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Non-vanishing modular forms

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]...
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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 ...
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4 votes
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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 ...
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7 votes
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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 $\operatorname{GL}_2 / K$ ...
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2 votes
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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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Mordell-Weil rank of some algebraic surface

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 ...
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