ICM 2026 Invited Section Lecture - Valentin Blomer

ICM 2026 Invited Section Lecture - Valentin Blomer

by Valentin Blomer · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Valentin Blomer 👥 60K 📅 September 21, 2026 ⏱ 40 min 👁 587 📄 original study 🧭 2026-09-22
Available in: English (current) Français

Keywords

automorphic formsdensity theoremRamanujan conjecturetrace formulaKloosterman sums

Summary

The lecture, given at ICM 2026, introduces automorphic density theorems, which quantify how often the Ramanujan conjecture may fail for automorphic representations. Blomer begins with a simple number theory problem: how much must the entries of a matrix be changed to lift a congruence determinant condition to an exact determinant one, highlighting the expected optimal size q^(1+1/n) and its failure near p-adic identity matrices. He then explains density theorems in the classical setting of the Riemann zeta function, where counting zeros off the critical line follows a linear interpolation between trivial bounds, known as the density hypothesis. The automorphic analogue concerns the distribution of exceptional eigenvalues of the Laplacian on hyperbolic surfaces, leading to Sarnak’s density hypothesis. The talk moves to higher rank, setting up families of automorphic representations on GL(n) and measuring violations by the largest real part of local Langlands parameters. The main methods involve relative trace formulas, requiring deep analysis of generalized Kloosterman sums, which are now equipped with non-trivial bounds due to recent work. Several theorems are presented: for Hecke congruence subgroups, principal congruence subgroups, Borel subgroups, and Archimedean families, the density hypothesis holds, sometimes with improved constants for cuspidal forms. Applications include uniform counting of matrices in congruence subgroups, equidistribution of arithmetic submanifolds, and connections to problems similar to Bombieri–Vinogradov. The lecture underscores the fragility of Ramanujan conjecture outside cuspidality, while density theorems provide a robust substitute for many applications.

235 words

Critical Evaluation

Value of the Information & Strength of the Argument

The value of the information is high for specialists: it presents recent advances in a central topic of analytic number theory, including results from the speaker and collaborators. The argumentation is coherent, moving from simple illustrative problems to abstract formulations and then to concrete theorems. Blomer carefully explains the motivation behind density theorems — allowing a few exceptions while quantifying their rarity — and demonstrates their utility through multiple applications. His reasoning is mathematically sound, with open problems clearly distinguished from solved results. The slides are well-structured, and the speaker provides intuitive explanations of technical steps without oversimplifying the underlying difficulties.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is appropriate for an invited ICM lecture: theorems are clearly stated with hypotheses, and methods are attributed to specific researchers (Sarnak, Friedberg, Assing, Nelson, etc.). However, the lecture format doesn’t allow full proofs, so the audience must rely on the speaker’s authority. No external sources or links are provided in the description, limiting verification capacity. The title accurately encapsulates the content, and the lecture delivers exactly what is promised. The description on YouTube gives a one-sentence summary that matches the talk.

200 words

Title / Content Match

The title accurately reflects the content: an ICM 2026 invited lecture given by Valentin Blomer, focusing on automorphic density theorems and their applications.

Quality & Reliability

8/10

The lecture is delivered by a leading expert in analytic number theory, presenting recent research results in automorphic density theorems. The arguments rely on deep methods such as trace formulas and Kloosterman sums, and results are stated with mathematical precision. However, as a conference lecture, many technical details and proofs are necessarily abbreviated, limiting immediate verifiability.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Contribution & Novelties

This lecture provides a synthesis of recent results on automorphic density theorems, particularly on GL(n), emphasizing their robustness compared to the Ramanujan conjecture itself. Blomer highlights the surprising validity of density hypotheses in higher rank despite the failure of the full Ramanujan conjecture for non-cuspidal representations. The talk also showcases how recent breakthroughs in bounding generalized Kloosterman sums (with savings proportional to the Weyl group length) feed into the trace formula machinery. A distinctive application is the uniform counting of matrices in congruence subgroups, linking purely arithmetic questions to deep spectral theory.

Pour aller plus loin :

  • Ramanujan conjecture — Central to the topic, the conjecture predicts temperedness of automorphic representations; density theorems quantify its failure.
  • Trace formula — The main analytic tool used in the lecture to translate spectral sums into geometric/arithmetic ones.
  • Kloosterman sum — Classical exponential sums whose generalizations are key inputs on the geometric side of relative trace formulas.

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Radar Profile

The radar profile shows high technical depth and information quality, with relatively lower quantitative scope due to the 40-minute format. The talk achieves a balance between presenting new results and describing established frameworks, but the density of technical jargon limits accessibility to non-specialists.

Reliability 8/10