Explanatory materials accompanying maple programs for iterated descent
on elliptic curves  y^2=x(x^2+Ax+B).
http://www.math.niu.edu/~rusin/research-math/descent/
rusin@math.niu.edu  1997/3/13.

Right now this file is mostly a place-holder. I hope to write up something
with this outline:
	1) Review of ordinary 2-descent for elliptic curves/Q with 2-torsion
	2) References to work on higher descent in that case
		Birch, Swinnerton-Dyer II Crelle 1965
		Merriman, Siksek, Smart  Acta Arith 
	3) Formulary for computer code of (2)
	4) Special note on selecting collections of divisors  N1  of a
		number  N  making  x^2-Dy^2=N1  solvable
	5) Summary of work on solutions of ax^2+by^2+cz^2=0
	6) Review of rapid local solvability tests a la Siksek
	7) Short comments on relevant factorization problems (few now)
	8) Examples of applications of the program (I've had good luck with
		my ~96 curves  y^2=x(x^2-x+(1-T^10)/5), for example)
	9) Caveats, bug reports, etc.

For now I should at least comment on #9.

One obvious bug is that the program screws up when there is more than one
2-torsion point. This is disappointing since an obvious class of examples is
the set of curves  y^2=x(x^2-D^2). I'll turn to this soon.

A major deficiency is that I haven't worked at all on a search routine for
the new quartics. (I have nothing to add to existing programs, and I'm not
even sure I'm making the new quartics in a reasonable way).

There are still two theoretical stumbling blocks to efficient performance.
(A) I haven't yet ruled out an impediment in NORMLLL() which might make it
	fail to find a point on a conic rapidly. It's never failed yet.
	I do know of two polynomial-time techniques but have coded neither.
(B) The analysis in (5) above is incomplete; see WHOSANORM().
Neither problem has caused any grief. 

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A few comments on the other proposed topics:

1) Review of ordinary 2-descent for elliptic curves/Q with 2-torsion
2) References to work on higher descent in that case
	Siksek's original letter explaining this to me is in a separate file.
3) Formulary for computer code of (2)
	Code is supposed to be more or less self-explanatory
4) Special note on selecting collections of divisors  N1  of a
		number  N  making  x^2-Dy^2=N1  solvable
	Again, there are some comments in the code.
5) Summary of work on solutions of ax^2+by^2+cz^2=0
	Some sort of a report for #5 will be fashioned from mail I sent several
	people; copies are in another file in this directory.
6) Review of rapid local solvability tests a la Siksek
7) Short comments on relevant factorization problems (few now)
8) Examples of applications of the program (I've had good luck with
		my ~96 curves  y^2=x(x^2-x+(1-T^10)/5), for example)

For #8 let me simply observe that in the space of 10 minutes or
so I brought the upper bounds for the rank of
	y^2 = x (x^2 - x + (1-(53/5)^10)/5 )
down as low as the lower bound provided by MWRANK point searches + parity
conjectures. I think this curve represents a hard test case for programs of
this type. More details on this family of curves and our success in studying
them with these procedures is in a separate file in this directory.

9) Caveats, bug reports, etc.
	Above