Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In page 109 of de Weger's paper, he says that for coprime $A, B, C$ the conductor $N$ of the Frey-Hellegouarch curve $$ E: y^2 = x(x - A)(x + B) $$ equals $N(A,B,C)$ (product of primes dividing $ABC$ without multiplicity, and where $A + B = C$) times an absolutely bounded power of $2$. Why is this the case?

Also, on page 114, he says that the conductor $N_q$ of the twisted curve

$$ E_q : qy^2 = x(x - A)(x+B) $$

where $q$ is a squarefree integer (ie the quadratic twist of $E$), is $lcm(N,q^2)$ and the difference in the power of $2$ is at most $2^8$. Why is this the case?

Thanks

share|improve this question

1 Answer 1

up vote 3 down vote accepted
  • The computation of the conductor of the Frey curve can be found in one of the early chapters (maybe the first) of Cornell--Silverman--Stevens.

  • The highest power of $2$ that can divide the conductor of an elliptic curve is $8$, if I remember correctly.

  • The formula for the conductor $N_q$ (which is valid provided that $q$ is coprime to $N$) can be checked directly from the definition of the conductor in terms of $\ell$-adic Tate modules. You can also think of it in terms of how the conductor of a newform changes when you make a twist. This is discussed in classical language in the article of Atkin and Lehner. It is also easily verified using representation-theoretic language.

share|improve this answer
    
Thanks again Professor Emerton! –  Eugene May 28 '12 at 20:54
    
Professor Emerton, do you mind if I include you in the acknowledgements for my thesis? –  Eugene May 31 '12 at 3:27
    
@Eugene: Dear Eugene, Thank you for asking; I don't mind at all. Best wishes, –  Matt E May 31 '12 at 3:49

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.