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

Abstract

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 nth (Fourier) coefficient of one was always larger than or equal to the nth coefficient of the other, then the maximal orders are actually the same. We prove that this is the case.

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

“…The study of sign changes of Fourier coefficients of a single cusp form has been the focus of much recent study (cf. [1,3,4,8,7,13,10]) due to their various number theoretic applications (see for instance [5]) and his long history which goes back at least to Siegel [17]. The question of simultaneous sign change of Fourier coefficients of modular forms was first considered by Kohnen and Sengupta in [9].…”