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Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions?

Christopher Deninger

TL;DR

This work proposes a Birch–Swinnerton-Dyer–type conjecture for Dedekind zeta functions of number fields by articulating a conjectural cohomological formalism that attaches to each number field a vector space $V_K$ with a symplectic structure and a star operator, whose dimension encodes the order of vanishing of $\zeta_K(s)$ at $s=\tfrac12$. Under Serre's vanishing conjecture, the paper shows that such functors $K \mapsto V_K$ exist abstractly and are all isomorphic, with a common automorphism group that is $2$-torsion and abelian. The framework relies on a conjectural cohomology theory $H^i(\mathcal{Y}_K, \mathcal{C})$ with a derivation $\theta$ and a motivic variant $H^1_{\mathcal{M}}(\mathcal{Y}_K, \mathbb{C}(1/2))$, linking vanishing orders to $\theta$-eigenspaces and Galois representations, particularly symplectic representations with root number $-1$. Theorem 2.15 establishes the existence of such functors, while Theorem 2.17 shows their essential uniqueness up to isomorphism and describes the automorphism group, thereby isolating structural constraints and pointing to a concrete, yet unresolved, realization problem. The discussion also analyzes obstacles from exponential motives and Fresán–Jossen results, clarifying what would be required to obtain a concrete candidate for $V_K$ and the leading coefficient, and highlighting open directions, including potential extensions via twisted motivic theories and Zagier-type conjectures.

Abstract

A Birch and Swinnerton-Dyer conjecture for number fields $K / \mathbb{Q}$ would assert that $dim V_K = ord_{s = 1/2} ζ_K (s)$ for some vector space functorially attached to $K$. Presently there is no natural candidate for the $V_K$'s. However, assuming $V_K$ is of a cohomological nature and assuming a conjecture of Serre on the vanishing order of $ζ_K (s)$ at $s = 1/2$ we show that such functors $K \mapsto V_K$ (with natural extra structures) exist and are all isomorphic. Their common automorphism group is $2$-torsion and abelian.

Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions?

TL;DR

This work proposes a Birch–Swinnerton-Dyer–type conjecture for Dedekind zeta functions of number fields by articulating a conjectural cohomological formalism that attaches to each number field a vector space with a symplectic structure and a star operator, whose dimension encodes the order of vanishing of at . Under Serre's vanishing conjecture, the paper shows that such functors exist abstractly and are all isomorphic, with a common automorphism group that is -torsion and abelian. The framework relies on a conjectural cohomology theory with a derivation and a motivic variant , linking vanishing orders to -eigenspaces and Galois representations, particularly symplectic representations with root number . Theorem 2.15 establishes the existence of such functors, while Theorem 2.17 shows their essential uniqueness up to isomorphism and describes the automorphism group, thereby isolating structural constraints and pointing to a concrete, yet unresolved, realization problem. The discussion also analyzes obstacles from exponential motives and Fresán–Jossen results, clarifying what would be required to obtain a concrete candidate for and the leading coefficient, and highlighting open directions, including potential extensions via twisted motivic theories and Zagier-type conjectures.

Abstract

A Birch and Swinnerton-Dyer conjecture for number fields would assert that for some vector space functorially attached to . Presently there is no natural candidate for the 's. However, assuming is of a cohomological nature and assuming a conjecture of Serre on the vanishing order of at we show that such functors (with natural extra structures) exist and are all isomorphic. Their common automorphism group is -torsion and abelian.

Paper Structure

This paper contains 4 sections, 4 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Dedekind zeta functions at $s = 1 / 2$ and cohomology
  3. Proofs
  4. Further remarks

Key Result

Theorem 2.16

There is a covariant functor with the following properties, where $\tilde{V} (\alpha)$ is the adjoint of $V (\alpha)$ 1) $\tilde{V} (\alpha) V (\alpha) = \mathrm{id}$ on $V_K$ for $\alpha : K \hookrightarrow L$. 2) If $L / K$ is Galois with group $G$, then 3) If $K / {\mathbb{Q}}$ is Galois with group $G$, then as ${\mathbb{C}} [G]$-modules

Theorems & Definitions (4)

  • Theorem 2.16
  • Theorem 2.17
  • Proposition 3.1
  • Theorem 4.1: J. Fresán and P. Jossen