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I will answer Q2: N=2: Denote by $Y^1(2)$ the moduli of elliptic curves with point of order 2 and fixed invariant differential. It is not hard to show that $Y^1(2) = \mathrm{Spec}\, \mathbb{Z}[\frac12]... • 11.4k 2 votes ### Reference for universal elliptic curves To complete Lennart Meier's nice answer, Baaziz has computed explicit equations for the universal elliptic curve over$Y_1(N)$up to$N=51$. The method used and the data up to$N=20$can be found in ... • 19.7k 4 votes Accepted ### Reference for universal elliptic curves For any$n\geq 1$, one can define a functor$\mathcal{M}_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a ... • 11.4k 7 votes Accepted ### Non-modular elliptic curves It is a widely believed conjecture that all elliptic curves, over any number field$K$, are modular (in the sense that there exists an automorphic representation [*]$\pi$of$\operatorname{GL}_2 / K$... • 31.5k 2 votes Accepted ### Torsors over elliptic curves Indeed this map is an isomorphism. There is a diagram of five-term exact sequences, arising from the Leray spectral sequence for the maps$E\to \text{Spec}(k), C\to \text{Spec}(k)$, from$0\to H^1(k, ...
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I can prove the generic rank over $\overline{\mathbb{Q}}$ is 4, however, I can just show the generic rank over $\mathbb{Q}$ is at most 2. In the following, we will first find the structure of singular ...