objective

• To establish equidistribution of multidimensional unipotent flows in the moduli space of genus $g$ principally polarized abelian varieties ($\mathcal{A}_g$).
• To analyze the asymptotic behavior of $\Gamma_g$-automorphic forms under unipotent averaging toward the codimension-one boundary of $\mathcal{A}_g$.
• To demonstrate that the error term in the equidistribution estimate is directly linked to the supremum of the real parts of the non-trivial zeros of the Riemann zeta function.
• To prove that iterated unipotent averaging preserves modularity under $\Gamma_{g-r}$, generalizing Zagier's result for $SL(2,\mathbb{Z})$.
• To unify the uniformization of modular integrals through unipotent flow dynamics and automorphic forms.

Proposed method

• Utilizes Iwasawa decomposition of $Sp(2g,\mathbb{R})$ to parametrize the Siegel upper half-space $\mathcal{H}_g$ in terms of unipotent, diagonal, and orthogonal components.
• Applies the Rankin-Selberg method to study $L^2$-norms and spectral properties of automorphic forms under unipotent averaging.
• Employs a $(g-r)$-corank block decomposition of $\tau_g$ to isolate unipotent directions corresponding to $2g-1$ real parameters $(w_{11}, \underline{w}, \underline{u})$.
• Derives the Riemannian metric on $\mathcal{H}_g$ in Iwasawa coordinates and computes the inverse metric components $G^{v_iv_i}$ to determine the Laplacian operator $\Delta$.
• Computes the action of the Laplacian on the Eisenstein series $\boldsymbol{E}_{g,1}(\tau,s)$, showing it satisfies a hypergeometric-type eigenvalue equation.
• Uses the invariance of the Laplacian under $\Gamma_g$ to extend eigenvalue results to the full automorphic form space.

Experimental results

Research questions

• RQ1 How does the error term in the equidistribution of unipotent flows on $\mathcal{A}_g$ relate to the Riemann Hypothesis?
• RQ2 Can the modularity of unipotent averages of automorphic forms be preserved under successive averaging over unipotent subgroups?
• RQ3 What is the spectral behavior of the Laplacian on $\mathcal{H}_g$ in Iwasawa coordinates, and how does it relate to automorphic $L$-functions?
• RQ4 How does the Iwasawa parametrization facilitate the analysis of unipotent flows and their equidistribution?
• RQ5 To what extent does the unipotent averaging process lead to uniformization of modular integrals via dynamical systems?

Key findings

• The error estimate in the equidistribution of unipotent flows is controlled by $\Theta = \sup\{\Re(\rho) \mid \zeta^*(\rho) = 0\}$, the supremum of the real parts of the non-trivial zeros of the Riemann zeta function.
• The unipotent average $\langle f \rangle_{v_1}(\tau_{g-1})$ of a $\Gamma_g$-invariant automorphic function $f$ is invariant under the action of $\Gamma_{g-1}$, proving $\Gamma_{g-1}$-modularity of the averaged function.
• The Laplacian $\Delta$ acts on the Eisenstein series $\boldsymbol{E}_{g,1}(\tau,s)$ as $\Delta \boldsymbol{E}_{g,1}(\tau,s) = 2^{\frac{g-1}{g+1}} s(g-s) \boldsymbol{E}_{g,1}(\tau,s)$, establishing a spectral eigenvalue equation.
• The metric component $G^{v_iv_i} = 2^{\frac{g-1}{g+1}} v_i^2$ is derived from the Iwasawa parametrization and the determinant of the Riemannian metric on $\mathcal{H}_g$.
• The method generalizes Zagier’s result on long horocycle averages for $SL(2,\mathbb{Z})$ to higher-rank symplectic groups and higher-genus moduli spaces.
• The unipotent averaging process iteratively yields $\Gamma_{g-r}$-modular functions, demonstrating a uniformization of modular integrals via dynamical flow.