arXiv subjects

Sedric Nkotto Nkung Assong

Publications and source records attributed to Sedric Nkotto Nkung Assong.

2 recordsLinked to original sources

arXiv2020

Orders occurring as endomorphism ring of a Drinfeld module in some isogeny classes of Drinfeld modules of higher ranks

The question we propose to answer throughout this paper is the following: Given an isogeny class of Drinfeld modules over a finite field, what are the orders of the corresponding endomorphism algebra (which is an isogeny invariant) that occur as endomorphism ring of a Drinfeld module in that isogeny class? It is worth mentioning that this question is different from the ones investigated by the authors Kuhn, Pink in [6] and Garai, Papikian in [3]. The former authors rather provided an answer to the question, given a Drinfeld module ϕ, how does one efficiently compute the endomorphism ring of ϕ? The importance of our question resides in the fact that it might be very helpful to better understand isogeny graphs of Drinfeld modules of higher rank (r > 2) and may be reopen the debate concerning the application to isogeny-based cryptography. We answer that question for the case whereby the endomorphism algebra is a field by providing a necessary and sufficient condition for a given order to be the endomorphism ring of a Drinfeld module. We apply our result to rank r = 3 Drinfeld modules and explicitly compute those orders occurring as endomorphism rings of rank 3 Drinfeld modules over a finite field.

math.NT↗

arXiv2020

Explicit description of isogeny and isomorphism classes of Drinfeld modules over finite field

When travelling from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r > 2. In this paper we investigate the classes of those Drinfeld modules of higher rank r > 2. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules for any rank r > 2. our explicit description of the Weil polynomials depends heavily on Yu's classification of isogeny classes (analogue of Honda-Tate at abelian varieties). Actually Yu has also explicitly did that work for r = 2. To complete the classification, we define the new notion of fine isomorphy invariants for any rank r Drinfeld module and we prove that the fine isomorphy invariants together with J-invariants completely determine the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.

math.NT↗