九大代数学セミナー：2024年度

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最終更新： 2025年3月11日

2025年1月24日 (金) 16:00-17:00 印刷用プログラム

• 井上 翔太 氏 (日本大学)
  "Riemannゼータ関数の非自明零点の間隔について"


• 本講演ではRiemannゼータ関数の非自明零点の虚部について議論する。有名なRiemann予想は非自明零点の実部に関するものだが、ここで議論する零点の虚部についても、素数分布などに応用がある重要な対象である。講演者は、Riemannゼータ関数の極大評価についての結果を零点の虚部の分布に応用することで、零点の間隔についての結果を得た。この結果は、Selbergにより言及された主張である一方、30年以上証明が分かっていなかった主張である。本講演では、この研究の歴史的背景について話し、その後に結果の証明について紹介したい。

2024年12月13日 (金) 16:00-17:00 印刷用プログラム

• 翁 林 氏 (九州大学)
  "Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves"


• For elliptic curves over rationals, there are a well-known conjecture of Sato-Tate and a new computational guided murmuration phenomenon, for which the abelian Hasse-Weil zeta functions are used. In this talk, we show that both the murmurations and the Sato-Tate conjecture stand equally well for non-abelian high rank zeta functions of elliptic curves over rationals. We establish our results by carefully examining the asymptotic behaviors of the p-reduction invariants a_{E/F_q} (n≥1), the rank n analogous of the rank one a-invariant a_{E/F_q}=1+q-N_{E/F_q} of elliptic curve E/F_q. Such asymptotic results are based on the counting miracle of the so-called α_{E/F_q,n}- and β_{E/F_q,n}-invariants of E/F_q in rank n, and a remarkable recursive relation on the β_{E/F_q,n}-invariants, established by Weng and Zagier, previously. This is a joint work with Zhan Shi.

2024年12月3日 (火) 16:00-17:00 印刷用プログラム

• 中村 健太郎 氏 (九州大学)
  "A local sign decomposition for symplectic self-dual Galois representations"


• Around 1987, Rubin envisioned a signed Iwasawa theory for CM elliptic curves at supersingular primes $p$ over the anticyclotomic $¥mathbb{Z}_p$-extension of the CM field, conjectural on a fundamental sign decomposition of the local Iwasawa cohomology. This conjecture was resolved by A. Burungale, K. Ota and S. Kobayashi in 2021. We generalize this decomposition to arbitrary families of two dimensional symplectic self dual Galois representations of $¥mathbb{Q}_p$ using Kato's local epsilon conjecture, which was resolved by the speaker. This is a joint work (in progress) with A. Burungale, K. Ota, and S. Kobayashi.

2024年10月25日 (金) 16:00-17:00 印刷用プログラム

• 山口 永悟 氏 (東京工業大学)
  "Anabelian geometry and $m$-step solvable reconstruction"


• In Anabelian geometry, we have an important conjecture by A. Grothendieck, which states that the geometric properties of (algebraic) hyperbolic curves can be determined group-theoretically by studying their arithmetic fundamental groups. H. Nakamura, A. Tamagawa, and S. Mochizuki proved this conjecture for finitely generated fields over $\mathbb{Q}$. This talk focuses on one of the remaining problems related to this conjecture, called the $m$-step solvable Grothendieck conjecture, which concerns the group-theoretical reconstruction of geometric properties of hyperbolic curves by the maximal geometrically $m(\geq 2)$-step solvable quotient of their arithmetic fundamental groups. This talk will explain the $m$-step solvable Grothendieck conjecture and a part of its proof as obtained by the speaker, focusing on the case where $g=0$.

2024年10月4日 (金) 16:00-17:00 印刷用プログラム

• Vaidehee Thatte 氏 (King's College London)
  "Ramification Theory for Henselian Valued Fields"


• Ramification theory serves the dual purpose of a diagnostic tool and treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of "nice" fields splits up neatly into the product of two well-understood numbers (ramification index and inertia degree) that encode how the base field changes. In the general case, however, a third factor called the defect (or ramification deficiency) can pop up. The defect is a mysterious phenomenon and the main obstruction to several long-standing open problems, such as obtaining resolution of singularities. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is non-trivial. I will discuss my previous and ongoing work in ramification theory that allows us to understand and treat the defect.

2024年7月12日 (金) 16:00-17:00 印刷用プログラム

• Didier Lesesvre 氏(Lille大学)
  "Poles of Rankin-Selberg L-functions"


• The Rankin-Selberg L-function associated with two automorphic cusp forms is known to have a pole at s=1 if and only if the two forms are essentially the same. This result is central in applications, and has been first proved using the method of integral representation of L-functions. However, such integral representations naturally exist only for generic representations, leaving aside important cases. This motivated Langlands' approach that goes "beyond endoscopy", replacing the use of integral representations by the one of trace formulas. I will present such an approach in the case of automorphic forms of GL(2) of different types, underlining the uniformity of the proof between the different types. This is based on results of Ganguly and Mawia, and a joint project in progress aiming at going beyond.

2024年6月7日 (金) 16:00-17:00 印刷用プログラム

• 馬 昭平 氏 (東京工業大学)
  "Mixed Hodge structures of locally symmetric varieties"


• I will talk about the mixed Hodge structures on the cohomology of locally symmetric varieties. In the middle degree, I relate the weight filtration to the Siegel operators for certain modular forms. This has application to a classical problem on the Siegel operators. In the general degrees, I construct a spectral sequence which converges to the edge components in the Hodge triangle, and whose E1 page is expressed by some simple geometric invariants associated to the cusps. This already degenerates at E1 in a certain range. MHS approach also has application to the restriction map to the Borel-Serre boundary.

2024年5月10日 (金) 16:00-17:00 印刷用プログラム

• 埴原 紀宏 氏 (九州大学)
  "Cohen-Macaulay representations in triangulated categories"


• Representation theory of algebras aims at understanding various categories associated with a given ring. Rings we are interested in are, for example, finite dimensional algebras over fields and commutative Cohen-Macaulay rings. In this talk, I would like to talk about representation theory of these rings, especially in triangulated categories such as derived categories, singularity categories, cluster cagtegories, and their differential graded enhancements. We will present some results which connects the representation theories of finite dimensional algebras and commutative rings through these triangulated categories.

2024年4月26日 (金) 16:00-17:00 印刷用プログラム

• Kang Zuo 氏 (Wuhan University)
  "p-adic Simpson correspondence, motivic Higgs bundles and Katz convolution"


• I shall explain our recent work, jointly with Jinbang Yang, on constructing rank-2 motivic local systems on 4-punctures projective line over complex numbers. The proof relies on non-abelian p-adic Hodge theory, p-adic Simpson correspondence for periodic Higgs bundles, Deligne's p to \ell companions, Yu's work on numerical characteristic p Simpson correspondence, Drinfeld's work on Langlands correspondence for rank-2 \ell-adic local systems with cusps and Grothendieck-Messing-Kato deformation theorem for log abelian schemes. It is remarkable that recently Lam-Litt provided a totally different approach to the above results using Katz middle convolution.

2024年4月2日 (火) 16:30-17:30 印刷用プログラム

• Christopher Deninger 氏 (University of Münster)
  "Hasse Weil zeta functions as dynamical zeta functions"


• For any system of diophantine equations or more generally for any separated scheme X of finite type over the integers, there is a zeta function originating from the works of Hasse and Weil. The Riemann zeta funtion is a special case. On the other hand, Ruelle introduced a zeta function for certain dynamical systems where the role of the primes is played by the periodic orbits. In the talk we explain the construction of a dynamical system whose Ruelle zeta function is the Hasse Weil zeta function of X.

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