# Bidisha Roy, Frobenius constants for families of elliptic curves (Friday, July 22)

:::{.remark}
See Frobenius constants, defined for homotopy classes of paths and has something to do with analytic continuation of solutions to DEs?
Zeta values appear in these constants.
For an elliptic curve $y^2 = x(x-1)(x-t)$, there is a period integral
\[
\int_1^\infty {1\over \sqrt{x(x-1)(x-t)}} \dx
.\]
If $L$ is a "geometric" (regular singular point, etc) differential operator (e.g. Picard-Fuchs), its solution spaces have a $\QQ\dash$structure spanned by "geometric" functions (period functions).
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:::{.remark}
Some phenomenon with the Frobenius constants is known for all hypergeometric DEs.
See Bloch-Vlasenko 2019: if $c\neq 0,\infty$ is a special reflection point of a hypergeometric DE and $\gamma: 0\to c$ is a path, then there is a meromorphic function on $\CC$ such that 
\[
\kappa(s) = \sum_{m\geq 0} \kappa_{\alpha, m} (s-\alpha)^m = {I(s) \over R(\exp(-2\pi i s))} \Gamma_{\sigma, \phi}(s), \qquad \Gamma_{\sigma, \phi}(s) \da \int_\sigma t^{s-1}\phi(t)\dt
,\]
where the latter term is a *generalized gamma function* and is entire.
These are *motivic* when $L$ is of Picard-Fuchs type and $\phi(t)$ is a period function.
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:::{.remark}
See Beauville stable families of elliptic curves, Apery families of K3s.
Theorem from this talk: computes $\kappa_{2+m}$ as an iterated integral in certain special cases, where the integral involves a modular form of weight 3.
Regularizable functions: $F: \HH\to\CC$ if $F(z) = p(z) + \bigo(\Im(z)^{-N})$ for $p \in \CC[z]$ as $\Im(z)\to \infty$. 
The regularized value is $F(z)_{\mathrm{reg}} \da p(0)$.

> Lots of interest in multiple zeta values, vs classical single zeta values.

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