3.3 Eisenstein series

Eisenstein series are key objects in modular forms and automorphic representations. They connect complex analysis, number theory, and algebraic geometry, serving as building blocks for more general modular forms and helping us understand their properties.

These series come in classical and generalized forms, with various convergence conditions and modularity properties. They have important applications in studying L-functions, zeta functions, and the spectral decomposition of modular forms, bridging different areas of mathematics.

Definition of Eisenstein series

• Fundamental objects in the study of modular forms and automorphic representations
• Play a crucial role in arithmetic geometry by connecting complex analysis, number theory, and algebraic geometry
• Serve as building blocks for constructing more general modular forms and understanding their properties

Classical Eisenstein series

Top images from around the web for Classical Eisenstein series

• 

  Rankin–Eisenstein classes in Coleman families | SpringerLink View original

  Is this image relevant?

• 

  Eisensteinserie – Wikipedia View original

  Is this image relevant?

• 

  Rankin–Eisenstein classes in Coleman families | SpringerLink View original

  Is this image relevant?

• 

  Rankin–Eisenstein classes in Coleman families | SpringerLink View original

  Is this image relevant?

• 

  Eisensteinserie – Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Classical Eisenstein series

• 

  Rankin–Eisenstein classes in Coleman families | SpringerLink View original

  Is this image relevant?

• 

  Eisensteinserie – Wikipedia View original

  Is this image relevant?

• 

  Rankin–Eisenstein classes in Coleman families | SpringerLink View original

  Is this image relevant?

• 

  Rankin–Eisenstein classes in Coleman families | SpringerLink View original

  Is this image relevant?

• 

  Eisensteinserie – Wikipedia View original

  Is this image relevant?

1 of 3

• Non-holomorphic modular forms defined on the upper half-plane $\mathbb{H} = \{z \in \mathbb{C} : \text{Im}(z) > 0\}$
• Constructed using the sum $E_k(z) = \sum_{(m,n) \neq (0,0)} \frac{1}{(mz + n)^k}$ for even integers $k \geq 4$
• Weight $k$ determines the transformation properties under the modular group $\text{SL}_2(\mathbb{Z})$
• Provide examples of modular forms that are not cusp forms

Generalized Eisenstein series

• Extended definition to broader contexts, including higher-dimensional spaces and more general groups
• Include adelic Eisenstein series defined on adelic groups
• Constructed using characters of parabolic subgroups
• Allow for the study of automorphic forms on more general reductive groups

Properties of Eisenstein series

Convergence conditions

• Absolute convergence for classical Eisenstein series when the weight $k > 2$
• Conditional convergence for $k = 2$, requiring careful summation techniques
• Convergence rates depend on the imaginary part of $z$ and the weight $k$
• Generalized Eisenstein series converge under specific conditions on the defining parameters

Modularity properties

• Transform under the action of the modular group $\text{SL}_2(\mathbb{Z})$ according to their weight
• Satisfy the equation $E_k(\gamma z) = (cz + d)^k E_k(z)$ for $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}_2(\mathbb{Z})$
• Invariant under translations $z \mapsto z + 1$
• Generalize to automorphy properties for Eisenstein series on other groups

Fourier expansions

• Admit Fourier expansions of the form $E_k(z) = 1 + \frac{2}{\zeta(k)} \sum_{n=1}^{\infty} \sigma_{k-1}(n) q^n$, where $q = e^{2\pi i z}$
• $\sigma_{k-1}(n)$ denotes the sum of the $(k-1)$-th powers of the divisors of $n$
• Constant term relates to special values of the Riemann zeta function
• Fourier coefficients encode important arithmetic information (divisor sums)

Applications in arithmetic geometry

• Eisenstein series bridge complex analysis, number theory, and algebraic geometry
• Provide concrete examples of modular forms with explicit arithmetic properties
• Used to construct more general automorphic forms and study their properties

Modular forms

• Eisenstein series form a subspace of the space of modular forms
• Complement the space of cusp forms in the spectral decomposition
• Used to construct basis elements for spaces of modular forms
• Provide explicit examples for studying the arithmetic of modular forms

L-functions

• Eisenstein series associated with L-functions of Dirichlet characters
• Mellin transforms of Eisenstein series yield products of L-functions
• Used to study analytic properties of L-functions (functional equations, special values)
• Provide a link between automorphic forms and arithmetic objects like elliptic curves

Zeta functions

• Closely related to the Riemann zeta function and its generalizations
• Eisenstein series for $\text{SL}_2(\mathbb{Z})$ involve values of the Riemann zeta function
• Used to study properties of more general zeta functions associated with number fields and varieties
• Provide a geometric interpretation of special values of zeta functions

Eisenstein series vs cusp forms

Spectral decomposition

• Space of modular forms decomposes into Eisenstein series and cusp forms
• Eisenstein series represent the continuous spectrum of the Laplacian operator
• Cusp forms correspond to the discrete spectrum of the Laplacian
• Spectral theory of automorphic forms built on this fundamental decomposition

Growth behavior

• Eisenstein series exhibit polynomial growth near the cusps
• Cusp forms decay exponentially near the cusps
• Growth behavior reflected in the Fourier expansions of these functions
• Distinct growth properties lead to different arithmetic and analytic applications

Algebraic properties

Hecke operators

• Eisenstein series are eigenfunctions of Hecke operators
• Eigenvalues of Hecke operators on Eisenstein series relate to Dirichlet L-functions
• Hecke algebra action on Eisenstein series simpler than on cusp forms
• Used to study the action of Hecke operators on more general automorphic forms

Petersson inner product

• Eisenstein series are not square-integrable with respect to the Petersson inner product
• Orthogonal to the space of cusp forms under a regularized inner product
• Lack of square-integrability related to their growth behavior at cusps
• Petersson inner product used to study relationships between Eisenstein series and cusp forms

Geometric interpretation

Lattice points

• Classical Eisenstein series related to counting lattice points in the complex plane
• Sum over lattice points (excluding the origin) in the definition of Eisenstein series
• Geometric interpretation connects to the theory of quadratic forms
• Lattice point interpretation generalizes to higher-dimensional settings

Fundamental domains

• Eisenstein series defined on the upper half-plane modulo the action of $\text{SL}_2(\mathbb{Z})$
• Fundamental domain for this action given by the standard modular domain
• Behavior of Eisenstein series on the boundary of the fundamental domain crucial for understanding their properties
• Generalizes to more complex fundamental domains for other groups

Eisenstein series on other groups

Siegel modular forms

• Generalization of classical Eisenstein series to higher-dimensional symplectic groups
• Defined on the Siegel upper half-space of symmetric complex matrices
• Used to study arithmetic properties of abelian varieties
• Provide examples of automorphic forms on $\text{Sp}_{2n}(\mathbb{R})$

Hilbert modular forms

• Eisenstein series defined for the Hilbert modular group
• Associated with totally real number fields
• Used to study arithmetic properties of abelian varieties with real multiplication
• Provide examples of automorphic forms on products of copies of $\text{SL}_2(\mathbb{R})$

Connections to number theory

Class numbers

• Fourier coefficients of Eisenstein series relate to class numbers of imaginary quadratic fields
• Used in explicit class number formulas for certain imaginary quadratic fields
• Provide a link between modular forms and algebraic number theory
• Class number relations generalize to higher-dimensional settings

Kronecker limit formula

• Relates the constant term in the Laurent expansion of Eisenstein series to special values of Dedekind eta function
• Connects Eisenstein series to the theory of complex multiplication
• Used to study arithmetic invariants of elliptic curves with complex multiplication
• Generalizes to limit formulas for Eisenstein series on other groups

Computational aspects

Algorithms for evaluation

• Efficient algorithms developed for computing Fourier coefficients of Eisenstein series
• Use of fast Fourier transform techniques to speed up computations
• Modular symbols methods adapted for Eisenstein series calculations
• Computational challenges increase with weight and level of the Eisenstein series

Numerical approximations

• Techniques for high-precision numerical evaluation of Eisenstein series
• Truncation methods for infinite sums with error bounds
• Use of theta functions and other special functions for improved convergence
• Numerical computations crucial for exploring properties and generating conjectures

Advanced topics

Eisenstein cohomology

• Cohomological interpretation of Eisenstein series in the context of locally symmetric spaces
• Relates to the study of cohomology of arithmetic groups
• Used to understand the structure of cohomology groups of arithmetic manifolds
• Connects Eisenstein series to topological and geometric properties of modular varieties

Eisenstein series on p-adic groups

• p-adic analogues of classical Eisenstein series
• Defined on p-adic symmetric spaces and Bruhat-Tits buildings
• Used in the study of p-adic L-functions and Iwasawa theory
• Provide examples of p-adic automorphic forms with arithmetic applications