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mostly corrections per WP:MOS and WP:MOSMATH; note that TeX in section headings fails to appear in the table of contents, so more works is needed |
→Additional Case: l = 2: another such correction |
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If <math>q</math> turns out not to be a square modulo <math>l</math> or if the equation does not hold for any of <math>w</math> and <math>-w</math>, our assumption that <math>(x^{q^{2}}, y^{q^{2}}) = +\bar{q}(x, y)</math> is false, thus <math>(x^{q^{2}}, y^{q^{2}}) = - \bar{q}(x, y)</math>. The characteristic equation gives <math>t_l=0</math>.
===Additional
If you recall, our initial considerations omit the case of <math>l = 2</math>.
Since we assume <math>q</math> to be odd, <math>q + 1 - t \equiv t \pmod 2</math> and in particular, <math>t_{2} \equiv 0 \pmod 2</math> if and only if <math>E(\mathbb{F}_{q})</math> has an element of order 2. By definition of addition in the group, any element of order 2 must be of the form <math>(x_{0}, 0)</math>. Thus <math>t_{2} \equiv 0 \pmod 2</math> if and only if the polynomial <math>x^{3} + Ax + B</math> has a root in <math>\mathbb{F}_{q}</math>, if and only if <math>\gcd(x^{q}-x, x^{3} + Ax + B)\neq 1</math>.
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