Arithmetic Geometry of CM Abelian Surfaces and the Distribution of L-function Zeros

Executive Summary

The research presented in arXiv:1806.03826 provides a comprehensive framework for classifying principal polarizations on abelian surfaces that are isogenous to the product of an elliptic curve with itself. By focusing on curves with Complex Multiplication (CM), the paper establishes an explicit bridge between the algebraic geometry of genus 2 curves and the arithmetic of imaginary quadratic orders. The core insight lies in the systematic enumeration of discriminants and class numbers, which are fundamentally governed by the distribution of zeros of Dirichlet and Hecke L-functions. This approach is promising because it transforms abstract questions about the Riemann Hypothesis (RH) into concrete, checkable properties of hyperelliptic curves and their Jacobian varieties.

Introduction

The Riemann Hypothesis remains the most significant challenge in number theory, asserting that the non-trivial zeros of the zeta function lie on the critical line. Extending this to L-functions of algebraic varieties—the Generalized Riemann Hypothesis (GRH)—requires a deep understanding of how arithmetic invariants like class numbers and discriminants behave. The source paper arXiv:1806.03826 addresses this by constructing explicit models for genus 2 curves whose Jacobians possess CM by a specific order. These constructions are not merely theoretical; they provide a data-rich environment for studying the analytic behavior of L-functions associated with quartic fields and their quadratic subfields.

Mathematical Background

The study centers on an imaginary quadratic field K with an order O of discriminant Δ < 0. The class group Cl(O) and its size, the class number h, determine the number of distinct CM elliptic curves. For an abelian surface A = E x E, the paper classifies principal polarizations using explicit Hermitian matrices. A key matrix form used in the analysis is:

M = ((n + Norm(alpha)/n, (x+y)alpha), ((x+y)conj(alpha), x^2 n + y^2 Norm(alpha)/n))

The distribution of these polarizations is linked to the Analytic Class Number Formula, which relates h to the value of the L-function at s = 1. Under the Riemann Hypothesis, the growth of h is strictly controlled, and the absence of "Siegel zeros" near the edge of the critical strip ensures that the discriminants listed in arXiv:1806.03826 (such as -163, -67, and -43) follow a predictable density.

Main Technical Analysis

Spectral Properties and Zero Distribution

The Jacobian of a genus 2 curve is an abelian surface. For the curves identified in the paper, the L-function factors into a product of Hecke L-functions. The Katz-Sarnak philosophy suggests that the zeros of these L-functions should follow specific statistical distributions. By providing explicit hyperelliptic equations, the paper allows for the numerical verification of these distributions. Examples include:

• Δ = -8: y^2 = x^5 + x
• Δ = -11: y^2 = 2x^6 + 11x^3 - 22
• Δ = -163: y^2 = x^6 + 1635420402x^5 + 5379x^4 + 533147051052x^3 - 876777x^2 + 43450919081142x - 4330747

The coefficients of these curves, particularly for Δ = -163, are governed by modular forms evaluated at CM points. The magnitude of these coefficients provides a direct measure of the arithmetic complexity, which is constrained by the vertical bias of zeros on the critical line.

Algebraic Structures and L-functions

The paper categorizes discriminants by their class group structure, specifically focusing on cases where the class group is a power of 2. This is significant for the Riemann Hypothesis because such structures are rare and often indicate the presence of specific symmetries in the associated L-functions. The Birch and Swinnerton-Dyer (BSD) Conjecture, a relative of RH, suggests that the rank of these Jacobians is determined by the behavior of the L-function at the central point s = 1. The explicit equations in arXiv:1806.03826 enable the calculation of root numbers, which dictate the parity of the rank and the vanishing of L-zeros.

Novel Research Pathways

1. Effective Lower Bounds for CM Discriminants

Using the explicit curve equations, researchers can investigate the Deuring-Heilbronn phenomenon. By analyzing the coefficients of the hyperelliptic models, one can derive effective lower bounds for the discriminants of quartic CM fields. This pathway seeks to prove that as the discriminant increases, the class number must also increase, a property that is effectively computable only if the Generalized Riemann Hypothesis holds.

2. Sato-Tate Distributions for CM Families

The curves provided in the source paper allow for a large-scale study of Frobenius traces. By counting points on these curves over finite fields F_p, one can compare the distribution of traces to the predicted Sato-Tate measure. The rate at which these distributions converge is directly linked to the error term in the Prime Number Theorem for curves, which is equivalent to the Riemann Hypothesis for the associated L-functions.

Computational Implementation

(* Section: Class Number and Zeta Zero Correlation *)
(* Purpose: Demonstrate the link between discriminants in 1806.03826 and L-function zeros *)

Module[{discs, hValues, zeros, plot},
  (* Highlighted discriminants from the paper *)
  discs = {-8, -11, -19, -43, -67, -163};
  
  (* Calculate approximate class numbers using the Analytic Class Number Formula *)
  hValues = Table[
    {d, Round[(Sqrt[Abs[d]] / Pi) * NSum[KroneckerSymbol[d, n]/n, {n, 1, 10^5}]]},
    {d, discs}
  ];
  
  Print["Discriminant vs Computed Class Number:", Grid[hValues]];

  (* Compute the first 20 zeros of the Riemann Zeta Function on the critical line *)
  zeros = Table[Im[ZetaZero[n]], {n, 1, 20}];
  
  (* Plot the magnitude of the Zeta function along the critical line *)
  plot = Plot[Abs[Zeta[1/2 + I*t]], {t, 0, 50}, 
    PlotRange -> {0, 5}, 
    Filling -> Axis, 
    PlotLabel -> "Zeta Magnitude on Critical Line",
    AxesLabel -> {"t", "|Zeta(1/2 + it)|"}];
    
  Print[plot];
  Print["First 5 Zeta Zeros (Imaginary Parts):", Take[zeros, 5]]
]

Conclusions

The explicit classification of CM abelian surfaces in arXiv:1806.03826 serves as a bridge between the geometry of curves and the analytic properties of L-functions. The most promising avenue for further research is the utilization of these explicit hyperelliptic models to test zero-spacing conjectures and effective class number bounds. By grounding abstract analytic theories in the concrete algebraic data of genus 2 curves, we move closer to a unified understanding of the Riemann Hypothesis and its generalizations across higher-dimensional varieties.

References

Source Paper: arXiv:1806.03826

• Katz, N. M., and Sarnak, P. (1999). Random Matrices, Frobenius Eigenvalues, and Conjugacy Classes.
• Silverman, J. H. (2009). The Arithmetic of Elliptic Curves.