Dear Reza Zomorrodian,

I suggest you to see links mybe you can found an inspiration for solve your problem.

https://macsphere.mcmaster.ca/bitstream/11375/9044/1/fulltext.pdf

https://www.semanticscholar.org/paper/EQUATIONS-OF-RIEMANN-SURFACES-OF-GENUS-4-%2C-5-%2C-AND-Conder-Jong/aef56b682ca19dd6664f2510295de4566b59220f

https://www.researchgate.net/publication/232357212_Fermionic_center_in_the_superconformal_algebra_on_the_supertorus

https://dokumen.pub/riemannian-geometry-and-geometric-analysis-7nbsped-3319618598-9783319618593.html

https://www.researchgate.net/publication/263893575_On_the_curve_ynxmxynxmx_over_finite_fields

Best regards

Thank you Mohamed for the refences to the existing literature in those areas.

But I needed someone to do the mathematics for me if it is possible.

I started working by myself and so far, I have come up with the following genus formula on that type of Riemann surfaces, but it is not complete. it works only on certain equations x^m * y^n + x^n * Y^m = 1 with a special property on m and n.

My found formula is as follows:

g = (m-1)(m-2)/2 - (n-1)(n-2)/2 + x,

where x = 1, 2, 3, 4, ... depends on a very special property on m and n.

I check the genus of many Riemann surfaces with the above equation and in most cases x =1 works perfectly.

Any idea what type relation x must have to do with m & n?

Thank you all, I really appreciate this. I need that for something very important.

Professor R. Zomorrodian

I am also interested to find a formula for the genus g (m, n) of a Riemann Surface with equation:

Xm *Yn + Ym *Zn +Zm *Xn = 1

Some examples of the unknown formula are as follows:

g(3, 1) = 3 Klein's Surface.

g(9, 8) = 64 By myself

9(9, 7) =60 By myself

g(9, 5) =52 By myself

g(8, 2) = 31 By myself

g(26, 13) = 469 By myself

g(91, 13)= 4629 By myself

g(4, 2) = 7 Macbeath

An idea anyone if there can be made a general formula for g(m, n)?

visit this link and get the source file

https://cds.cern.ch/record/406883/files/ext-99-020.ps.gz

Hello to all researchers in automorphism Groups of Riemann surfaces of certain genus, who has worked with "MAGMA".

Is it true that MAGMA lately has some bugs and it runs much slower than a couple of months ago. I did some computation of high genus for certain high degree surfaces a couple of months ago which lasted only 6 hours and now I have been waiting for the same computation of genus of the surface:

u^91*v^416 + u^416*v^91 = 1

for almost two days and still the answer hasn't come out.

Please answer if you know anything about this.

Thank you all, I really appreciate it.