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Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is connected since it is the quotient of the upper half plane by the action of a subgroup (called $\Gamma_1(N)$) of $SL(2, \mathbb{Z})$.

Does anybody know a simple and algebraic proof of the irreducibility of this moduli space, not necessarily over $\mathbb{C}$ (but in characteristic not dividing $N$)?

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Well, not necessarily over $\mathbb{C}$, it's not necessarily irreducible. In particular "it" (which needs to be defined properly) isn't irreducible in characteristics dividing $N$. Are you only interested in characteristic zero? – Ramsey Nov 4 '11 at 15:32
"Does anybody know..." Interesting! Now Math asks questions about us instead of the other way around.... – Cam McLeman Nov 4 '11 at 15:34
Yes, I am interested in characteristic zero, and I would like to know if one can produce a simple, direct algebraic proof of this fact (without resorting to Lefschetz's principle or similar nonsense). – Math Nov 4 '11 at 15:39
I have edited my question, thank you Ramsey. – Math Nov 4 '11 at 15:42

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