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Frey Curve

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Let a^p+b^p=c^p be a primitive nonzero solution to Fermat's last theorem, with p>=5. After permuting and changing signs, take b even and a=-1 (mod 4). The corresponding Frey curve is

y^2=x(x-a^p)(x+b^p).
(1)

Ribet (1990a) showed that such curves cannot be modular. The Taniyama-Shimura conjecture would therefore rule them out, proving Fermat's last theorem. Frey curves are semistable. Direct calculation from the displayed Weierstrass form gives the elliptic discriminant

Delta=2^4a^(2p)b^(2p)c^(2p).
(2)

Its minimal discriminant is

Delta_(min)=2^(-8)a^(2p)b^(2p)c^(2p).
(3)

Its elliptic curve conductor is the radical of abc (Sutherland 2023),

N_E=product_(l|abc)l=rad(abc).
(4)

The j-invariant is

j=(2^8(a^(2p)+b^(2p)+a^pb^p)^3)/(a^(2p)b^(2p)c^(2p))=(2^8(c^(2p)-a^pb^p)^3)/((abc)^(2p)).
(5)

See also

Elliptic Curve, Elliptic Curve Conductor, Fermat's Last Theorem, j-Invariant, Taniyama-Shimura Conjecture, Weierstrass Form

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References

Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. Math. Monthly 101, 3-14, 1994.Gouvêa, F. Q. "A Marvelous Proof." Amer. Math. Monthly 101, 203-222, 1994.Ribet, K. A. "From the Taniyama-Shimura Conjecture to Fermat's Last Theorem." Ann. Fac. Sci. Toulouse Math. 11, 116-139, 1990a.Ribet, K. A. "On Modular Representations of Gal(Q^_/Q) Arising from Modular Forms." Invent. Math. 100, 431-476, 1990b.Sutherland, A. V. "Why the Frey-Hellegouarch Curve Should Not Exist." Lecture 25 in 18.783 Elliptic Curves. Cambridge, MA: Massachusetts Institute of Technology, 2023. https://math.mit.edu/classes/18.783/2023/LectureNotes25.pdf.

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Frey Curve

Cite this as:

Weisstein, Eric W. "Frey Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FreyCurve.html

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