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Register a new userExceptional cases of adelic surjectivity for Drinfeld modules of rank 2
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Authors
Chien-Hua Chen
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In this paper, we study the surjectivity of adelic Galois representation associated to Drinfeld $mathbb{F}_q[T]$-modules over $mathbb{F}_q(T)$ of rank $2$ in the cases when $q$ is even or $q=3$.
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Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity. Specifically, we provide an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and prove several basic properties. As an application, we give a numerical example of an asymptotically optimal tower of Drinfeld modular curves.
We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof.
We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular polynomials attached to Drinfeld modules.
We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $Theta_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-complet
Suppose we are given a Drinfeld Module $phi$ over $mathbb{F}_q(t)$ of rank $r$ and a prime ideal $mathfrak{l}$ of $mathbb{F}_q[T]$. In this paper, we prove that the reducibility of mod $mathfrak{l}$ Galois representation $${rm{Gal}}(mathbb{F}_q(T)^{rm{sep}}/mathbb{F}_q(T))rightarrow {rm{Aut}}(phi[mathfrak{l}])cong {rm{GL}}_r(mathbb{F}_mathfrak{l})$$ gives a bound on the degree of $mathfrak{l}$ which depends only on the rank $r$ of Drinfeld module $phi$ and the minimal degree of place $mathcal{P}$ where $phi$ has good reduction at $mathcal{P}$. Then, we apply this reducibility bound to study the Drinfeld module analogue of Serres uniformity problem.
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