On Aug 27, 2014 6:19 AM, "David Leon Gil" <coruus@gmail.com> wrote:

>

> Cryptographers have, for quite a while, abused notation and identified=
=C2=A0different, isomorphic, elliptic forms. Well, abuse of notation strike=
s back.

>

> Part of the problem is that calling a=C2=A0transfer of=C2=A0co=C3=B6rd=
inates=C2=A0to a different form a 'co=C3=B6rdinate transform' is co=
nfusing. (I suspect that most people don't have=C2=A0sufficient topolog=
y to realize that different forms have different=C2=A0topologies.)

They all have the same topology. They don't have the sam= e geometry. The subject you are looking for is algebraic geometry. If we= 9;re going to be annoyingly precise, we should be annoyingly precise.

>

> Mathematicians have a perfectly good terminology already; why not simp=
ly adopt that, and be explicit about the maps=C2=A0used to transfer between=
=C2=A0forms?

Well, there is a slight problem: Let X be the curve. The= n X has no coordinates: rather coordinates are a choice of injective map to= P^2 or P^1 \times P^1 or something like that. So when someone says "t= ake x+486662/3 to map onto a curve of the form blah blah" they mean th= at there is a commutative triangle with X over P^2 and X over P^2, with the= image of the downwards morphisms lying in the zero locus of the given equa= tion homogenized, and the thing commutes. (It also all goes over F_p for so= me F_p)

The reason X has no coordinates is that a scheme isn't defined to in= clude a canonical choice, any more than an atlas is a choice of charts in d= ifferential geometry. The definition has already elided all coordinates.

To avoid all this verbiage, which everyone understands (with the usual c= riteria about what "everyone" means) we speak in the informal lan= guage of coordinates, and then mentally translate into the language of EGA.= I don't see what's missing from DJBs email in this regard.

Sincerely,

Watson Ladd

>

>

> On Monday, August 25, 2014, D. J. Bernstein <djb@cr.yp.to> wrote:

>>

>> All relevant coordinate systems already have standard names in the=

>> literature, and I would suggest sticking to those names whenever i=
t's

>> necessary to discuss the coordinate systems per se:

>

>

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