# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

1,348
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### Which endomorphisms of the Tate algebra are “algebraic”?

For an abelian variety $A$ over a field $k$ with characteristic different from $\ell$ and Galois group $G = Gal(\overline k/k)$, there is always an injective map of the form:
$$\mathbb Q_\ell\otimes ...

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45 views

### Eta quotient and order of a cuspidal divisor

Let $X_0(N)$ be the modular curve associated to a congruence subgroup $\Gamma_0(N)$. If $N=p$ is a prime, then there are two cusps $0$ and $\infty$ on $X_0(N)$. Suppose that $p>7$ so that the genus ...

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338 views

### Durov approach to Arakelov geometry and $\mathbb{F}_1$

Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...

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107 views

### Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields

Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...

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321 views

### Is Hilbert's tenth problem undecidable for multilinear polynomials in $\mathbb Z[x_1,\dots,x_n]$?

A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.
Is there no general purpose algorithm for ...

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57 views

### Isomorphism between 2nd symmetric product and Jacobian

Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map
$$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...

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119 views

### Is a birational morphism between normal projective varieties residually separable?

My goal is to use a "normal Bertini" theorem (see https://link.springer.com/article/10.1007%2Fs000130050213)
More specifically, let $k$ be a field (you may assume that k is infinite but it should be ...

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241 views

### Can someone help in how to approach reading Mordell-Weil Theorem for abelian varieties?

I was thinking to start reading the proof of Mordell-Weil Theorem for abelian varieties over number fields, after getting done with the proof in the case of elliptic curves over number field and I ...

**11**

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836 views

### Bounded Torsion, without Mazur’s Theorem

Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it ...

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52 views

### Can someone help in understanding this isomorphism helpful in proving finiteness of n-Selmer group?

This is from J.S.Milne's Elliptic Curves book.
We have $H^1(\mathbb{Q}, E_2) \cong (\mathbb{Q}^× / \mathbb{Q}^{×2})^2$ because $Gal(\mathbb{Q}^{al} / \mathbb{Q})$ acts trivially on $E_2(\mathbb{Q}^{...

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102 views

### Having difficulty in understanding a result that'll help in proving the finiteness of Selmer group

I'm reading and trying to understand the proof of the finiteness of n-Selmer group from J.S.Milne's Elliptic Curves book but having difficulty in understanding it. Here's a screenshot from the book-
...

**2**

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**1**answer

283 views

### Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.
Question: The arthur said that the de Rham and crystalline period rings implicitly ...

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245 views

### How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is ...

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540 views

### Classify 2-dim p-adic galois representations

Recently I have known how to classify 1-dim p adic Galois representations $\phi$. The p-adic Galois representations mean that a representation $G_K$ on a p-adic field $E$, where $K$ is also a p-adic ...

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177 views

### Complex isomorphism class of abelian varieties and $L$-functions

In his famous Mordell paper, Faltings proved that two abelian varietes $A_1, A_2$ defined over a number field $K$ are isogenous if and only if the local $L$-factors of $A_1, A_2$ are equal at every ...