Veranstaltungen

Conference on "Progress and Emergent Theories in Zeta and L-Functions" (PRETZL), July 21-25, 2025

There has been an enormous amount of progress in the analytic theory of zeta and L-functions over the past decade. Our understanding in the classical theory of the value distribution of the Riemann zeta function has achieved new heights with spectacular breakthroughs on moments and extreme values (both local and global) leading to a better understanding of families of L-functions more generally.

The theory of perfectoid towers, introduced by Ishiro-Nakazato-Shimomoto, provides an axiomatic approach to perfectoid theory in commutative algebra through tower-theoretic approximations. In contrast, Bhatt and Scholze introduced prisms as a "deperfection" of perfectoid rings. Our main result shows that a "gradual perfection" of a prism becomes a perfectoid tower. As a consequence, we prove that any p-torsion-free p-adically complete delta-ring that is reduced modulo p admits a perfectoid tower.

In 1978, Apery proved that zeta(3) was irrational. Perhaps unexpectedly, his methods have
proven exceptionally hard to generalize. In recent work with Dimitrov and Tang, we have found a new framework with which to study these ideas, with a number of applications both to modular forms and irrationality. Our aim will be to give a gentle overview of our ideas, as well as highlight a number of basic open questions about G-functions which Riemann would understand.