# Tagged Questions

31 views

### Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
80 views

### 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 ...
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### Varieties with infinitely many topological covers of finite degree

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers. If $\dim X =1$, this holds if and only if the genus of $X$ is positive. Do we have a similar ...
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### Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
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### Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
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### Factorizing rational functions of curves

Let $f:X\to \mathbf{P}^1$ be a rational function of degree $d\geq 2$ on a curve $X$. Let $n\geq 2$ be a divisor of $d$. Does there exist a curve $Y$ with a rational function $g:Y\to \mathbf{P}^1$ of ...
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### Does de Franchis' theorem hold over any base field

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$. Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$. I know this statement holds over $k=\mathbf{C}$. In ...
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### Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
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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 can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension ...
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### Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let ...
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### Rational points on singular curves and their normalization

Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable. Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?
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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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### Why are these curves not defined over a smaller field

Let $K$ be a number field and let $\pi$ be an element in $K$. Assume that $\pi$ is not contained in a subfield of $K$. Consider the curve $y^2 = x^{2g+1}+\pi$. This defines (after homogenization and ...
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### Smallest genus example of a non planar curve

A curve is a smooth projective connected curve over an algebraically closed field. Every curve of genus 2 is planar. Also, every curve of genus 3 is planar. But what about curves of genus 4? What ...
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### Is there a fundamental domain for $\Gamma(2)$ contained in the following strip

Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane. Does it have a fundamental domain contained in the ...
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### Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I ...
Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of ...
Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers. Does this imply that $X$ itself can be ...