passport representatives

[ciaran-schembri]

The passport representatives s_1,s_2,s_3 of https://beta.lmfdb.org/Belyi/6T3/%5B6%2C2%2C2%5D/6/222/2211/g0/a do not satisfy s_3s_2s_1=1. We have s_2s_3s_1 =1 but the function BelyiMap([s_1,s_3,s_2]) produces the error that it has failed a sanity check, but (s_2s_3s_1)^-1 = 1 and BelyiMap([s_1^-1, s_2^-1, s_3^-1]) works fine.

[jvoight]

Please help:

sigma := [Sym(6) | (1,2,3,4,5,6), (1,4)(2,3)(5,6), (1,5)(2,4)]; sigma[3]sigma[2]sigma[1]; Id($) BelyiMap(sigma); Chebyshev spherical Belyi map ...verified! Curve over Rational Field defined by 0 -1/(16x^6 - 24x^4 + 9*x^2 - 1)

Seems OK to me.

[jvoight]

There is an issue with a different triple in the passport.

sigma := [Sym(6) | (1, 2)(4, 5), (1, 4)(2, 6)(3, 5), (1, 6, 2, 4, 3, 5) ]; sigma[3]sigma[2]sigma[1]; Id($) BelyiMap(sigma); Chebyshev spherical Belyi map Local Sanity Check Failed for p=9721: sigma = [ (1, 2)(4, 5), (1, 4)(2, 6)(3, 5), (1, 6, 2, 4, 3, 5) ]. supp(phi) = [ Place at (0 : 1), Place at (7086 : 1), Place at (2635 : 1), Place at (9720 : 1), Place at (4861 : 1), Place at (4860 : 1), Place at (1 : 1) ] [ 2, 2, 2, -1, -2, -2, -1 ]. supp(phi-1) = [ Place at (1 : 0), Place at (9720 : 1), Place at (4861 : 1), Place at (4860 : 1), Place at (1 : 1) ] [ 6, -1, -2, -2, -1 ]. Curve over Rational Field defined by 0 (16x^6 - 24x^4 + 9x^2)/(16x^6 - 24x^4 + 9x^2 - 1) Runtime error: Failed sanity check, so the Belyi map is wrong. This shouldn't happen.

The sanity check fails but not because the map is wrong: it looks like it wants the ramification at a different place, so the usual S_3 issue.

[michaelmusty]

Yeah it looks like I implemented the local version of the sanity check without a lax (up to S_3 action) option. I see 3 options:

1. redo the hardcoded spherical map in nonhyperbolic.m to match the triple.
2. add ability to test lax isomorphism in sanity_check_local.m
3. for spherical maps it should be OK to use "regular" (non-local) sanity check which has optional lax := true argument.

Mike

On Tue, Apr 21, 2020 at 1:15 PM John Voight notifications@github.com wrote:

There is an issue with a different triple in the passport.

sigma := [Sym(6) | (1, 2)(4, 5), (1, 4)(2, 6)(3, 5), (1, 6, 2, 4, 3, 5) ]; sigma[3]

sigma[2]sigma[1]; Id($) BelyiMap(sigma); Chebyshev spherical Belyi map Local Sanity Check Failed for p=9721: sigma = [ (1, 2)(4, 5), (1, 4)(2, 6)(3, 5), (1, 6, 2, 4, 3, 5) ]. supp(phi) = [ Place at (0 : 1), Place at (7086 : 1), Place at (2635 : 1), Place at (9720 : 1), Place at (4861 : 1), Place at (4860 : 1), Place at (1 : 1) ] [ 2, 2, 2, -1, -2, -2, -1 ]. supp(phi-1) = [ Place at (1 : 0), Place at (9720 : 1), Place at (4861 : 1), Place at (4860 : 1), Place at (1 : 1) ] [ 6, -1, -2, -2, -1 ]. Curve over Rational Field defined by 0 (16x^6 - 24x^4 + 9x^2)/(16x^6 - 24x^4 + 9x^2

• 1) Runtime error: Failed sanity check, so the Belyi map is wrong. This shouldn't happen.

The sanity check fails but not because the map is wrong: it looks like it wants the ramification at a different place, so the usual S_3 issue.

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[ciaran-schembri]

Yes sorry I meant a different triple in the passport. The examples I came across where sigma[3]sigma[2]sigma[1] is not the identity but BelyiMap() works are:

s1:=[ Sym(3) | (1, 3, 2), (2, 3), (1, 2) ];

&Reverse(s1); (1, 2, 3) BelyiMap(s1); Curve over Rational Field defined by 0 2/(4x^3 - 3x + 1) s2:=[ Sym(6) | (1, 3, 2)(4, 6, 5), (1, 4)(2, 6)(3, 5), (1, 5)(2, 4)(3, 6) ]; &Reverse(s2); (1, 2, 3)(4, 5, 6) BelyiMap(s2); Curve over Rational Field defined by 0 -4x^3/(x^6 - 2x^3 + 1) s3:=[ Sym(6) | (1, 6, 2, 4, 3, 5), (1, 4)(2, 6)(3, 5), (1, 2)(4, 5) ]; &Reverse(s3); (1, 2, 3)(4, 5, 6) BelyiMap(s3); Curve over Rational Field defined by 0 -1/(16x^6 - 24x^4 + 9x^2 - 1) s4:=[ Sym(12) | (1, 9, 2, 7, 3, 8)(4, 12, 5, 10, 6, 11), (1, 4)(2, 6)(3, 5)(7, 10)(8, 12)(9, 11), (1, 11)(2, 10)(3, 12)(4, 8)(5, 7)(6, 9) ]; &Reverse(s4); (1, 2, 3)(4, 5, 6)(7, 8, 9)(10, 11, 12) BelyiMap(s4); Curve over Rational Field defined by 0 -4x^6/(x^12 - 2*x^6 + 1)