Lagmental Vicfred

Frobenius Satisfies a Quadratic Equation on an Elliptic Curve by Vicfred

tags: mathematics, elliptic curves, isogenies, moduli

Over F_q, the Frobenius endomorphism has trace a_q determined by the point count. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Objects and notation

An elliptic curve \(E/K\) is a smooth projective genus-one curve with a chosen point. In short Weierstrass form \(y^2=x^3+Ax+B\), smoothness is encoded by the discriminant.

$$ a_q=q+1-\#E(\mathbf F_q) $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \pi^2-a_q\pi+q=0\quad\text{in }\operatorname{End}(E) $$

Push the symbols

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \begin{pmatrix}0&-q\\1&a_q\end{pmatrix}\ \text{has characteristic polynomial }T^2-a_qT+q,\qquad|a_q|\le2\sqrt q $$

Structural reading

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad a_q=q+1-\#E(\mathbf F_q),\\[5pt] \mathsf{C}\;&:\quad \pi^2-a_q\pi+q=0\quad\text{in }\operatorname{End}(E). \end{aligned} $$

A hypothesis worth keeping

Short Weierstrass form requires characteristic different from \(2\) and \(3\). Torsion and isogeny formulas also change in inseparable characteristic.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \pi^2-a_q\pi+q=0\quad\text{in }\operatorname{End}(E) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Wed 26 February 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.