Adolph Hurwitz 1859-1919

1. Adolph Hurwitz 1859-1919

2. Adolph HurwitzTimeline • 1859 born • 1881 doctorate under Felix Klein • Frobenius’ successor, ETH Zurich, 1892 • Died 1919, leaving many unpublished notebooks. • George Polya drew attention to the contents. • 1926 Fritz Gassmann (double s, double n, no r) • published one set of Hurwitz’s notes followed by Gassmann’s • interpretation of what Hurwitz meant.

3. In Gassmann’s paper, the following group-theoretic condition appeared for two subgroups H1, H2 of a group G: |gG ∩ H1| = |gG ∩ H2| for every conjugacy class gG in G.

4. A group G and subgroups H1, H2 form a Gassmann triple when Gassmann’s Criterion holds: (I) |gG ∩ H1| = |gG ∩ H2| for every conjugacy class gG in G. Let χi(g) = number of cosets of G/Hi (i = 1,2) fixed by left-multiplication by g. Reformulation: Gassmann’s criterion (1) holds if and only if (2) χ1(g) = χ2(g) for all g in G.

5. Reformulation: • (1) holds iff • (3) Q[G/H1] and Q[G/H2] are isomorphic Q[G]-modules. • Reformulation: • (1)holds iff • (4) There isbijection H1  H2 which is a local conjugation in G. • When any of these criteria hold, then (G:H1) = (G:H2). • Conjugate subgroups H1, H2 of G are always Gassmann equivalent; • this is the case of trivial Gassmann equivalence. • We are interested in nontrivial Gassmann equivalent subgroups.

6. Applications: • Gassmann triples (G, H1, H2) can be used to produce • pairs of arithmetically equivalent number fields (identical zeta functions); • pairs of isospectral riemannian manifolds; • pairs of nonisomorphic finite graphs with identical Ihara zeta functions; • I thought it would be interesting to collect some results about Gassmann triples. • Exercise: Translate each of the statement below into a statement about • arithmetically equivalent number fields, about isospectral manifolds, and about graphs with the same Ihara zeta functions.

7. Organization: • Small index • Solvable groups • Prime index • Index p2, p prime • Index 2p+2, p an odd prime. • Beaulieu’s construction • Involutions with many fixed points

8. Small index: (P, 1978, de Smit-Bosma, 2005) • Number of faithful, nontrivial Gassmann triples of index (G:H1) = n.

9. 2. Solvable Groups • The Lenstra-de Smit Theorem (1998): • Let n be a positive integer. Then the following are equivalent: • There exists a nontrivial solvable Gassmann triple of index n • There are prime numbers p, q, r (possibly equal) with • pqr | n and p | q(q-1)

10. Prime Index • Feit’s Theorem (1980): • Let (G, H1, H2) be a nontrivial Gassmann triple of prime index n=p. • Then either • p = 11 • or • p = (qk – 1) / (q – 1) • for some prime power q and some k ≥ 3.

11. Index p2, p a prime • Guralnick’s Theorem ( 1983): Let p be a prime. • There is a nontrivial Gassmann triple of index p2 iff • pe = (qk –1) / (q-1) for some e≤2, k≥3, and some prime-power q.

12. 5. Index 2p+2, p an odd prime. • de Smit’s Construction, (2003): • For every odd prime p • there is a nontrivial Gassmann triple of index n=2p+2.

13. Beaulieu’s Construction • Beaulieu’s Theorem (1996): • Let (G, H, H') be a faithful, nontrivial triple of index n • having no automorphism σ in Aut(G) taking H to H'. • Let π (resp. π') : G  Sn be the permutation representations coming • from left translation of G on G/H (resp. of G on G/H'). • Set G1= Sn , H1 = π(G), and H1′ = π′(G). Then (G1, H1, H1′) is a faithful • nontrivial triple of index > n with no outer automorphism taking H1 to H1′ . • ************************************************************************************ • Iteration gives infinitely many triples arising canonically from the first triple.

14. Involutions with many fixed-points • The Chinburg-Hamilton-Long-Reid Theorem (2008): • Every Gassmann triple (G, H1, H2) of index n, and • containing an involution δ with χ1(δ) = n-2, is trivial. • One Interpretation: • If K it a number field of degree n over Q having exactly n-2 real embeddings, • then K is determined (up to isomorphism) by its zeta function.