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Jacobian graphs

Arthur Forey, Javier Fresán, Emmanuel Kowalski, Yuval Wigderson

Abstract

We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the geometric properties of generalized jacobians of curves and on general equidistribution theorems for character sums over finite fields.

Jacobian graphs

Abstract

We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the geometric properties of generalized jacobians of curves and on general equidistribution theorems for character sums over finite fields.
Paper Structure (6 sections, 13 theorems, 99 equations)

This paper contains 6 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Sum graphs: basic properties and spectrum
  3. Jacobian graphs: basic properties
  4. Jacobian graphs: semicircular spectrum
  5. Complements on generalized jacobians
  6. Non-technical overview of generalized jacobians

Key Result

Theorem 1.2

With notation as above, assume that $k$ is a finite field. For $n\geqslant 1$, denote by $k_n$ the extension of degree $n$ of $k$ in a fixed algebraic closure $\bar{k}$ of $k$. Denote further Assume that $\dim(J_{\mathfrak{m}})=2$ and the genus of $C$ is either $1$ or $2$.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • ...and 26 more