Automorphic Forms and Algebraic Structures

Automorphic forms lie at the heart of modern number theory and arithmetic geometry, linking analysis on symmetric spaces with deep algebraic structures. Broadly speaking, an automorphic form is a complex-valued function on a reductive algebraic group or its associated symmetric domain that satisfies invariance properties under a discrete arithmetic subgroup. Classical examples include modular forms on the upper half-plane, which encode rich information about elliptic curves, partition functions and L-series. Ongoing research explores generalisations to higher rank groups, leading to connections with Langlands functoriality, representation theory and the theory of L-functions. Algebraic structures such as Hecke algebras organise the action of correspondences, while Shimura varieties provide geometric realisations of automorphic representations and furnish arithmetic invariants via their special points and cycle classes.

The study of theta lifts and Green functions has proven instrumental in understanding special values of L-series and in formulating arithmetic intersection theories on modular and orthogonal Shimura varieties. These developments have yielded finiteness results for height pairings of algebraic cycles and new instances of reciprocity laws. In parallel, advances in p-adic and harmonic Maass forms are revealing refined congruence phenomena and integral structures in Fourier expansions. Thanks to these interconnections, automorphic methods now inform progress in cryptography, mathematical physics and coding theory, underscoring the global significance and practical reach of this vibrant field.

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Automorphic Forms and Algebraic Structures publication trend

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Technical terms

Automorphic form: A function on a reductive algebraic group or symmetric space invariant under a discrete arithmetic subgroup, often characterised by analytic, algebraic and growth conditions.

Modular form: A holomorphic automorphic form on the upper half-plane transforming under a subgroup of SL(2,ℤ), with prescribed behaviour at cusps.

Shimura variety: A higher-dimensional generalisation of modular curves parametrising abelian varieties with additional endomorphism and level structures, realising automorphic representations geometrically.

Theta lift: An integral transform pairing automorphic forms on dual reductive pairs via a theta kernel, used to transfer automorphicity between groups.

Mock modular form: A holomorphic function on the upper half-plane whose nonholomorphic completion transforms like a modular form, with shadows given by unary theta functions.

Green function: A fundamental solution to a Laplace-type operator on a symmetric domain, used to define arithmetic intersection pairings on Shimura varieties.

Fourier coefficient: A component in the Fourier expansion of an automorphic form, encoding arithmetic data such as representation numbers or values of L-series.

References

1. Massive theta lifts. Journal of High Energy Physics (2023).
2. CM values of higher automorphic Green functions for orthogonal groups. Inventiones Mathematicae (2021).
3. A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles. Research in Number Theory (2015).
4. Mock modular forms with integral Fourier coefficients. Advances in Mathematics (2022).