From Spec Z to Zeta spectral triplesOberseminar Global Analysis and Operator Algebras

by Alain Connes (IHES Paris)

Thursday Jul 2, 2026, 9:00 AM → 11:00 AM Europe/Berlin

Lipschitzsaal (Mathezentrum)

In this extended two-hour seminar (joint work with Katia Consani for the first two parts and also with Henri Moscovici for the last part), we will explore the global geometric structure of the absolute curve--the one-point compactification $\overline{\text{Spec } \mathbb{Z}}$--and its profound implications for the Riemann Hypothesis.
    In the first part, we expound recent developments demonstrating that the adele class space arises naturally as the Picard monoid of this compactified curve. We subsequently construct a tentative global geometric structure $C=({\text{Spec } \mathbb{Z}})_{\mathbb{F}_1}$ for ${\text{Spec } \mathbb{Z}}$ over $\mathbb{F}_1$.Taking the geometric points of $C$ over a perfectoid field of characteristic $p$ provides a foundational compatibility check: it successfully recovers  the moduli space of untilts, yielding a precise geometric realization of the "Scholze heuristic" as formulated by J. Lurie, while illuminating the boundaries of local vs. global expectations at other primes.
    The geometry of the adele class space transitions us, in the second half of the talk, to the spectral properties of the Riemann Zeta function. Starting from Riemann's explicit formulas, we show how the action of the idele class group on the adele class space gives the zeros of $L$-functions as an absorption spectrum, with the explicit formulas emerging as a Lefschetz formula carried by the image of the curve through the Abel-Jacobi map. We then detail the translation from an absorption to an emission spectrum via the semilocal adele class space.
Finally, we present a remarkable computational and analytic convergence: using a "zeta spectral triple" constructed from only the first five finite primes ($2, 3, 5, 7, 11$), one can access the first fifty nontrivial zeros of the Riemann Zeta function with incredible precision. Crucially, the spectral triple truncated to finitely many primes is self-adjoint, ensuring these approximate zeros lie strictly on the critical line. This analytic architecture ultimately reduces the proof of the Riemann Hypothesis to demonstrating the convergence of the approximation of the lowest eigenvector of the Weil quadratic form by prolate spheroidal wave functions.