Automorphic Forms and L-Function Theory

Automorphic forms are highly symmetric functions on arithmetic quotients of Lie groups that generalise classical modular forms and encode deep number-theoretic and representation-theoretic information. They arise as eigenfunctions of commuting operators and can be organised into families known as cuspidal representations. L-functions, built from automorphic data via Dirichlet series and Euler products, serve as analytic bridges to prime distributions, arithmetic geometry and spectral theory. Central objectives include establishing analytic continuation, locating zeros and poles, and obtaining bounds on special values, all of which underpin results in prime number theory, equidistribution phenomena, and Diophantine applications. The Langlands programme predicts vast networks of correspondences—functorial lifts—between automorphic spectra on different groups, while analytic methods such as amplification and trace formulas propel progress on subconvexity and zero-free regions. Together, automorphic forms and L-functions form a unifying framework linking algebraic, analytic and geometric aspects of modern number theory.

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Automorphic Forms and L-Function Theory publication trend

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Technical terms

Automorphic form: A smooth function on a quotient of a reductive Lie group by a discrete arithmetic subgroup, satisfying invariance, growth and eigenvalue conditions under Hecke operators.

L-function: A complex analytic function formed by a Dirichlet series with Euler product, attached to arithmetic or spectral data, encoding global arithmetic properties and conjecturally satisfying a functional equation.

Cuspidal representation: An irreducible automorphic representation corresponding to square-integrable automorphic forms that vanish on all boundary components of the quotient.

Functoriality: A principle predicting that natural operations on representations (such as symmetric powers or tensor products) correspond to transfers between automorphic spectra on different groups.

Subconvexity: A bound on the growth of an L-function in the critical strip that improves upon the general convexity (or trivial) estimate, crucial for applications to equidistribution and Diophantine problems.

References

1. Symmetric power functoriality for holomorphic modular forms. Publications mathématiques de l'IHÉS (2021).
2. Symmetric power functoriality for holomorphic modular forms, II. Publications mathématiques de l'IHÉS (2021).
3. Standard zero-free regions for Rankin–Selberg L-functions via sieve theory. Mathematische Zeitschrift (2018).
4. Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $t$-aspect. Journal of the European Mathematical Society (2021).