ICTS

Automorphic forms sit at the crossroads of modern mathematics, forming a central component of the Langlands program, which unites number theory, representation theory, and the study of L-functions. They generalize classical modular forms and play a pivotal role in understanding deep arithmetic structures. Historically, congruences among modular forms, such as Ramanujan’s relation between the Delta function and an Eisenstein series, revealed rich arithmetic patterns. Theorems by Hida, Ribet, and Wiles demonstrated how such congruences are governed by special values of L-functions and the cohomology of
arithmetic groups.

From a representation theoretic perspective, automorphic forms arise naturally as functions exhibiting symmetry under certain groups, including real and p-adic groups. Extending congruence phenomena to higher-rank groups leads to fundamental questions about distinguishing congruences intrinsic to a group from those inherited from smaller subgroups.

Automorphic forms also bridge number theory and geometry through their connection with elliptic curves and abelian varieties. Isogenies—morphisms between elliptic curves that preserve algebraic structure—define relationships central to modular curves. The height of an elliptic curve, such as its Faltings height or height of modular j-invariant, quantifies its arithmetic complexity. Isogeny height estimates measure how these heights vary between isogenous curves and bound the degrees of minimal isogenies. These estimates have far-reaching implications in Diophantine geometry, offering tools to analyze the arithmetic and geometric complexity of modular curves, their Faltings heights, and the self-intersection properties of their dualizing sheaves and modular polynomials.

The following four mini-courses in this program will focus on these topics:

• Congruences and Eisenstein cohomology by A. Raghuram
• Congruence numbers and special values of L-functions by Jacques Tilouine
• Isogeny estimates and modular curves by Fabian Pazuki
• Introduction to the theory of heights and dynamical systems by José Ignacio Burgos Gil

Eligibility Criteria: The program is primarily meant for graduate students, postdocs and faculty in Number Theory and related fields. A small number of highly motivated senior undergraduates can
also be considered. A basic understanding of algebraic number theory and graduate level algebra is a prerequisite for the courses.

ICTS is committed to building an environment that is inclusive, non-discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.

Program registration start date: February 01, 2026

Registration deadline for participants: March 31, 2026