Francisco García Cortés

My email is: (underlined parts of my name)3 at alum.us.es

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About me

I am a first year PhD student at Depto. de Álgebra of Universidad de Sevilla, under the direction of Francisco Castro Jiménez and Antonio Rojas-León, funded by Ministry of Science of Spain (FPU 23). I earn a Bachelor degree in Mathematics (graduated with Honors), a M.Sc. degree in Mathematics (graduated with Honors) and a M.Ed. degree on Teaching Mathematics, conferred by Universidad de Sevilla.

Research interests

I am interested in algebraic and arithmetic geometry. More precisely, I am interested in applications of $\ell$-adic and rigid cohomology to exponential sums, (arithmetic) $\mathcal{D}$-modules, convolution and monodromy of local systems. I am also learning about geometric representation theory, the Langlands program (both geometric and for function fields), Yun's rigidity method and Deligne—Lusztig's theory.

Preprints / Publications

  1. (w/ Benjamin Church) $\mathrm{SL}_2$-character varieties of $2$-generated groups and failure of weak integrality. arXiv:2410.23233
    Abstract. (click to reveal/hide) Let $\ell$ be a prime number, $k$ a positive integer and consider the group $\Gamma_{\ell^k}=\langle a, b\ \vert\ a^{\ell^k(\ell^k-1)}ba^{-\ell^k}b^{-2}\rangle$. We prove that $\Gamma_{\ell^k}$ 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 $a^{n_1}b^{m_1}a^{n_2}b^{m_2}=1$.
  2. (w/ Antonio Rojas-León) Finite monodromy of some two-parameter families of exponential sums. arXiv:2406.10385
    Abstract. (click to reveal/hide) We determine the set of polynomials $f(x)\in k[x]$, where $k$ is a finite field, such that the local system on $\mathbb{G}^2_m$ which parametrizes the family of exponential sums $(s,t)\mapsto \sum_{x\in k}\psi(sf(x)+tx)$ has finite monodromy, in two cases: when $f(x)=x^d+\lambda x^e$ is a binomial and when $f(x)=(x−\alpha)^d(x−\beta)^e$ is of Belyi type.

Some talks

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