**abstract:**
The theory of Drinfeld modules runs parallel to the theory of Elliptic curves, and our result was motivated by a similar result for Elliptic curves.
Let $A=\mathbb{F}_{qT$} be the polynomial ring over $\mathbb{F}_{q$,} and
$F$ be the field of fractions of $A$. Let $\phi$ be a Drinfeld $A$-module of rank $r$ over $F$.
For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $\phi$ modulo $\mathfrak{p}$
to obtain a Drinfeld $A$-module $\phi\otimes\mathbb{F}_{{\mathfrak{p}}$} of rank $r$ over $\mathbb{F}_{{\mathfrak{p}}=A\mathfrak{p}$.}* It is known that the
endomorphism ring $\mathcal{E} _{{\mathfrak{p}}=\text{End}
}
_{{\mathbb{F}}_{{\mathfrak{p}}}}(
\phi\otimes\mathbb{F}_{{\mathfrak{p}})$} is an order in an imaginary field extension $K$ of $F$ of degree $r$.
Let $\mathcal{O}_{\mathfrak{p}$} be the integral closure of $A$ in $K$, and let $\pi_{\mathfrak{p}\in} \mathcal{E}_{\mathfrak{p}$} be the Frobenius endomorphism of $\phi\otimes\mathbb{F}_{\mathfrak{p}$.} Then we have the inclusion of orders $A\pi_\mathfrak{p}\subset \mathcal{E}_{\mathfrak{p}\subset} \mathcal{O}_{\mathfrak{p}$} in $K$. In a joint work with my advisor, Mihran Papikian, we showed that if $\phi$ is a Drinfeld Module without complex multiplication, then for arbitrary non-zero ideals $\mathfrak{n}, \mathfrak{m}$ of $A$, there are infinitely many $\mathfrak{p}$
such that $\mathfrak{n}$ divides the index $\chi(\mathcal{E}_{{\mathfrak{p}}}*A\pi_\mathfrak{p})$ and $\mathfrak{m}$ divides the index $\chi(\mathcal{O}

Tue 6 Nov, 14:30 - 15:00, Aula Dini

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