Joint Linnik problems

TL;DR

This paper proves a conjecture related to joinings of Linnik problems involving quaternionic embeddings and small split primes, extending to classical cases like Gauss's construction linking Linnik points and CM points.

Contribution

It establishes the conjecture of Michel--Venkatesh in a new setting and addresses a non-equivariant version relevant to classical number theory constructions.

Findings

Proved Michel--Venkatesh conjecture for quaternionic embeddings.

Extended results to classical Gauss construction and CM points.

Identified splitting conditions holding for most discriminants.

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.