3.2 Cusp forms

Cusp forms are a special type of modular form in arithmetic geometry. They vanish at cusps of the upper half-plane and have Fourier expansions without constant terms, encoding crucial arithmetic information about elliptic curves and other algebraic varieties.

These forms combine analytic properties with deep number-theoretic significance. They're central to automorphic form theory, providing insights into the structure of modular curves, L-functions, and Galois representations, with applications ranging from the modularity theorem to the Langlands program.

Definition of cusp forms

• Cusp forms represent a specialized subset of modular forms in arithmetic geometry
• Play a crucial role in understanding the arithmetic properties of elliptic curves and other algebraic varieties
• Provide deep insights into the structure of certain number-theoretic objects

Modular forms vs cusp forms

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• Modular forms transform predictably under the action of modular groups
• Cusp forms exhibit additional vanishing behavior at cusps of the upper half-plane
• Satisfy stricter growth conditions compared to general modular forms
• Possess Fourier expansions with specific properties

Fourier expansion of cusp forms

• Expressed as power series in $q = e^{2\pi i z}$ where z belongs to the upper half-plane
• Coefficients of the expansion encode important arithmetic information
• General form: $f(z) = \sum_{n=1}^{\infty} a_n q^n$
• Absence of constant term distinguishes cusp forms from other modular forms

Vanishing condition at cusps

• Cusp forms approach zero as z approaches any rational point on the real line
• Mathematically expressed as $\lim_{y \to \infty} f(x + iy) = 0$ for all real x
• Ensures integrability of cusp forms on the fundamental domain
• Crucial for defining the Petersson inner product and spectral theory

Properties of cusp forms

• Cusp forms exhibit unique characteristics that set them apart in arithmetic geometry
• Combine analytic properties with deep number-theoretic significance
• Form a central object of study in the theory of automorphic forms

Weight and level

• Weight k determines transformation behavior under scaling
• Level N specifies the congruence subgroup under which the form is invariant
• Notation $S_k(\Gamma_0(N))$ denotes the space of cusp forms of weight k and level N
• Higher weights generally correspond to more complex arithmetic information

Holomorphicity and modularity

• Cusp forms are holomorphic functions on the upper half-plane
• Satisfy the modularity condition: $f(\frac{az+b}{cz+d}) = (cz+d)^k f(z)$ for matrices in the congruence subgroup
• Holomorphicity extends to the cusps, a key property distinguishing them from other automorphic forms
• Modularity connects cusp forms to the geometry of modular curves

Growth conditions

• Cusp forms exhibit polynomial growth as Im(z) approaches infinity
• Specifically, $|f(z)| \leq C(Im(z))^{k/2}$ for some constant C
• This growth condition ensures absolute convergence of certain series involving cusp forms
• Crucial for defining L-functions associated with cusp forms

Hecke operators on cusp forms

• Hecke operators Tn act on spaces of cusp forms, preserving weight and level
• Define an algebra of operators that commute with each other
• Eigenforms under Hecke operators have arithmetic significance (newforms)
• Hecke eigenvalues relate to Fourier coefficients: $T_n f = \lambda_n f \implies a_n = \lambda_n a_1$

Spaces of cusp forms

• Form finite-dimensional vector spaces over complex numbers
• Exhibit rich structure related to modular curves and their geometry
• Provide a framework for studying arithmetic properties systematically

Dimension formulas

• Dimension of $S_k(\Gamma_0(N))$ depends on k, N, and genus of the modular curve
• For k ≥ 2, dim $S_k(\Gamma_0(N)) = (k-1)(g-1) + \frac{k}{2}\sum_{p|N} (1-\frac{1}{p})$ where g is the genus
• Special cases: dim $S_2(\Gamma_0(N))$ equals the genus of X0(N)
• Dimension zero for small weights and levels (k = 2, N ≤ 10)

Basis and generators

• Eisenstein series and theta series generate spaces of modular forms
• Delta function Δ(z) generates the space of cusp forms of weight 12 and level 1
• Basis elements can be constructed using modular symbols or trace formulas
• Hecke eigenforms often serve as a particularly useful basis

Petersson inner product

• Defines an inner product on the space of cusp forms
• Given by $\langle f, g \rangle = \int_{\mathcal{F}} f(z)\overline{g(z)}y^{k-2} dx dy$ where $\mathcal{F}$ is a fundamental domain
• Allows for orthogonalization of bases and spectral decomposition
• Relates to special values of L-functions through Rankin-Selberg method

Applications in number theory

• Cusp forms serve as a bridge between analytic and algebraic aspects of number theory
• Provide powerful tools for studying arithmetic properties of various mathematical objects
• Play a central role in modern approaches to longstanding conjectures

L-functions and cusp forms

• L-functions associated to cusp forms encode deep arithmetic information
• Defined as Dirichlet series: $L(s,f) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ where an are Fourier coefficients
• Satisfy functional equations relating values at s and k-s
• Critical values of L-functions often have arithmetic significance (periods, regulators)

Modularity theorem

• States that every elliptic curve over Q is modular, i.e., associated to a cusp form
• Proved by Wiles, Taylor, et al., leading to the proof of Fermat's Last Theorem
• Establishes deep connection between geometric objects (elliptic curves) and analytic objects (cusp forms)
• Generalizations to higher-dimensional varieties form active area of research

Serre's conjecture

• Relates odd, irreducible Galois representations to cusp forms
• Predicts existence of a cusp form for every such 2-dimensional representation
• Proved by Khare and Wintenberger, building on modularity theorem techniques
• Provides powerful tool for studying Galois representations via cusp forms

Computational aspects

• Efficient algorithms for computing with cusp forms crucial for applications
• Combine techniques from complex analysis, linear algebra, and number theory
• Enable exploration of conjectures and discovery of new phenomena

Algorithms for cusp forms

• Methods for computing bases of cusp form spaces (modular symbols, trace formulas)
• Algorithms for evaluating cusp forms at points in the upper half-plane
• Techniques for computing Hecke eigenvalues and Atkin-Lehner eigenvalues
• Implementations available in computer algebra systems (Sage, Magma, PARI/GP)

Modular symbols method

• Represents cusp forms using homology of modular curves
• Allows for efficient computation of Hecke operators and their eigenvalues
• Particularly effective for weight 2 cusp forms
• Generalizes to higher weights using vector-valued modular symbols

q-expansions and precision

• Cusp forms often computed and represented by their q-expansions
• Precision requirements depend on the application (bounds on coefficients, Sturm bound)
• Techniques for extending precision of q-expansions (Newton iteration, modular equations)
• Trade-offs between symbolic and numerical methods in computations

Generalizations and variations

• Cusp forms generalize in various directions, each capturing different aspects of arithmetic
• Provide framework for studying more general automorphic forms
• Connect to representation theory and harmonic analysis on more general groups

Vector-valued cusp forms

• Transform according to representations of the modular group
• Arise naturally in the study of modular forms on higher-dimensional domains
• Include examples like Siegel modular forms and Hilbert modular forms
• Provide tools for studying arithmetic of higher-dimensional varieties

Half-integral weight cusp forms

• Modular forms with half-integer weights (k/2 where k is odd)
• Require introduction of metaplectic group to define transformation properties
• Closely related to theta series and quadratic forms
• Shimura correspondence relates them to integral weight forms

Maass cusp forms

• Non-holomorphic analogues of cusp forms
• Eigenfunctions of the hyperbolic Laplacian on modular curves
• Conjectured to always have transcendental Fourier coefficients
• Connect to spectral theory and quantum chaos

Connections to other areas

• Cusp forms interface with diverse areas of mathematics and physics
• Provide concrete realizations of abstract concepts in representation theory
• Serve as testing ground for conjectures in arithmetic geometry

Cusp forms and elliptic curves

• Modularity theorem establishes bijection between rational elliptic curves and weight 2 newforms
• L-functions of elliptic curves coincide with those of corresponding cusp forms
• Allows transfer of information between analytic and geometric perspectives
• Generalizes to higher-dimensional abelian varieties (Sato-Tate conjecture)

Automorphic representations

• Cusp forms correspond to certain irreducible representations of adelic groups
• Provide concrete realizations of abstract representation-theoretic objects
• Allow application of harmonic analysis techniques to number-theoretic problems
• Connect to theory of Lie groups and symmetric spaces

Langlands program and cusp forms

• Cusp forms play central role in formulation and evidence for Langlands conjectures
• Relate to Galois representations via compatible systems of l-adic representations
• Functoriality conjectures predict relationships between cusp forms on different groups
• Provide concrete examples and testing ground for general conjectures in the program