Weber’s formula for the bitangents of a smooth plane quartic

Abstract

In a section of his 1876 treatise Theorie der Abelschen Functionen vom Geschlecht 3 Weber proved a formula that expresses the bitangents of a non-singular plane quartic in terms of Riemann theta constants (Thetanullwerte). The present note is devoted to a modern presentation of Weber's formula. In the end a connection with the universal bitangent matrix is also displayed.

Plane quartics over $\mathbb {Q}$ with complex multiplication

“…Indeed, normally that algorithm involves certain normalization constants k i . However, in the current situation [16,Cor.2] shows that these constants are automatically equal to 1 for our choices of a ji in (2.26), which leads to a computational speedup. Let…”

“…Its points parametrize the moduli space of smooth plane quartics with full level two structure [19]. From an Aronhold system of bitangents, one can reconstruct a plane quartic following Weber's work [62, p.93] (see also [46,16]). We take advantage here of the particular representative (a 1i , a 2i , a 3i ) of the projective points (a 1i : a 2i : a 3i ) to simplify the algorithm presented in loc.…”

“…In the subsequent Section 2.2 we indicate how these values allow us to obtain the Dixmier-Ohno invariants of our smooth plane quartic curves. This is based on formulas obtained by Weber [62,16].…”

“…(ix) r 5 ← ϑ 2 0 (0, 2τ ) × B 3 ( 1, ϑ 2 0,32,2,34,8,40,10,42 (0, 2τ )/ϑ 2 0 (0, 2τ ) ). (x) r 6 ← ϑ 2 0 (0, 2τ ) × B 3 ( 1, ϑ 2 0,16,8,24,4,20,12,28 (0, 2τ )/ϑ 2 0 (0, 2τ ) ). (xi) r 7 ← ϑ 2 0 (0, 2τ ) × B 3 ( 1, ϑ 2 0,8,16,24,32,40,48,56 (0, 2τ )/ϑ 2 0 (0, 2τ ) ).…”

“…+ 3 y 2 z 2 + 2 y z 3 + z 4 = 0 with disc X 15 = 2 −45 · 3 −27 · I min 10422787 x 3 y − 1171743077 x 3 z + 272093232 x 2 y 2 + 894539212 x 2 y z + 1758438152 x 2 z 2 − 239684773 x y 3 − 3355325973 x y 2 z + 21854285561 x y z 2 + 213880974126 x z 3 + 731104019 y 4 − 6282157788 y 3 z − 38790710054 y 2 z 2 + 288506848419 y z 3 + 1153356733618 z 4 = 0with disc X16 = 2 −45 ·3 −27 ·19 18 ·I min 27 where I min 27 = 2 5 ·3 35 ·19 7 ·37 14 ·79 14 ·13373064392147 14 . X 17 : 3717829 x 4 − 1434896 x 3 y + 19525079 x 3 z − 23623031 x 2 y 2 + 55253545 x 2 y z + 168545160 x 2 z 2 + 36024736 x y 3 − 64558785 x y 2 z + 379342822 x y z 2 − 329255097 x z 3 + 42096963 y 4 + 115245505 y 3 z − 817353798 y 2 z 2 + 498157725 y z 3 − 34967215 z 4 = 0 with disc X 17 = 2 −36 · 3 −27 · I min 27 where I min 27 = 2 20 · 3 77 · 19 7 · 1229 14 · 3913841117 14 .…”