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one time my sister was paying the tab at a bar and she briefly made flirty eye contact with someone on the other side of the room. she wanted to go chat them up after paying the bill.

except there was no other side of the room. the wall behind the bar was mirrored, and she just saw her own reflection like "who's she??👀 "

anyway i'm partway through making fun of her for the rest of my life, and that process requires telling this story publicly every so often. thx for reading! have a good day!

Now that I have only 8h of classes each week, having 4h in the same single day feels quite overwhelming

Using the 10 minute pause in-between classes to power nap 👌

Thinking about going to the European Congress of Mathematics 8ecm.si/ but also don't want to spend too much money :(

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Another one. Here, flocks of arrowheads are tracing out the Herschel graph.
The Herschel graph is something I keep returning to, because our building is named after its inventor.

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Kudos to the person that first discovered EC Division polynomials (en.wikipedia.org/wiki/Division).

They tell us so much but the computations are tedious.

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The p-power Frobenius has trivial kernel as well, so it is purely inseparable... And that's essentially the whole story: every isogeny α can be decomposed as
α = α'∘πⁿ,
where α' is a separable isogeny. We define the separability degree of α as the degree of α'.

Again, if we did this with function fields, the separability/inseparability degrees would match this facts 👍.

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The Frobenius endomorphism has trivial kernel, and fixes exactly the Fq-rational points.

This leads to the concepts of separable and (purely) inseparable isogenies.

An isogeny is said to be separable if the cardinal of its kernel is equal to its degree, and inseparable if it has a smaller kernel (it can't be bigger). And it is purely inseparable if it has trivial kernel.

Then, for fields of positive characteristic one has Frobenius endomorphisms. These work the same as for extensions L/Fq, raising each coordinate to the q-th power. It is sometimes written as
π(x:y:z)=(x^q : y^q : z^q),
and so it has degree q (as probably was expected).

The composition of isogenies is obviously an isogeny, and using the function field definition of degree one has deg(φ∘ψ) = degφ × degψ.

From here we define the degree of the isogeny as the maximum degree between p and q.

If we did this more abstractly, we would define the degree of φ as the extension degree of the function fields of E and E' (φ:E→E'), i.e. [K(E) : φ*K(E')].

Of course, it turns out that both are equivalent.

Currently studying the basics of elliptic curve isogenies. I feel like summarizing some parts here, so there you go...

An isogeny is just a morphism between two elliptic curves, but because an EC is both a group and an algebraic curve, it has to respect both structures.

The most basic property combines both aspects, namely the fact that every isogeny can be written as

φ(x,y) = (p(x)/q(x), y s(x)/t(x)),

with p,q,s,t polynomials in x.

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I just discovered this :gauss: emoji, I'm so happy right now

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One thing I really like about mathematics is that I can grab a book from 50 years ago and not be worried that it's too old to be relevant.

Hey, Javascript programmers, you ever get tired of worrying that the code you haven't maintained for two years being outdated and unusable, try mathematics instead. Our theorems from 2000 years ago are still as good as new.

Even our API is stable. The last big changes in notation were about 200 years ago. Read some :gauss: or :euler: originals, no problem.

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