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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.
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
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
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
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