Oberseminar Algebra und Geometrie

Winter term 2026/27: Arithmetical Similarities

Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.

All talks take place on Fridays at 14:30 in 25.22.03.73.

If you want to get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.

Infos für Studierende

Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können zu promovieren. Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.

Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.

Aims and Content

Non-isomorphic number fields can have the same prime decomposition behavior: the residue degrees above every rational prime can agree, giving identical Dedekind zeta functions. This phenomenon, called arithmetical equivalence, is governed by finite group theory. In a common Galois closure, the corresponding subgroups have equal permutation characters, even though they need not be conjugate. Such Gassmann triples form the central link between arithmetic and group theory in this seminar. Starting with prime decomposition, Frobenius automorphisms and zeta functions, we develop this link through explicit examples, including the point and line actions on the Fano plane. We then introduce local fields and adele rings, examine what these similarities reveal about class numbers and regulators, and conclude with Sunada's construction of isospectral manifolds.
You can find a more detailed program here.
Klingen's book (password protected)

Schedule

(The dates of the talks are temporary and might still change.)

16.10.2026: D. G.: Prime decomposition and Frobenius
Introduce rings of integers, prime ideals, ideal norms, ramification indices and residue degrees. Explain the relation \(\sum_i e_i f_i = [K:k]\), then the transitive Galois action on primes, decomposition and inertia groups, and the cyclic quotient \(D/I\). Define Frobenius and its conjugacy class. Use a quadratic field as a running example. State the Dedekind-domain and finiteness results; concentrate proofs on the Galois action and the construction of Frobenius.
References: I §1a-b, pp. 3-11 (stop before §1c); I.1.1-1.10.
23.10.2026: H. U.: From splitting primes to permutation characters
Identify the \(k\)-embeddings of \(K=N^H\) into \(N\) with the coset action \(G/H\). Prove that Frobenius cycle lengths are residue degrees. Introduce permutation modules and their characters as fixed-point counts; show how the values \(\chi(g^j)\) recover every cycle length. Explain that subgroups \(H\) and \(H'\) of \(G\) are conjugate if and only if the transitive \(G\)-sets \(G/H\) and \(G/H'\) are isomorphic, and that \(\chi_H=\chi_{H'}\) if and only if \(\mathbb{C}[G/H]\cong_{\mathbb{C}G}\mathbb{C}[G/H']\). Discuss why the latter condition is potentially weaker. Work out the action of \(S_3\) on three letters and relate it to the factorization of \(X^3-X-3\) at the primes 2, 3, and 7 (the discriminant is \(-239\), the splitting field has Galois group \(S_3\)).
References: I §1c, pp. 11-14; Theorem I.1.12, Corollary I.1.13 and Theorem I.1.14.
30.10.2026: E. R.: Dedekind zeta functions and Chebotarev density
Define the Dedekind zeta function and its Euler product. Prove that its Euler factors determine all residue degrees over \(\mathbb{Q}\) (I.2.1). Introduce Dirichlet density and state Chebotarev; calculate splitting-type densities in the \(S_3\) example. Deduce from Chebotarev that two number fields with the same splitting types outside a set of primes of density zero have the same splitting type at every unramified prime. Briefly define the ideal class group, class number, signature, roots of unity and regulator, and state the analytic class number formula for later use.
References: I §2a-b, pp. 15-20; I.2.1-2.3. Preparation only: III §1a, p. 81, equation (11).
Additional reference for the arithmetic invariants: Jürgen Neukirch, Algebraic Number Theory, Springer, 1999.
06.11.2026: H. H.: Gassmann triples and arithmetical equivalence
Work over the ground field \(\mathbb{Q}\) and define arithmetical equivalence. Prove the equivalence between identical splitting types almost everywhere, equal permutation characters, and equal cardinalities of the intersections of \(U\) and \(U'\) with every conjugacy class of \(G\) (III.1.3). Call such \((G,U,U')\) a Gassmann triple. For ramified primes, restrict the permutation modules to decomposition groups and take inertia invariants; recover residue degrees from Frobenius on inertia orbits, using Appendix A of the PDF program. This replaces the Artin \(L\)-function step on p. 78. Deduce equality of splitting types at every prime and, by the Euler product, of Dedekind zeta functions. Use I.2.1 for the converse. Leave the common arithmetic invariants to Talk 5.
References: III §1a, pp. 75-79; Definition III.1.1, Theorem III.1.2 and Theorem III.1.3(i)-(v). For the ramified-prime step: Appendix A, using I §1b, pp. 7-11, and I.1.13, pp. 12-14.
13.11.2026: M. L.: Gassmann’s example and common invariants
Work out Gassmann's two Klein four-subgroups in \(S_6\): compare their intersections with conjugacy classes, prove non-conjugacy by their fixed letters, and obtain degree-180 coset actions. State the existence of Galois realizations with group \(S_6\). Derive equal degrees, Galois closures, maximal Galois subfields, roots of unity and signatures for arithmetically equivalent fields. For maximal Galois subfields, use equality of the normal closures of \(H\) and \(H'\) in \(G\); for signatures, use complex conjugation. State discriminant equality and equality of the sets of ramified primes. Use the analytic class number formula to deduce \(h_K R_K = h_{K'} R_{K'}\).
References: III §1b, p. 85, Example III.1.7 (omit the subsequent Kronecker-equivalence remark); III §1a, pp. 79-81, Theorem III.1.4a-j. Omit III.1.4k and III.1.5.
20.11.2026: R. S.: An explicit Gassmann pair of degree eight
Construct the common splitting field of \(\mathbb{Q}(\sqrt[8]{3})\) and \(\mathbb{Q}(\sqrt[8]{48})\). Prove that \(X^8-3\) remains irreducible over \(\mathbb{Q}(\zeta_8)\), using Eisenstein at a prime above 3, and identify its Galois group with the affine group of \(\mathbb{Z}/8\mathbb{Z}\). Verify that the two fixed subgroups are \(H=\{\pm x,\,\pm 3x\}\) and \(H'=\{\pm x,\,\pm 3x+4\}\). Match their elements by conjugacy class and prove that the subgroups are not conjugate, using the absence of a common fixed point for \(H'\). Apply Talk 4 to obtain equal zeta functions for non-isomorphic fields. Explain the same construction for suitable radicands \(a\), including \(a=97\) for Talk 9, and preview the degree-seven example.
References: III §1b, pp. 86-87, Example III.1.9; use the direct affine-group proof on p. 87 and omit the Kronecker-equivalence discussion. Preview: III.1.10, pp. 87-88.
27.11.2026: C. G.: The Fano plane and a first look at local fields
Draw the Fano plane and construct its point and line stabilizers in \(\mathrm{GL}_3(\mathbb{F}_2)\). Prove non-conjugacy and equality of permutation characters by comparing fixed vectors and fixed covectors. Present the degree-seven fields defined by \(X^7-7X+3\) and \(X^7+14X^4-42X^2-21X+9\); take their stated Galois realization as input. Then introduce \(p\)-adic absolute values, the completions \(\mathbb{Q}_p\) and their valuation rings \(\mathbb{Z}_p\), and completions of number fields at prime ideals. Define uniformizers and residue fields, state the local degree formula \(ef\), and relate its ramification index and residue degree to those from Talk 1.
References: III §1b, Example III.1.10, pp. 87-88. Local foundations: VI §2a, p. 235, including the local-degree formula (3); supplement with Lang, Algebraic Number Theory, Ch. II §1 (Klingen [67]).
04.12.2026: A. B.: Local extensions and adele rings
Build on the local foundations from Talk 7. State uniqueness of unramified extensions of a prescribed degree, Hensel's lemma in its valuation form, and the Eisenstein criterion for total ramification. Prepare Talk 9 explicitly: use Hensel at \(x=5\) for \(X^8-97\) to show that 97 has an eighth root in \(\mathbb{Q}_2\). Factor \(X^8-1\) into cyclotomic factors and \(X^8-16\) into \(X^2\pm 2\) and \(X^2\pm 2X+2\); use shifts where needed to make the cyclotomic factors Eisenstein. Then define the adele ring as a restricted product, including the real and complex places. Explain its basic open sets, the compact-open valuation rings, and the diagonal embedding of the number field.
References: VI §2a-b, pp. 235-237: unramified extensions and the definition of adeles before VI.2.3. Background: Lang, Algebraic Number Theory, Ch. II §§1,4 (Klingen [67]); Neukirch, as in Talk 3.
11.12.2026: M. G.: Local fields: what splitting types fail to determine
Work over \(\mathbb{Q}\). Prove VI.2.1: arithmetical equivalence matches residue fields at every prime, and matches the quotient rings modulo \(p\) at all but finitely many primes. Using the uniqueness of unramified local extensions from Talk 8, show that this is equivalent to matching completions over \(\mathbb{Q}_p\) for almost every p. Define local isomorphism at every finite prime. For \(K=\mathbb{Q}(\sqrt[8]{97})\) and \(K'=\mathbb{Q}(\sqrt[8]{16\cdot 97})\), explain how factorization over \(\mathbb{Q}_2\) gives the completions above 2. Scale using the eighth root from Talk 8 and deduce ramification indices \((1,1,2,4)\) and \((2,2,2,2)\), although all four residue degrees are one in both fields. Discuss why equal zeta functions and discriminants do not determine the completions.
References: VI §2a, pp. 234-236; VI.2.1-2.2 and the example on p. 235. Recall I §1b, pp. 7-10.
(18.12.2026: No talk - pre-Christmas break)
08.01.2027: D. E.: Adeles and locally isomorphic number fields
Recall the restricted product from Talk 8 and prove that number fields are locally isomorphic over \(\mathbb{Q}\) exactly when their adele rings are isomorphic as topological rings. Identify the closed maximal ideals as kernels of coordinate projections and recover the completions as quotients; use the density of the elements with finite support for the key step. Explain how equal signatures supply the archimedean factors in the converse. State Komatsu's congruence criterion for the degree eight pairs from Talk 6 and compare \(a=97\) with \(a=-33\). The latter gives \(\mathbb{Q}(\sqrt[8]{-33})\) and \(\mathbb{Q}(\sqrt[8]{-528})\) with isomorphic adeles. Explain that such an isomorphism need not carry one diagonally embedded number field onto the other.
References: VI §2b, pp. 236-240; VI.2.3b, the closed-maximal-ideal part of VI.2.4, and the topological parts of VI.2.5. State VI.2.6 for degree eight; VI.2.7 leads to Talk 11.
15.01.2027: S.Y.: Class numbers and regulators: what zeta functions miss
Recall ideal classes and describe the regulator using the logarithmic embedding of units, stating Dirichlet's unit theorem. Illustrate with \(\mathbb{Q}(\sqrt{2})\), then revisit the equality of the products \(hR\) from Talk 5. Take the following theorem as a black box: for arithmetically equivalent fields in a common Galois extension with group \(G\), \(\operatorname{Cl}(K)_p\cong\operatorname{Cl}(K')_p\) whenever \(p\nmid |G|\). For the degree eight family, conclude that the odd parts agree and the class-number quotient is a power of two. Present the de Smit-Perlis pair \(\mathbb{Q}(\sqrt[8]{-15})\), \(\mathbb{Q}(\sqrt[8]{-240})\). Take their computation \(h_K=2h_{K'}\) as input and deduce \(R_{K'}=2R_K\). Finally use VI.2.7 to show that even isomorphic adele rings do not determine the class number.
References: III.1.4j, p. 81; IV.2.2, p. 151, with IV.1.2c, p. 134 (consequence stated without proof); IV.3.1, p. 164; VI.2.7, p. 240. Additional reference: de Smit-Perlis [119], pp. 213-215.
(22.01.2027: Reserve slot - flexibility / postponed talk)
29.01.2027: J. H.: Sunada’s construction: Gassmann triples in geometry
Explain the dictionary between field extensions and finite Riemannian coverings, and between prime splitting and lifts of closed geodesics. State the length-spectrum result. Prove Sunada's Laplace-isospectrality theorem by viewing each eigenspace on a common finite regular cover as a \(G\)-module and comparing \(H\)- and \(H'\)-invariants. Reuse the \(S_6\) or Fano-plane triple to explain the construction of isospectral examples. Discuss separately the additional geometric argument needed for non-isometry: non-conjugacy alone does not prove it for an arbitrary metric.
References: VI §5a-b, pp. 250-254; VI.5.1-5.5, with the main emphasis on VI.5.5.
Additional references: Sunada, Riemannian coverings and isospectral manifolds, Ann. Math. 121 (1985), 169-186 (Klingen [123]); Bérard, Bourbaki exposé 705 (1988/89), pp. 127-154 (Klingen [9]).
(05.02.2027: Discussion of next semester’s program)

References:

[Kli98] N. Klingen, Arithmetical Similarities: Prime Decomposition and Finite Group Theory, Clarendon Press, Oxford, 1998.
[Neu99] J. Neukirch, Algebraic Number Theory, Springer, 1999.
[Lan] S. Lang, Algebraic Number Theory, Ch. II, Sections 1 and 4 (Klingen [67]).
[dSP94] B. de Smit and R. Perlis, Zeta functions do not determine class numbers, Bull. AMS 31 (1994), 213-215.
[Sun85] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Math. 121 (1985), 169-186.
[Ber89] P. Bérard, Bourbaki exposé 705 (1988/89), 127-154.