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Joint Linnik problems

Valentin Blomer, Farrell Brumley, Maksym Radiwiłł

Abstract

We prove a conjecture of Michel--Venkatesh on joinings of distinct Linnik problems, in the setting of simultaneous quaternionic embeddings of imaginary quadratic fields having sufficiently many small split primes. This splitting condition is known to hold for all but $O((\log\log X)^{1 + o(1)})$ discriminants up to $X$. We also treat a non-equivariant form of this conjecture proposed by Aka--Einsiedler--Shapira, which in particular applies to the classical Gauss construction joining Linnik points on the sphere with CM points on the modular surface.

Joint Linnik problems

Abstract

We prove a conjecture of Michel--Venkatesh on joinings of distinct Linnik problems, in the setting of simultaneous quaternionic embeddings of imaginary quadratic fields having sufficiently many small split primes. This splitting condition is known to hold for all but discriminants up to . We also treat a non-equivariant form of this conjecture proposed by Aka--Einsiedler--Shapira, which in particular applies to the classical Gauss construction joining Linnik points on the sphere with CM points on the modular surface.
Paper Structure (38 sections, 29 theorems, 229 equations)

This paper contains 38 sections, 29 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Examples, historical context, and the main theorem
  3. Orthogonal complement of Gauß
  4. Simultaneous equidistribution
  5. General context
  6. Strategy, outline, and auxiliary results
  7. Reduction to number theoretic statements
  8. Weyl criterion
  9. Reduction to short arithmetic expressions
  10. Ensuring growth
  11. Twisted moments of $L$-functions
  12. Proof of Theorem \ref{['thm12']}: cuspidal case
  13. Proof of Theorem \ref{['thm12']}: Eisenstein case
  14. Proof of Theorem \ref{['thm:unconditional']} : the equivariant case
  15. Heuristic
  16. ...and 23 more sections

Key Result

Theorem 1.1

The simultaneous equidistribution conjecture of Michel--Venkatesh, as well as a quadratic variant due to Aka--Einsiedler--Shapira, hold for quaternionic varieties over $\mathbb{Q}$ at almost maximal level, for discriminants $-D$ such that the quadratic Dirichlet $L$-function $L(s, \chi_{-D})$ has no

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 3.1
  • Remark 3.3
  • Theorem 3.4
  • ...and 49 more