Remarks on a paper by El-Guindy and Papanikolas
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Remarks on a paper by El-Guindy and Papanikolas Takehiro Hasegawa1 Received: 26 September 2018 / Accepted: 25 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract There are similarities between Drinfeld modules and abelian varieties. The purpose of this paper is to investigate these similarities in terms of supersingularity. More specifically, we provide explicit formulas of Hasse invariants (or, equivalently, of supersingular polynomials) for Drinfeld modules, which is a continuation of the 2013 paper by Ahmad El-Guindy and Matthew A. Papanikolas. We present several supersingular Drinfeld modules as an application. Keywords Hasse invariants · Supersingular polynomials · Supersingular Drinfeld modules · Partition of a set Mathematics Subject Classification 11G09 (11F52 · 11R58 )
1 Introduction It is well-known that Drinfeld modules (more precisely, T -modules) are a functionfield analogue of abelian varieties. This was first noted by Drinfeld [3,4], and has been studied by many researchers since then (see, for example, [8,11,18]). It is thus natural to investigate similarities and differences between Drinfeld modules and abelian varieties. This paper does so in terms of supersingularity. This section presents our main theorem. First, for background, we recall some facts about elliptic curves and abelian varieties. Let p ≥ 3 be a prime number. It is known that every elliptic curve is isomorphic (over the algebraic closure F¯ p ) to an elliptic curve in the Legendre form E λ : y 2 = x(x − 1)(x − λ), where λ is an element of F¯ p with λ not 0 or 1 (see [17, Chap. III, Proposition 1.7] for a proof). Let m ≥ 1 be an integer, and let E λ (F¯ p )[m] denote the m-torsion subgroup of E λ . The curve E λ
The author was supported by JSPS KAKENHI Grant Number 19K03400.
B 1
Takehiro Hasegawa [email protected] Shiga University, Otsu, Shiga 520-0862, Japan
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T. Hasegawa
is called supersingular when E λ (F¯ p )[ p] = 0. Now, we regard λ as an indeterminate element, and define the Deuring polynomial: H p (λ) = H p(2) (λ) :=
( p−1)/2 i=0
( p − 1)/2 i
2 λi ∈ F p [λ].
This was first defined by Deuring [2]. The following fact (Ell) is known. (Ell): The elliptic curve E λ is supersingular if and only if H p (λ) = 0 (see [17, Chap. V, Theorem 4.1(b)]). Supersingular abelian varieties were studied in [12–14] and in many other papers. These varieties are given by the common roots of several polynomials. These results are generalizations of (Ell). For example, we have proven the following. (Abel): The Jacobian variety of genus-3 curve y 2 = x 8 + (2 − 4λ)x 4 + 1 is supersingular (more precisely, superspecial) if and only if λ ∈ F¯ p is a common root of H p(2) (λ) and H p(4) (λ) :=
( p−1)/4 i=0
p − 1 − ( p − 1)/4 i
( p − 1)/4 i λ i
(see [13, Theorem 4.5]). Here, for each real number x, we define x := min{n ∈ Z | x ≤ n}. We know the following function-field analogues of (Ell) mentioned above, which are results for “rank-2” Drinfeld modules. Gekeler has
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