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A function field tower is a sequence of function fields $$\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \dots \subset \mathcal{F}_{n} \subset \mathcal{F}_{n+1} \subset \dots $$ over a base field (in our case the finite field $F_{q}$ say) such that each extension is finite and separable, and the genus tends to infinity with $n$. (Such towers are constructed and studied with the aim of ensuring that the number of rational points also grows with the genus (or better still, are optimal), and such towers have applications in codes and cryptography.) Simple constructions of towers are "recursive" in the sense that we use one equation of the form $f(Y) = h(X)$ as the defining equation of the tower, and then $\mathcal{F}_{n+1}$ would be obtained from $\mathcal{F}_{n}$ as $f(X_{n+1}) = h(X_{n})$, with $\mathcal{F}_0 = F_{q}(X_0)$. (We would need to choose the defining equation appropriately so that this defines a valid tower with good properties).

My question is whether there exist any known tower constructions with a larger "depth" or "memory", i.e. where $X_{n+1}$ depends not just on $X_n$ but also $X_{n-1}$ or more. This would involve a defining equation which is not just bivariate.


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Not exactly what you are looking for, but in this thesis the author consider different defining equations at each step. – Matemáticos Chibchas Oct 31 '15 at 1:46

I'm pretty sure that there are no papers in the literature addressing such constructions.

Maybe one can produce cheap examples by starting with a tower built from an equation $f(X_{n+1})=h(X_n)$ and then rewriting $h(X_n)$ as a function of the previous few $X_i$'s? Of course, you'll want to avoid towers like this that are just disguised forms of simple recursive towers.

As you surely know, loads of examples of recursive towers are in papers by Garcia and Stichtenoth. For steps towards a theoretical investigation of all such towers, I recommend the paper "Asymptotically good towers and differential equations" by Peter Beelen and Irene Bouw (Compositio Math. 141 (2005), 1405-1424).

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Update: I found this result in which it is mentioned that they construct towers where $x_{n+1}$ depends on $x_n$ and $x_{n-1}$. Not sure if its what I have in mind, since the theory is a bit more involved unlike the constructions of Garcia and Stichtenoth. But do have a look if you're interested. Thanks. – BharatRam Feb 26 '14 at 7:38
Interesting. In light of that paper, and the earlier paper by Bassa, Beelen, Garcia, and Stichtenoth, it seems that an interesting research direction in this subject is towers of Drinfeld modular curves and towers of curves on Drinfeld modular varieties. – Michael Zieve Feb 26 '14 at 8:01

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