Serre's theorem on Galois Representations

peccavi_2006 · **Posted:** Sat, 24 Dec 2011 01:24:36 UTC

Serre's Theorem states (according to this textbook):

Let C be an elliptic curve given by a Weierstrass equation with rational coefficients. Assume that C does not have complex multiplication. Then there is an integer N $\geq$ 1, depending on the curve C, such that for any relatively prime integer n the Galois representation $$\rho_{n} : Gal( \mathbb{Q}(C[n])/ \mathbb{Q}) \rightarrow GL_{2}(\frac{\mathbb{Z}}{n \mathbb{Z}})$ is an isomorphism

What the text doesn't explain is how N depends on C, only that it does. So effectively the text is telling me that, for some mystical number N, $\rho_{n}$ is an isomorphism, but finding N is down to black magic.

So I was wondering: if it isn't too difficult, how can I find the integer N? Though saying that, if it requires pages and pages of incredibly advanced mathematics, I'll just assume the reason it was only mentioned in passing was because it is beyond anything I should be able to do.

Thank you!

Shadow · **Posted:** Sat, 24 Dec 2011 01:39:15 UTC

peccavi_2006 wrote:

Serre's Theorem states (according to this textbook):

Let C be an elliptic curve given by a Weierstrass equation with rational coefficients. Assume that C does not have complex multiplication. Then there is an integer N $\geq$ 1, depending on the curve C, such that for any relatively prime integer n the Galois representation $$\rho_{n} : Gal( \mathbb{Q}(C[n])/ \mathbb{Q}) \rightarrow GL_{2}(\frac{\mathbb{Z}}{n \mathbb{Z}})$ is an isomorphism

What the text doesn't explain is how N depends on C, only that it does. So effectively the text is telling me that, for some mystical number N, $\rho_{n}$ is an isomorphism, but finding N is down to black magic.

So I was wondering: if it isn't too difficult, how can I find the integer N? Though saying that, if it requires pages and pages of incredibly advanced mathematics, I'll just assume the reason it was only mentioned in passing was because it is beyond anything I should be able to do.

Thank you!

What makes you think you can find N with any degree of ease?

peccavi_2006 · **Posted:** Sat, 24 Dec 2011 01:54:03 UTC

As with most things in maths, absolutely nothing. I was just wondering whether or not it was particularly difficult, and therefore whether or not it was worth going into in great detail or similarly simply mention it in passing

outermeasure · **Posted:** Sat, 24 Dec 2011 10:25:43 UTC

Isn't this one of those problems where, if you assume the generalised Riemann hypothesis, you get a reasonable polynomial (in the height of C) bound (for all elliptic curves defined over any number field K, not just $\mathbb{Q}$ ), otherwise the current best is an absolutely silly exponential bound (with a practically useless exponent like 10^something huge)?

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Serre's theorem on Galois Representations

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