Complexity of elliptic functions and the maximal compact subgroup of certain anti-affine groups.

Harry Schmidt (Oxford)

Frank Adams 1,

For any lattice in \(\mathbb{C}\) Weierstrass constructed a meromorphic function \(\wp\) that is periodic with respect to that lattice. This function satisfies the infamous differential equation \[ \dot{\wp}^2 = 4\wp^3 -g_2\wp - g_3. \]

Macintyre discovered that the real and imaginary parts of the inverse of \(\wp\) can be locally defined by a Pfaffian chain of differential equations. In this talk we will discuss how to express the graph of the real and imaginary parts of \(\wp\) as zero-sets of a fixed explicit number of functions that are part of a Pfaffian chain of fixed complexity.

Time permitting we will discuss how this can be used to establish explicit uniformity in a generalization of a result of Corvaja-Masser-Zannier on the maximal compact subgroup of universal additive extensions of elliptic curves. This is joint work with Gareth Jones.

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