We say a tame Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that OL is a free OK [G]-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l so that for each there is a tame Galois field extensio...

Last time we saw (see Proposition 6.5 in those lecture notes) that an abelian variety A of dimension g over K, the fraction field of a henselian dvr R, acquires semistable reduction over K(A[`]) for ` equal to 4 or an odd prime, with ` not divisible by the residue characteristic. (The same circle of ideas yielded the non-obvious fact that over K the intersection of two “semistable field extensi...

Let K be a field, and let K be a separable closure of K. Let C be an elliptic curve over K. For each g in the Galois group G := Gal(K/K), let C be the elliptic curve obtained by conjugating C by g. One says that C is an elliptic K-curve if all the elliptic curves C are K-isogenous to C. Recall that a subfield L of K is said to be a (2, . . . , 2)-extension of K if L is a compositum of a finite ...

Given a Galois extension L/K of number fields with Galois group G, a fundamental problem is to describe the (unramified) primes p of K whose Frobenius automorphisms lie in a given conjugacy class C of G. In particular, all such primes have the same splitting type in a sub-extension of L/K. In general, all that is known is that the primes have density |C|/|G| in the set of all primes, this being...