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Frey Curve

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Let a^p+b^p=c^p be a solution to Fermat's last theorem. Then the corresponding Frey curve is

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

Ribet (1990a) showed that such curves cannot be modular, so if the Taniyama-Shimura conjecture were true, Frey curves couldn't exist and Fermat's last theorem would follow with b even and a=-1 (mod 4). Frey curves are semistable. Invariants include the elliptic discriminant

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

The minimal discriminant is

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

the j-conductor is

N=product_(l|abc)l,
(4)

and 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, Fermat's Last Theorem, Taniyama-Shimura Conjecture

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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.

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Frey Curve

Cite this as:

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

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