Equidistribution and Monodromy

Supporting material for the talk at VII Congreso de Jóvenes Investigadores de la RSME 2025. This work was partially supported by grants PID2020-117843GB-I00 and PID2020-114613GB-I00 funded by MCU/AEI/10.13039/501100011033.

Let $p$ be a prime number, $d$ an odd prime to $p$ positive integer and $r$ a positive integer. For every $t\in\mathbb{F}_{p^r}$ consider the exponential sum \[S_{p^r,d}(t):= \frac{-1}{{p}^{r/2}}\sum_{x\in\mathbb{F}_{p^r}}\mathsf{exp}\big(2\pi i \mathsf{Trace}_{\mathbb{F}_{p^r}/\mathbb{F}_p}(x^d+tx)/p\big).\] Observe that, since $d$ is odd, these exponential sums are real numbers with absolute value less than $1$ after Deligne's proof of the Weil Conjectures.

Below you will find the histogram of the set $\{S_{p^r,d}(t)\ \vert\ t\in\mathbb{F}_{p^r}\}$ for different values of the parameters $p,d,r$; computed using SageMath and Julia Programming Language.

The shape of the histogram is determined by the geometric monodromy group of the corresponding $\ell$-adic local system on $\mathbb{A}^1/\mathbb{F}_p$. These pictures suggest that the monodromy group $G_{\mathsf{geom}}$ may be finite in some cases (for $(p,d)\in \{(2,3),(2,5),(3,5),(5,3)\}$), and in other cases we obtain the pushforward by the trace of the Haar measure of a compact maximal subgroup of $\mathsf{Sp}_{d-1}(\mathbb{C})$ (for $p=7$ and $d=3,5$), i.e. $G_{\mathsf{geom}}=\mathsf{Sp}_{d-1}(\mathbb{C})$.

$p=2$; $d=3$

$p=2$; $d=5$

$p=3$; $d=5$

$p=5$; $d=3$

$p=7$; $d=3$

$p=7$; $d=5$

References

1. Katz, Nicholas M.; Tiep, Pham Huu. Exponential sums, hypergeometric sheaves, and monodromy groups. Annals of Mathematics Studies, 220. Princeton University Press, Princeton, NJ, 2025. ix+584 pp. MR4901926. Zbl 7982378. Available at first author's website.
2. Katz, Nicholas M. Rigid local systems on $\mathbb{A}^1$ with finite monodromy. With an appendix by Pham Huu Tiep. Mathematika 64 (2018), no. 3, 785--846. MR3867322. Zbl 1456.11232. doi.org/10.1112/S0025579318000268.
3. Katz, Nicholas M.; Sarnak, Peter. Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999. xii+419 pp. MR1659828. Zbl 0958.11004. Available at first author's website.
4. Lachaud, Gilles. On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius. Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, 185--221, Contemp. Math., 663, American Mathemathical Society, Providence, RI, 2016. MR3502944. Zbl 1417.11123. doi.org/10.1090/conm/663/13355.
5. Rojas-León, Antonio. Finite monodromy of some families of exponential sums. J. Number Theory 197 (2019), 37--48. MR3906488. Zbl 1412.11087. doi.org/10.1016/j.jnt.2018.06.012.
6. Sutherland, Andrew V. Sato—Tate distributions in genus 1 to 3. Author's web page: Genus 1, Genus 2 and Genus 3.