How do sub-convex bounds turn back into number theory? As of around 2010, we get estimates of L-functions like this:
$$ L(\tfrac{1}{2}, \pi \times \chi ) \ll\_{F, \epsilon, \pi} C(\chi)^{\frac{1}{2}} $$
$\pi$ is a "cuspidal automorphic representation", and $\chi$ is a character. The twisted L-function would look like this:
$$ L(s, \pi \times \chi) = \sum\_{\mathfrak{m}} \lambda\_{\pi \otimes \chi}(\mathfrak{m}) (N\mathfrak{m})^{-s} $$
where $\mathfrak{m}$ indexes the , and $\chi$ is a character of conductor $\mathfrak{q}$. The Hecke eigenvalues if they were to compute them look like this, are multiplicative:
$$ \lambda\_{\pi \otimes \chi}(\mathfrak{m}) = \lambda\_\pi(\mathfrak{m})\lambda\_\chi(\mathfrak{m}) $$


My question is basic and somewhat procedural... How does the automorphic function and the respresentation and the number theory problem match up? It looks like the number theory and the representation theory have been de-coupled.


Example $\pi$ could be a "[holomorphic Hilbert cusp form](https://www.lmfdb.org/ModularForm/GL2/TotallyReal/2.2.5.1/holomorphic/2.2.5.1-31.1-a)". How do we interpret the sub-convexity bounds in terms of more familiar objects? This seems to be related to the [Burgess](https://mathoverflow.net/questions/327455/explicit-version-of-the-burgess-theorem?rq=1) theorem.


When I search for the Burgess formula all I get are complicated sums of Dirichlet characters:
$$ \sum\_{N \leq n \leq N+H} \chi(n) \ll H^{1-\frac{1}{r}} q^{\frac{1}{4}+\epsilon}$$
There are no L-functions here or automorphic forms, yet. Burgess' [1962](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms/s3-12.1.193) result says:
$$ L(\frac{1}{2}, \chi) \ll k^{\frac{7}{32}+\epsilon} $$
where $\chi$ is a [Dirichlet character](https://en.wikipedia.org/wiki/Dirichlet_L-function) modulo $k$. The constant $\frac{7}{32}$ seems interseting but there is also $L(\frac{1}{2}, \chi)\ll k^{\frac{1}{2}}$ in [1931](https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/plms/s3-12.1.193).



How does the sub-convex bound $ L(\tfrac{1}{2}, \pi \times \chi ) \ll_{F, \epsilon, \pi} C(\chi)^{\frac{1}{2}} $ relate to "elementary" number theory?
The most straightforward application of subconvexity is to bounds to sums of the form


$$ \sum\_{ n < X} \chi(n) a\_n(f) $$ where $f(q) = \sum\_n a\_n(f) q^n$ is a modular form, or to the analogous sequences of Fourier coefficients arising from other modular forms. It's slightly easier to work with sums of the form $$ \sum\_{n < X} \Psi(\frac{n}{X} ) \chi(n) a\_n (f)$$ where $\Psi$ is a smooth function supported on $[0,1]$.


These functions have bounds of the form $ O ( \min (X, k))$ (where $k$ is the conductor of the character and a classical modular form $f$ is fixed), and subconvexity allows you to improve this in the range $X \approx k$.


This is perfectly elementary, although, if you think the sum $\sum\_{n=N}^{N+H} \chi(n)$ is complicated, maybe you will not like this.


One perspective on this sum, primarily seen in the work of Fouvry, Kowalski, and Michel, is that one should replace $\chi(n)$ here by a more general function that is periodic mod $k$, for example $\chi$ composed with a polynomial and expect, and hope to prove, similar results.




---


A second application of subconvexity is by period integral formulas which relate special values of $L$-functions to integrals of modular forms or other automorphic forms. Subconvexity gives good bounds for these integrals, which often show up in practical problems. Lucia has explained examples of this beautifully in the comments, so I won't attempt to.




---


Finally, a broad hope is that any subconvexity result represents progress on the Riemann hypothesis, and proving subconvexity in all these different cases might lead us to a clue which helps to prove the Riemann hypothesis. (More generally, cases of the Riemann hypothesis are used in a wide variety of arguments, and subconvexity results can remove some of them, rendering the results unconditional.)


