12.2 L-functions and their properties

L-functions are complex-valued functions that encode crucial arithmetic information. They're associated with various mathematical objects, from the Riemann zeta function to Galois representations, and are defined using intricate products and determinants.

These functions have fascinating analytic properties, including convergence in specific regions and meromorphic continuation. Their zeros and poles hold key insights, with applications ranging from prime distribution to the Langlands Program and modularity theorems.

L-functions and Their Analytic Properties

L-functions of Galois representations

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• L-functions encode arithmetic information as complex-valued functions associated with arithmetic objects (Riemann zeta function)
• Galois representations map absolute Galois group to general linear group as continuous homomorphisms over finite-dimensional vector spaces (cyclotomic character)
• L-function for a Galois representation defined as $L(s, \rho) = \prod_{p \text{ prime}} \det(I - \rho(\text{Frob}_p)p^{-s})^{-1}$ where $s$ complex variable, $\rho$ Galois representation, $\text{Frob}_p$ Frobenius element at prime $p$

Analytic properties of L-functions

• Convergence occurs in right half-plane, represented as Euler product
• Meromorphic continuation extends L-function beyond initial convergence domain for complex plane analysis (Riemann zeta function)
• Functional equation relates L-function values at $s$ to $1-s$, involving gamma factors and conductor
• Representation-theoretic methods employ character theory and Fourier analysis on adelic groups
• Proof steps involve local factor decomposition, Archimedean and non-Archimedean place analysis, Poisson summation formula application
• Meromorphic continuation enables special value study, crucial for conjecture formulation and proof (Riemann Hypothesis)

Zeros and poles in L-functions

• Critical zeros lie on critical line $\text{Re}(s) = \frac{1}{2}$, Riemann Hypothesis analog posits all non-trivial zeros on this line
• Poles, usually finite, correspond to important arithmetic information
• Prime Ideal Theorem describes asymptotic distribution of prime ideals, related to associated zeta function zeros
• Chebotarev Density Theorem generalizes Dirichlet's theorem on primes in arithmetic progressions using Frobenius elements in Galois groups
• Applications include studying Galois extensions of number fields and analyzing prime splitting behavior

Significance in arithmetic geometry

• L-functions encode arithmetic properties (class numbers, regulators, special values related to periods)
• Birch and Swinnerton-Dyer Conjecture connects elliptic curve rank to L-function vanishing order
• Langlands Program links number theory and representation theory with L-functions central to correspondence formulation
• Modularity Theorem proves elliptic curves over Q are modular using L-functions
• Sato-Tate Conjecture describes Frobenius element statistical distribution, proven for elliptic curves with non-integral j-invariant
• Applications to algebraic varieties involve Hasse-Weil zeta functions, finite field point study, and étale cohomology connections