Let $Y(N),$ for $N\ge3,$ be the modular curve over $\mathbb C$ with respect to the $\Gamma(N)$-level structure. Let $f:E\to Y(N)$ be the universal elliptic curve. Then $R^1f_*\mathbb Q$ is a semisimple local system on $Y(N).$ Is it irreducible? And how to prove it or where can I find a reference? Thank you.
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$\begingroup$ Dear shenghao, can you please elaborate how you can construct from $f: E \rightarrow Y(N)$ a representation of $\Gamma(N)$. What is $R^1$? Can you prove that the representation has a congruence subgroup in its kernel? I am really interested in this construction. $\endgroup$– Marc PalmCommented Jun 5, 2011 at 12:04
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3$\begingroup$ The representation is just given by the inclusion $\Gamma(N)\subseteq \mathrm{SL}(\mathbb Q)$. As a representation of $\mathrm{SL}(\mathbb Q)$ $\mathbb Q^2$ is irreducible and as $\Gamma(N)$ is Zariski dense in $\mathrm{SL}(\mathbb Q)$ so is the restriction to it. $\endgroup$– Torsten EkedahlCommented Jun 5, 2011 at 13:33
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1$\begingroup$ Dear pm, you may find a definition of right derived functors of pushforward in any reference on sheaves (e.g., Kashiwara-Schapira). $\endgroup$– S. Carnahan ♦Commented Jun 5, 2011 at 15:34
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$\begingroup$ Thanks Torsten! This answers my question. Now I remember that Deligne used a similar argument for Sato-Tate of elliptic curves in Weil II. @pm: Thanks for the question. As Scott said, $R^1f_*$ is the derived functor of $f_*,$ and in this case $R^1f_*\mathbb Q$ is the family of $H^1$ of the fiber elliptic curves of $f,$ and this local system corresponds to a rep. of $\pi_1(Y(N))=\Gamma(N).$ $\endgroup$– shenghaoCommented Jun 6, 2011 at 10:14
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