On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring

TL;DR

This paper proves a conjecture by Chevyrev and Galbraith, confirming that their algorithm for constructing supersingular elliptic curves always terminates by verifying a key theta series dominance conjecture.

Contribution

The paper verifies Chevyrev and Galbraith's conjecture, ensuring the algorithm's universal termination for constructing supersingular elliptic curves with a given endomorphism ring.

Findings

Confirmed the theta series dominance conjecture.

Proved the algorithm always halts.

Validated the correctness of the construction method.

Abstract

Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal order. They proved that their algorithm is correct whenever it halts, but did not show that it always terminates. They did however prove that the algorithm halts under a reasonable assumption which they conjectured to be true. It is the purpose of this paper to verify their conjecture and in turn prove that their algorithm always halts. More precisely, Chevyrev and Galbraith investigated the theta series associated with the norm maps from primitive elements of two maximal orders. They conjectured that if one of these theta series "dominated" the other in the sense that the $n$th (Fourier) coefficient of one was always larger than or equal to the $n$th coefficient of the other, then the maximal orders are actually the same. We prove that this is the case.