Stronger arithmetic equivalence

Stronger arithmetic equivalence, Discrete Analysis 2021:23, 23 pp.

An algebraic number field is a subfield \(K\) of \(\mathbb C\) that is finite-dimensional when considered as a vector space over \(\mathbb Q\), which implies that every element of \(K\) is algebraic of degree at most the dimension of this space (since otherwise there would exist \(x\) such that the numbers \(1,x,x^2,\dots,x^d\) were linearly independent). The ring of integers \(O_K\) of \(K\) is the set of elements of \(K\) that are roots of monic polynomials with integer coefficients. Given an ideal \(I\) in this ring, \(N_{K/\mathbb Q}(I)\) denotes the index of \(I\) in \(O_K\). The Dedekind zeta function of \(K\) is then defined, for \(\Re(s)>1\), by the formula
\[\zeta_K(s)=\sum_I(N_{K/\mathbb Q}(I))^{-s},\]
where \(I\) ranges over all non-zero ideals of \(O_K\).

If \(K=\mathbb Q\), then \(O_K=\mathbb Z\), so the non-zero ideals are just the sets of the form \(I_n=\{rn:r\in\mathbb Z\}\) with \(n\in\mathbb N\). Since \(I_n\) has index \(n\), the formula reduces to the familiar formula for the Riemann zeta function in this case, and many of the basic properties of the Riemann zeta function can be generalized to all Dedekind zeta functions. For example, Dedekind zeta functions have an Euler product (the formula for which is easy to guess), and can be analytically continued to the whole of the complex plane, apart from a simple pole at 1 (this is a result of Hecke). They also have a functional equation, and the extended Riemann hypothesis states that they have their non-trivial zeros on the critical line.

The Dedekind zeta function of \(K\) encapsulates many important properties of \(K\), and it is therefore of interest to ask to what extent a number field \(K\) is determined by its Dedekind zeta function. It turns out that distinct number fields can have the same Dedekind zeta function: if this happens, they are called arithmetically equivalent.

The notion of arithmetic equivalence has been studied in detail. Although arithmetically equivalent fields can be distinct, they must have several important properties in common: for example, they are of the same degree over \(\mathbb Q\), and they have the same roots of unity. And it is known that pairs of arithmetically equivalent fields all arise in the same way, via the group-theoretic notion of a Gassmann triple, which is a triple \((G,H_1,H_2)\) of finite groups that satisfies one of the following three conditions, which can be shown to be equivalent.

1. The modules \(\mathbb C[G/H_1]\) and \(\mathbb C[G/H_2]\) are isomorphic.

2. The modules \(\mathbb Q[G/H_1]\) and \(\mathbb Q[G/H_2]\) are isomorphic.

3. For every conjugacy class \(C\subset G\), the intersections \(C\cap H_1\) and \(C\cap H_2\) have the same cardinality.

Two number fields are arithmetically equivalent if they have the same Galois closure with Galois group \(G\), and there are subgroups \(H_1,H_2\) of \(G\) such that one field is the set of numbers fixed by \(H_1\), the other is the set of numbers fixed by \(H_2\), and \((G,H_1,H_2)\) is a Gassmann triple.

The third condition for being a Gassmann triple trivially holds if \(H_1\) and \(H_2\) are conjugate, so the interesting Gassmann triples are the ones for which that is not the case. Such triples arise in several other contexts as well. For example, they have a bearing on the famous problem of whether one can hear the shape of a drum: non-trivial Gassmann triples can be used to construct different drums that sound the same.

In 2017, Dipendra Prasad introduced a notion that he called a refined Gassmann triple. This is one for which the modules \(\mathbb Z[G/H_1]\) and \(\mathbb Z[G/H_2]\) are isomorphic. Since this is a stronger requirement than asking for \(\mathbb Q[G/H_1]\) and \(\mathbb Q[G/H_2]\) to be isomorphic, the corresponding notion of arithmetic equivalence of number fields is also stronger. Prasad showed that two number fields arising out of a refined Gassmann triple not only have the same Dedekind zeta function, but they also have isomorphic adele groups, idele groups and idele class groups.

Of course, since the definition is stronger, non-trivial examples are harder to come by, and to date essentially only one non-trivial example is known of a refined Gassmann triple, a surprising construction due to Leonard Scott. (To give an idea of its complexity, the group \(G\) in Scott’s example is \(\text{PSL}_2(\mathbb F_{29})\). The subgroups \(H_1\) and \(H_2\) are both isomorphic to \(A_5\), and they are conjugate in \(\text{PGL}_2(\mathbb F_{29})\) but not in \(\text{PSL}_2(\mathbb F_{29})\).) Nevertheless, this suffices for Prasad’s results to yield distinct number fields that share many important invariants.

This paper considers two alternative strengthenings of the notion of arithmetic equivalence, which the author calls local integral equivalence and solvable equivalence. (The latter turns out to be strictly stronger than the former.) They have the advantage of being easier to check than Prasad’s notion, which the author calls integral equivalence. Furthermore, solvable equivalence, which the author shows does not imply integral equivalence, is nevertheless a sufficient condition to imply that the invariants considered by Prasad are equal. This opens the door to easier proofs of Prasad’s result, and lessens the reliance on Scott’s construction: the author finds a generalization of this construction that yields infinitely many examples of solvable equivalence.

The paper also contains several examples to clarify the relationships between the various different notions of equivalence. Some of these examples (which are mainly found with the help of a computer) answer open questions from the group theory literature.