Questions tagged [arithmetic-geometry]

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

36 questions
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what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
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The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
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Does ABC implies Fermat's last theorem?

I read from the newspaper that Mochizuki's proof of the ABC conjecture implies the Fermat's last theorem. Is it true? I think it implies the proof only for large enough exponents?
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The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
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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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For any elliptic curve $E/\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is ordinary.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and fix a Weierstrass equation for $E$ having coefficients in $\mathbb{Z}$. How do I see that there are infinitely many primes $p \in \mathbb{Z}$...
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Arithmetic-geometric series which includes Fibonacci

In connection with a problem I'm solving, I seem to be getting the series $$S = 4 \cdot \frac{F_1}{4}+5 \cdot \frac{F_2}{8}+6 \cdot \frac{F_3}{16}+7 \cdot \frac{F_4}{32}+ \cdots$$ where $F_i$ are the ...
Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$
Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn't find it in Diamond and Shurman, and neither could I find an explicit formula with a ...