# The Epsilon Conjecture In Fermat's Last Theorem

1. Jun 19, 2011

### Kevin_Axion

By supposing there is a solution to Fermat's Last Theorem then according to Frye you can create an elliptic curve that isn't modular. Taniyama-Shimura says that all elliptic curves are modular, so in proving that that Frye curve is not modular which was done by Ribet don't you disprove the Taniyama-Shimura conjecture and Fermat's Last Theorem?

I'm really confused.

2. Jun 19, 2011

### Dschumanji

I think the confusion stems from the idea that the Frey Curve must exist since it could be described and studied. However, its existence depends on whether or not Fermat's Last Theorem is actually true or false. If we do not know whether Fermat's Last Theorem is actually true or false then all we can do is assume it is either true or false which means the existence of the Frey Curve is uncertain. If the existence of the Frey Curve is uncertain, does it make sense to conclude with certainty that the Taniyama-Shimura Conjecture is false if all we do is assume Fermat's Last Theorem is false?

Last edited: Jun 19, 2011
3. Jun 19, 2011

### Kevin_Axion

Thanks for your answer, I found out my problem elsewhere right after I posted this.