Titles and Short Summaries of the Talks

24 Algebra
Final: 2013/2/7
67 Masakazu Yamagishi
(Nagoya Inst. of Tech.)
∗ Chebyshev polynomials, cyclotomic polynomials and twin primes · · · · · 10
Summary: In a series of papers, Stephen P. Humphries defined and investigated certain operators
to determine the geometric and algebraic intersection number functions associated to a simple
closed curve on a surface. A prominent role was played by Chebyshev polynomials. We are
particularly interested in number theoretical aspects of his results. Specifically, he indicated some
connections between Chebyshev polynomials and twin primes. We give short proofs of some key
results of Humphries’ by using Chebyshev polynomials of the third and fourth kinds and cyclotomic
polynomials. We also give counterexamples to a conjecture of his.
68 Hajime Kuroiwa (Kochi Univ.) ∗ An application of a remainder represented by a splitting behavior · · · · 15
Summary: We may discuss a splitting behavior of polynomials at primes. We defined a remainder
of x p − x divided by u(x). We used this, we obtained an answer of question. And we may discuss
an application of a remainder.
69 Yuuki Takai
∗ Indivisibility of relative class numbers of totally imaginary quadratic
(Univ. of Tokyo/Keio Univ.) extensions of totally real number fields · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: In this talk, we consider indivisibility of relative class numbers of CM quadratic
extensions. For a fixed totally real number field F which is a Galois extension over Q and a
sufficiently large prime number p, we show that there are infinitely CM quadratic extensions over
F whose relative class numbers are not divided by p. To obtain the result, we use Hilbert modular
forms of half-integral weight, these diagonal restrictions and Sturm’s theorem. Our method is a
generalization of Kohnen–Ono’s method.
70 Tsuyoshi Itoh (Chiba Inst. of Tech.) ∗ On the µ-invariant of tamely ramified Iwasawa modules · · · · · · · · · · · · · 15
Yu Takakura (Kyushu Univ.)
Summary: We consider the µ-invariant of the “tamely ramified Iwasawa module” for Zp-extensions
of imaginary quadratic fields.
71 Nao Takeshi (Tsuda Coll.) ♯ Elliptic curves with good reduction everywhere over cubic fields · · · · · · 10
Summary: It is known that there is no elliptic curve over Q having good reduction everywhere
by J. Tate. Concerning the existence or nonexistence of elliptic curves having good reduction
everywhere, we have many results for quadratic fields. In this talk, I will concentrate on the case of
cubic fields and report on the following results: (1) nonexistence of such elliptic curves over certain
cubic fields, which extends a result due to Bertolini–Canuto, and (2) the construction of certain
infinite families of cubic fields over which such elliptic curves exist.
72 Akinari Hoshi (Rikkyo Univ.) ♯ Krull–Schmidt theorem fails for dimension 5 · · · · · · · · · · · · · · · · · · · · · · · 10
Aiichi Yamasaki (Kyoto Univ.)
Summary: Let G be a finite subgroup of GL(n, Z) and MG be the corresponding G-lattices, i.e.
finitely generated Z-free Z[G]-module, of rank n. Theorem. (i) When n ≤ 4, the Krull–Schmidt
theorem holds for MG. (ii) When n = 5, the Krull–Schmidt theorem fails for MG if and only if G
is one of the 11 groups as in [HY, Theorem 4.6]. (iii) When n = 6, the Krull–Schmidt theorem fails
for MG if and only if G is one of the 149 groups as in [HY, Theorem 4.6]. See [HY, Section 4], A.
Hoshi, A. Yamasaki, Rationality problem for algebraic tori, arXiv:1210.4525v3.