[Paper Review] An algorithm for constructing certain differential operators in positive characteristic

Alberto F. Boix, Alessandro De Stefani+1 more|arXiv (Cornell University)|Mar 4, 2015

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper presents an algorithm to compute a differential operator δ in positive characteristic p that maps 1/f to 1/f^p for any non-zero polynomial f in a polynomial ring over a finite field. The key contribution is a characterization of supersingular elliptic curves via the level of the associated differential operator: supersingular curves correspond exactly to those with level two, while ordinary curves have level one.

ABSTRACT

Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $δ$ which raises $1/f$ to its $p$th power. For some specific families of polynomials, we also study the level of such a differential operator $δ$, i.e., the least integer $e$ such that $δ$ is $R^{p^e}$-linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator.

Motivation & Objective

• To develop an algorithm for constructing differential operators δ in positive characteristic p such that δ(1/f) = 1/f^p for a given non-zero polynomial f.
• To determine the level e of such a differential operator δ, defined as the smallest integer such that δ is R^{p^e}-linear.
• To characterize the level of differential operators associated with specific families of polynomials, including monomials and elliptic curve polynomials.
• To establish a precise link between the level of the differential operator and the supersingularity of elliptic curves over finite fields.
• To implement the algorithm in Macaulay2 for computational verification and application.

Proposed method

• The algorithm constructs δ ∈ D_R such that δ(1/f) = 1/f^p by leveraging the structure of differential operators in positive characteristic and the Frobenius map.
• It uses a filtration of the ring of differential operators D_R = ⋃_e End_{R^{p^e}}(R), where R^{p^e} is the subring of p^e-th powers.
• For monomials f = x_1^{a_1}⋯x_d^{a_d} with 0 ≤ a_i < p, the method explicitly computes δ_α that acts as Kronecker delta on monomials of degree < p.
• The algorithm checks whether f^{p^e - 1} lies in the p^e-th symbolic power of an ideal I_e(f^{p^e - 1}), using a condition-checking subroutine to verify monomial degrees.
• It employs a recursive application of differential actions via the DifferentialAction function, which computes the action of a differential operator on a given element.
• The implementation in Macaulay2 uses symbolic computation to handle multivariate polynomials, differential operators, and Frobenius powers.

Experimental results

Research questions

• RQ1 Can we algorithmically construct a differential operator δ such that δ(1/f) = 1/f^p in positive characteristic?
• RQ2 What is the level e of such a differential operator δ, i.e., the smallest e for which δ is R^{p^e}-linear?
• RQ3 How does the level of the differential operator associated with a monomial f relate to its exponents and the characteristic p?
• RQ4 For elliptic curves defined over F_p, how does the level of the associated differential operator relate to the curve’s supersingularity?
• RQ5 Can we characterize supersingular and ordinary elliptic curves via the level of the differential operator associated with their defining polynomials?

Key findings

• The paper provides a constructive algorithm to compute a differential operator δ ∈ D_R such that δ(1/f) = 1/f^p for any non-zero f in a polynomial ring over a finite field of characteristic p.
• For monomials f = x_1^{a_1}⋯x_d^{a_d} with 0 ≤ a_i < p, the level of f is exactly e = ⌈log_p(max{a_i + 1})⌉, and the associated differential operator δ is explicitly described.
• The ideal I_e(f^{p^e - 1}) of p^e-th roots of f^{p^e - 1} is characterized as the smallest ideal J such that f^{p^e - 1} ∈ J^{[p^e]}.
• For elliptic curves over F_p, the associated differential operator has level one if and only if the curve is ordinary, and level two if and only if the curve is supersingular.
• The algorithm is implemented in Macaulay2 and verified through computational examples, including monomials and elliptic curve polynomials.
• The characterization of supersingular elliptic curves via operator level provides a new algebraic-geometric invariant in positive characteristic.