4
votes
1answer
489 views

Does the Mordell conjecture imply the Shafarevich conjecture

The base field is a number field. It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin). Is the converse also true? Note that both conjectures are now ...
1
vote
1answer
181 views

Is there an easier argument to prove that almost all of these curves have no semi-stable reduction

Fix a number field $K$ and a polynomial $F(x)\in K[x]$ of degree at least $4$. For a squarefree integer $d$, define the curve $X_d$ over $K$ by the equation $dy^2 = F(x)$. Note that the curves $X_d$ ...
10
votes
1answer
533 views

Which curves have stable Faltings height greater or equal to 1

Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$. Question 1. Can one classify or describe the ...
10
votes
2answers
860 views

Families of curves for which the Belyi degree can be easily bounded

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above. The modular curves $X(n)$. They are ...