arXiv subjects

Francisco García-Cortés

Publications and source records attributed to Francisco García-Cortés.

2 recordsLinked to original sources

arXiv2024

$\mathrm{SL}_2$-character varieties of $2$-generated groups and failure of weak integrality

Let $\ell$ be a prime number, $k$ a positive integer and consider the group $Γ_{\ell^k} :=\langle a,b\ \vert\ a^{\ell^k(\ell^k-1)}ba^{-\ell^k}b^{-2}\rangle$. We prove that $Γ_{\ell^k}$ is not $\mathrm{SL}_2$-weakly integral with obstruction at exactly the prime $\ell$. We also give a general description of the character varieties of $2$-generated groups with a relation of the form $a^{n_1} b^{m_1} a^{n_2} b^{m_2} = 1$.

math.NT↗

arXiv2024

Finite monodromy of some two-parameter families of exponential sums

We determine the set of polynomials $f(x)\in k[x]$, where $k$ is a finite field, such that the local system on $\mathbb G_m^2$ which parametrizes the family of exponential sums $(s,t)\mapsto\sum_{x\in k}ψ(sf(x)+tx)$ has finite monodromy, in two cases: when $f(x)=x^d+λx^e$ is a binomial and when $f(x)=(x-α)^d(x-β)^e$ is of Belyi type.

math.NT↗