(1)ERRATA: ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS PETE L

ERRATA:

ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS

PETE L. CLARK AND JOHN VOIGHT

This note gives some errata for the articleAlgebraic curves uniformized by con- gruence subgroups of triangle groups [1]. Thanks to Juanita Duque Rosero and Michael Schein.

(1) Before Theorem C: replace “n-6abc” with “ncoprime to 6abc”.

(2) Below equation (5.21): replace “Letnbe the prime ofE=F(a, b, c) below N” with “Letn=ZE∩Nbe the prime ofEbelow N”.

(3) Remark 5.24: sign errors crept into the second generator. The correct orthogonal elements forB are

1,2δa−λ2a,(λ²_2a−4)δb+ (λ2aλ2b+ 2λ2c)δa−(λ²_2aλ2b+λ2aλ2c−2λ2b), not

1,2δ_a−λ_2a,(λ²_2a−4)δ_b+ (λ_2aλ_2b+ 2λ_2c)δ_a−(λ²_2aλ_2b−λ_2aλ_2c+ 2λ_2b).

In the corrected basis, we obtain the presentation:

B '

λ²_2a−4,−(λ²_2a−4)β F

'

λ²_2a−4, β F

whena6=∞.

(4) Proof of Theorem 9.1: the unipotent case should be allowed in the proof, whens=∞. Replace the start of the middle paragraph by:

Next, we show that orders ofg1, g2, g3area^], b^], c^]. Lets=a, b, c and write g for the corresponding element. We have trφ(δs) ≡

±λ2s (modP). If g = 1, then the image is commutative, and this possibility was just ruled out. Ifs=∞, then sinceg6= 1 and λ_∞= 2 we must havegunipotent, sog has orderp=s^]. ...

References

[1] Pete L. Clark and John Voight, Algebraic curves uniformized by congruence subgroups of triangle groups, Trans. Amer. Math. Soc.371(2019), no. 1, 33–82.

Date: March 16, 2022.

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