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 $p,d,r$ (the total area normalized to be 1); 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

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