John Carlos Baez on Nostr: 2024-02-26 https://youtu.be/7LGvunt8I8Q The clock arithmetic of slopes: fixing ℤ/n ...
2024-02-26
https://youtu.be/7LGvunt8I8Q
The clock arithmetic of slopes: fixing ℤ/n by introducing formal reciprocals for all elements that don't have them, or better yet: introducing formal reciprocals for all elements, but identifying them with the actual reciprocals when those exist. We don't get a ring, just a set with 2n - ϕ(n) elements, where where ϕ(n) is the number of elements of ℤ/n that don't have reciprocals, called 'Euler's totient function':
https://en.wikipedia.org/wiki/Euler%27s_totient_function
The resulting set acts like a projective line for ℤ/n: indeed it's the set of ℤ/n-points for the scheme P¹. Apparently this set can also be seen as SL(2,ℤ)/Γ₀(n) where Γ₀(n), the so-called 'Hecke congruence subgroup of level n', consists of 2×2 integer matrices where the lower left entry equals 0 mod n. For more on congruence subgroups, see:
https://math.berkeley.edu/~sander/speaking/24June2015%20UMS%20Talk.pdf
Next, compare SL(2,ℤ)/Γ₀(n) to the modular curve X₀(n) = H/Γ₀(n) where H is the hyperbolic plane. This sets up connections to the dessin d'enfant for X₀(n):
https://en.wikipedia.org/wiki/Dessin_d%27enfant
and also the Coxeter group
o--3--o--∞--o
namely PGL(2,Z). Drawing a picture of X₀(11). Kostant on Galois' last letter and PSL(2,11):
Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, http://www.ams.org/notices/199509/kostant.pdf
Erratum: 24 is *not* the largest number such that every number smaller than it and relatively prime to it is prime; that honor goes to 30:
https://mathstodon.xyz/@johncarlosbaez/112017268404654926
https://youtu.be/7LGvunt8I8Q
The clock arithmetic of slopes: fixing ℤ/n by introducing formal reciprocals for all elements that don't have them, or better yet: introducing formal reciprocals for all elements, but identifying them with the actual reciprocals when those exist. We don't get a ring, just a set with 2n - ϕ(n) elements, where where ϕ(n) is the number of elements of ℤ/n that don't have reciprocals, called 'Euler's totient function':
https://en.wikipedia.org/wiki/Euler%27s_totient_function
The resulting set acts like a projective line for ℤ/n: indeed it's the set of ℤ/n-points for the scheme P¹. Apparently this set can also be seen as SL(2,ℤ)/Γ₀(n) where Γ₀(n), the so-called 'Hecke congruence subgroup of level n', consists of 2×2 integer matrices where the lower left entry equals 0 mod n. For more on congruence subgroups, see:
https://math.berkeley.edu/~sander/speaking/24June2015%20UMS%20Talk.pdf
Next, compare SL(2,ℤ)/Γ₀(n) to the modular curve X₀(n) = H/Γ₀(n) where H is the hyperbolic plane. This sets up connections to the dessin d'enfant for X₀(n):
https://en.wikipedia.org/wiki/Dessin_d%27enfant
and also the Coxeter group
o--3--o--∞--o
namely PGL(2,Z). Drawing a picture of X₀(11). Kostant on Galois' last letter and PSL(2,11):
Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, http://www.ams.org/notices/199509/kostant.pdf
Erratum: 24 is *not* the largest number such that every number smaller than it and relatively prime to it is prime; that honor goes to 30:
https://mathstodon.xyz/@johncarlosbaez/112017268404654926