A finiteness property of abelian varieties with potentially ordinary good reduction (Q2892818)

The author makes the following conjecture, and proves if if \(p>2\) and \(F={\mathbb Q}\). Fix a number field \(F\), a rational prime \(p\) and a prime ideal \({\mathfrak p}\) of \(F\) over \(p\). Say that two \(F\)-simple non-CM abelian varieties \(A\) and \(B\) are twist equivalent if the \(L\)-functions \(L(s,\rho_A)\) and \(L(s,\rho_B)\) agree up to a twist by a Galois character, i.e.\ \(L(s,\rho_B)=L(s,\rho_A\otimes \chi)\) for some finite order character \(\chi: {\text{Gal}}(\overline{\mathbb Q}/F)\to \overline{\mathbb Q}^\times\). The conjecture is that there are only finitely many twist equivalence classes of \(F\)-simple non-CM abelian varieties of \({\text{GL}}(2)\) type having good reduction everywhere outside \({\mathfrak p}\) and potentially ordinary reduction at \({\mathfrak p}\).NEWLINENEWLINEThe precise meaning and importance of the result, and the reasons for formulating the conjecture in this precise form, are elaborated in the very readable introduction to the paper. Although there are serious difficulties in the way of going much beyond \(F={\mathbb Q}\), in a way the restriction to \(p>2\) is more fundamental and needs to be removed first. The author promises to try, and explains what else might be within reach. The rest of the paper is written for specialists, very concisely. The proof rests on earlier results of the author, about CM-types of families of modular forms and how to characterise them \(p\)-adically (this is where the assumption is made that \(p\) is not \(2\)) and on the theorem of Khare and Wintenberger, among other things.