# 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.

450 questions
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### What are some topics of advanced number theory every young geometers should know? (soft question)

By "advanced number theory", I mean topics like arithmetic/Diophantine geometry, modular/automorphic forms and Shimura varieties. I'm interested in derived/non-commutative algebraic geometry, some ...
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### Tate curve and cusps

I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve? Naive googling seems to suggest that level structures on the Tate curve (up to ...
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### A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
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### find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
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### What is the motivation behind the Hilbert Symbol?

The Hilbert Symbol it superficially similar to the Legendre Symbol: it measures whether or not solutions to some polynomial exist. In the case of the Legendre Symbol it was clear for me that it is a ...
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### Bad reduction at prime numbers for the elliptic curve $y^2+y=x^3-x^2+2x-2$

Consider the elliptic curve $$E:y^2+y=x^3-x^2+2x-2.$$ My goal is to compute the conductor of the elliptic curve, the example is from https://planetmath.org/conductorofanellipticcurve. My problem isn'...
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### Reduction map of elliptic curves

$\newcommand{\b}[1]{\overline{#1}}$ $\newcommand{\m}[1]{\,(\text{mod }#1)}$ $\def\red{\tilde{E}_{\mathfrak{p}}}$ $\def\p{\mathfrak{p}}$ $\def\at{_{\mathfrak{p}}}$ $\def\comp{K_{\mathfrak{p}}}$ I'm ...
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### Torsors under elliptic curves splitting over the same fields

I have a question somewhat related to my last question. Suppose $C$ and $C'$ are two genus $1$ curves (smooth, projective, geom conn.) over a perfect field $k$ with no $k$-rational points and that $C$ ...
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### genus of normalization of stable curve

Let $X$ be a stable curve of genus $g$ over the field $k$, i.e., a $k$-rational point of the Deligne-Mumford stack $M_g$. What is the genus of the normalization of $X$? Does it depend on the number ...
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### Twists of rational points

Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero. The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection ...
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### Qing Liu exercise 4.1.3: Non-affine Dedekind scheme

Exercise 4.1.3 of Qing Liu's Algebraic Geometry and Arithmetic Curves asks of us to prove, given a normal Noetherian local scheme $X$ of dimension $2$ with closed point $s$, that $X \backslash \{ s \}$...
### 4-Torsion on an elliptic curve embedded in $\mathbb{P}^{3}$ as the intersection of two quadrics
Let $X$ be a smooth projective variety over the complex numbers. Let $$c_1 : \text{Pic}(X)\to H^2(X,\mathbf{Z}(1))$$ Its kernel is the subgroup of homologically trivial divisor classes on $X$. How ...