Entwicklungen zur Transformation fünfter und siebenter Ordnung einiger specieller automorpher Functionen

Modular curves, invariant theory and $E_8$

“…In the present paper, we will give a different relation which connects E 8 with the modular curve X (13). We will show that the E 8 root lattice can be constructed from the modular curve X(13) by the invariant theory for the simple group PSL (2,13).…”

“…Let us begin with the invariant theory for PSL (2,13). Recall that the six-dimensional representation (the Weil representation) of the finite simple group PSL (2,13) of order 1092, which acts on the five-dimensional projective space P 5 = {(z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ) : z i ∈ C (i = 1, 2, 3, 4, 5, 6)}. This representation is defined over the cyclotomic field Q(e 2πi 13 ).…”

“…Recall that the six-dimensional representation (the Weil representation) of the finite simple group PSL (2,13) of order 1092, which acts on the five-dimensional projective space P 5 = {(z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ) : z i ∈ C (i = 1, 2, 3, 4, 5, 6)}. This representation is defined over the cyclotomic field Q(e 2πi 13 ). Put where ζ = exp(2πi/13).…”

Differential Equations for Invariant Curves Under Klein's Simple Group of Order 168

“…Then, from Lemma 1.1, π −1 (0) must be p 3 or p 3 . On the other hand, it is known in [Fr,p. 381] that p 3 = {F 4 = F 14 = 0}.…”

Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group, and Some Other Examples

“…1 By work of Shimura [Sh1], based on the classical work of Fricke [F1,F2], the group Γ (1)/{±1} ⊂ SL 2 (R) is the (2, 3, 7) triangle group (the group generated by products of pairs of reflections in the sides of a hyperbolic triangle of angles π/2, π/3, π/7). Hence X (1) is a curve of genus 0 with elliptic points of orders 2, 3, 7.…”