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Results tagged with riemann-surfaces
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Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.
1
vote
Accepted
Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k =...
The equations $\sum_{i=1}^6 y_i^k = 0$ for $k=1,2,4,7$ do not cut out a curve because the $k=1,2,4$ equations imply the one for $k=7$. (The 7th power sum is a polynomial in the power sums of degree 1 …
- 79.9k
4
votes
Accepted
A generalization of polynomial algebra on a Riemann surface
Counterexample to the first question ("is $A$ an algebra of functions?"):
Let $M$ be a vertical strip such as {$x + iy : 0 < x < 1$},
and define $f_1,f_2$ as the restriction to $M$ of the
entire func …
- 79.9k
5
votes
Accepted
Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms
Sure: let $Y$ be your favorite hyperelliptic curve $u^2=f(t)$
with many automorphisms, and let $X$ be the curve $u^2=f(t(s))$
for some "random" rational function $f$ of degree at least $2$.
For examp …
- 79.9k
47
votes
Accepted
Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$?
Yes, this Riemann surface, call it $C: y^2 = x^{11}-11x^6-x$,
is quite special: not only does it have the maximal number of automorphisms
for a hyperelliptic surface of genus $5$, but it is a modular …
- 79.9k
11
votes
Accepted
Bounding the modular discriminant of an elliptic curve in the j-invariant
The new $\| \Delta \|$, defined as $\mathop{\rm Im}(\tau)^6$ times the absolute value of the usual modular form $\Delta$, is invariant under the full modular group $\Gamma = {\rm PSL}_2({\bf Z})$ acti …
- 79.9k