Unexpected applications of transcendental number theory?

In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of the monodromy operator acting on the cohomology of a degenerating family of complex algebraic varieties).

The argument uses the base change compatibility and regularity of the Gauss-Manin connection to reduce the problem to the following statement: if $M$ is a complex square matrix such that, for every field automorphism $\sigma$ of $\mathbb{C}$, $\exp(2\pi i M^{\sigma})$ is conjugated to a matrix with integer coefficients, then $\exp(2\pi i M)$ is quasi-unipotent. This in turn is a simple consequence of Gelfond-Schneider theorem!

I always found this proof quite surprising and I was wondering if there aren't other unexpected applications of transcendental numbers out there.

• They are all over number theory, two of the central examples being Leopoldt's conjecture for abelian extensions of $\mathbb{Q}$ (due to Baker and Brumer) and the diophantine approximations solution to the Gauss class number one problem (due to Gelfond, Linnik and Baker). If you are asking for algebraic geometry specifically, are you familiar with Bombieri's higher dimensional generalization of the Gelfond-Schneider theorem? It is a theorem on transcendental numbers but the techniques of its proof are considered a major advance precisely in algebraic geometry. – Vesselin Dimitrov Feb 25 '16 at 1:44

There is also Simpson's proof that isolated points of the characteristic varieties of fundamental groups of projective manifolds are torsion. It also relies on Gelfond-Schneider Theorem.

The moduli space of representations of those fundamental groups on $\mathbb C^*$ admit three different algebraic/analytic structures. And in each of these the characteristic varieties are algebraic subspaces. This allows him to use Gelfond-Schneider Theorem to describe the nature of the isolated points.

• Siu's proof of the abundance conjecture, which cites Simpson's paper several times, could perhaps also be mentioned in response to this question. It is a striking use of Gelfond's technique, while a diophantine ingredient (Liouville's lower bound on non-zero algebraic quantities) becomes a crucial point in the proof. – Vesselin Dimitrov Feb 25 '16 at 2:11

Tools from transcendence theory have been crucial to the most significant recent advances in problems of unlikely intersections. The strategy was first dreamed up by Zannier, I believe, and has been applied with great success by Pila and Habegger, for example here: http://arxiv.org/abs/1409.0771 The authors prove some major cases of the the Zilber-Pink conjecture, some of them unconditionally, some conditional on conjectures in transcendence theory. For people who don't know, the Z-P conjecture is an extremely general and very strong statement in diophantine geometry. As an example of it's strength, Pink has given a very short argument that reproves Faltings's Theorem assuming only a very special case of the Z-P Conjecture.

In Zannier's monograph Some Problems of Unlikely Intersections in Arithmetic and Geometry one can find a great introduction to these problems, as well as some indications of the applications from transcendence theory. Transcendence techniques are also discussed in some of the appendices (by David Masser).

Very vaguely speaking, the strategy is to bound the height of elements in a set you want to show is finite, say, which lives inside some abelian variety, by looking at the preimage under the canonical map that sends $\mathbb{C}^g$ to your abelian variety. Counting points here requires techniques from transcendence theory and model theory (o-minimal structures), because the things you are looking at have "transcendental parts" which must be carefully dealt with.

Here is an application of transcendental number theory to differential geometry that I think would count as unexpected to all but a small group of experts in the area (who would probably view the application as being a very natural one).

Let $G$ be a connected absolutely simple real algebraic group and $\mathcal G=G(\mathbb R)$ the corresponding real Lie group. We'll call $\mathcal G$ absolutely simple. Let $\mathcal K$ be a maximal compact subgroup of $\mathcal G$ and $\mathfrak X=\mathcal K\backslash \mathcal G$ the symmetric space of $\mathcal G$.

If $\Gamma_1$ and $\Gamma_2$ are discrete subgroups of $\mathcal G$ then denote by $\mathfrak X_{\Gamma_1}=\mathfrak X/\Gamma_1$ and $\mathfrak X_{\Gamma_2}=\mathfrak X/\Gamma_2$ the associated locally symmetric spaces. We say that such a locally symmetric space is arithmetically defined if the corresponding discrete subgroup of $\mathcal G$ is arithmetic.

In their paper Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Gopal Prasad and Andrei Rapinchuk prove a number of interesting results about the spectral theory of such locally symmetric spaces, for instance:

Theorem (Prasad and Rapinchuk) Let $\mathfrak X_{\Gamma_1}$ and $\mathfrak X_{\Gamma_2}$ be two arithmetically defined locally symmetric spaces of the same absolutely simple real Lie group $\mathcal G$. If they are isospectral, then the compactness of one of them implies the compactness of the other.

Theorem (Prasad and Rapinchuk) Any two arithmetically defined compact isospectral locally symmetric spaces of an absolutely simple real Lie group of type other than $A_n (n > 1)$, $D_{2n+1} (n\geq 1)$, $D_4$ and $E_6$, are commensurable to each other.

While these results are unconditional for rank one locally symmetric spaces, for spaces of higher rank the results depend on Schanuel's conjecture in transcendental number theory.

As I mentioned above, I think that this geometric application would probably come as a complete surprise to non-experts, whereas to people working the field it is extremely natural, the idea being that the Laplace spectrum of such a space is related to the geodesic length spectrum (by results of Duistermaat and Guillemin), and the lengths of geodesics are in turn given by logarithms of algebraic numbers.