# Tagged Questions

A subject that lies at the intersection of algebraic geometry and number theory dealing with varieties, the Mordell conjecture, Arakelov theory, and elliptic curves.

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### Intuitively, what is the height of a point on an abelian variety?

I have been reading through Silverman's classic text on elliptic curves and I just can't seem to wrap my head around the height functions. It just kind of shows up. What exactly does the height ...
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### Elliptic curves over Spec Z

I want to show that there are only finitely many elliptic curves over Spec $\mathbf Z$ without appealing to Siegel's theorem or Shafarevich' theorem. Firstly, I think (but I am not sure) that such an ...
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### What is the intuition behind the Fontaine-Mazur Conjecture?

The Fontaine-Mazur conjecture (over $\textbf{Q}$ for simplicity) says that a (continuous irreducible) Galois representation  \rho: \text{Gal}(\overline{\textbf{Q}}/\textbf{Q}) \to GL_n(\overline{\...
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### Translation for EGA/SGA

People often recommend Grothendieck's EGA (Elements de Geometrie Algebrique) and SGA (seminaire de geometrie algebrique) as a good reference for learning arithmetic geometry. However, as the title ...
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### Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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### How to Compute Genus

How to compute the genus of $\{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3$? We know that the genus of $\{X^4+Y^4+Z^4=0\} \subset \mathbb{P}^3$ is 3 because the degree is 4. ...
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