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Register a new userAn Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications
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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
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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 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 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.
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 $q=3$.
We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f in mathbb{Z}[x]$. We use an explicit version of Mertens theorem for number fields to estimate a related sum over rational primes. For a given $f in mathbb{Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of $f$.
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