02 / Maths Lab
Bounds & Invariants.
What changes as the genus grows? Explore two quantitative results, with their hypotheses alongside.
A positive lower bound / Theorem 1.1
The Zhang–Kawazumi invariant
For every compact connected Riemann surface X of genus g ≥ 2:
In genus two this gives φ(X) > ½. The argument uses the canonical Arakelov metric and positivity of a bilinear form.
For hyperelliptic X, the stronger bound is φ(X) > (g/2)(Hg−1).
Compare the two bounds
Blue, solid: general. Red, dashed: hyperelliptic. Curves show lower bounds, not values of φ for particular surfaces.
Looper · Silverman · Wilms / Theorem 1.1
How does the bound grow with the genus?
Let K = k(B), where k is algebraically closed and B is a smooth projective connected curve. For a smooth projective geometrically connected, non-isotrivial curve X/K of genus g ≥ 2 and any degree-one divisor D,
c(2) = 76, c(3) = 231;
Blue: c(g). Dashed red: 16g² + 32g + 124. These are upper bounds, not numbers of torsion points known to occur.