Local–global problem for Drinfeld modules

Let <f>K</f> be a function field with an <f>A</f>-algebra structure. The ring <f>A</f> arises in the definition of the Drinfeld module <f>φ</f> over <f>K</f>. By <f>E(K)</f> we denote <f>K</f> together with the <f>A</f>-module structure induced on it by <f>φ</f>. For any principal prime ideal <f>(a)⊂A</f>, we study the question whether an element <f>x∈E(K)</f> which is an <f>a</f>-fold in <f>E(Kν)</f> for every place <f>ν</f> of <f>K</f>, is an <f>a</f>-fold in <f>E(K)</f>. In particular, we study the groupS(a,K)≔ker<fen><cp type="lpar" style="s">E(K)/aE(K)→∏lower limit ν E(Kν)/aE(Kν)<cp type="rpar" style="s"></fen>for Drinfeld modules of rank <f>2</f>. We show that this finite group is trivial in many cases, but can become arbitrarily large. [Copyright &y& Elsevier]