At 23:46 22/08/2002, Franktal Gallery wrote:

Could you please explain the "semistable" concept ?

Oof. Where to start (and where to finish?) See http://mathworld.wolfram.com/Semistable - following is a paraphrase. An elliptic curve can be written as: y^2 + a[1]xy + a[3]y = x^3 + a[2]x^2 + a[4]x + z[6] (That is to say, the set of solutions to this equation is known an elliptic curve). Often it can be rewritten (with a suitable change of variables) as y^2 = x^3 + Ax + B in which case the discriminant is given by -16(4A^3+27B^2) (So called, because questions about the curve's overall appearance is captured in the question of whether this value is negative, zero or positive - analogous to the way that "eccentricity" does for conics, or b^2-4ac does for quadratics.) An elliptic curve is said to be "semistable" when (to quote mathworld:) "When a prime l divides the discriminant of a elliptic curve E, two or all three roots of E become congruent (mod l). An elliptic curve is semistable if, for all such primes l, only two roots become congruent mod l (with more complicated definitions for p = 2 or 3)." The bit about "...more complicated definitions" covers the situation when the elliptic curve _can't_ be written in the simpler form. "congruent (mod l)" of course means that the roots have the same remainder on division by l. "Prime" has the usual meaning. "Roots" of course, refers to the roots of the equation. As noted previously, the Taniyama-Shimura conjecture was able to dispense with the requirement of semistability for modularity to hold. If you think "semistable" is abstruse, consider that the more formal statement of the TSC mentions "conductors", "Hecke operators" and assigns "weight" to modular forms. If this seems like an awfully small subject, remember that x and y aren't necessarily drawn from the field of real numbers: any field would do. Since all we're doing with them is adding and multiplying them together, anything that can be added and multiplied together is fair game. Morgan L. Owens "Mathematics. It all hangs together. Probably in an infinitely complex fashion."