Deligne conjecture (critical L-values)

Deligne’s conjecture predicts that critical values of a motive’s L-function, after normalization by specified powers of 2πi and motivic periods, are algebraic and encode arithmetic information.

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Critical value: A value of an L-function at an integer where its gamma factors are regular on both sides of the functional equation. The conjecture singles out these integers as the points where period-normalized values should be algebraic.
Motive: An abstract object intended to unify the cohomological invariants of algebraic varieties. Deligne formulates the conjecture for motives and their realizations.
Pierre Deligne: A Belgian mathematician whose work reshaped algebraic geometry, representation theory, and number theory. He formulated the conjecture on special values of motivic L-functions.
Rationality of special values of Dirichlet L-functions: The algebraicity of suitable normalized values of Dirichlet L-functions at nonpositive integers. These values furnish classical examples of critical-value algebraicity predicted by Deligne.
Motivic cohomology: Cohomology theories built from algebraic cycles that are intended to capture universal motivic information. Its relationship to regulators underlies broader conjectures about special L-values.
Motivic L-function: An L-function assembled from the realizations of a motive, encoding its arithmetic through Euler factors. Its special values are the quantities Deligne predicts periods can normalize.
Period (number theory): A complex number arising by comparing algebraic and analytic structures, often represented by an integral of algebraic differential forms. Motivic periods are the transcendental factors removed from critical L-values.
Valeurs de fonctions L et périodes d’intégrales: Pierre Deligne’s 1979 paper proposing period formulas for special values of motivic L-functions. This paper gives the conjecture’s foundational formulation.
Bernoulli numbers: A sequence of rational numbers governing power sums and appearing in special values of zeta and L-functions. They make normalized critical values explicit in basic Tate-motive examples.
Regulator map: A map from motivic or algebraic-cycle data to a cohomology group with real or complex periods. Regulators enter extensions of special-value formulas beyond Deligne’s critical values.