Deligne's Conjecture for Critical Motive Values

Deligne's Conjecture for Critical Motive Values

Definitions and mathematical statement

For a pure motive $M/{\mathbb{Q}}$ with coefficient field $E$, let $H_B^\pm(M)$ be the complex-conjugation eigenspaces of its Betti realization and $d_\pm$ their $E$-dimensions. The periods $c^\pm(M)$ are determinants of the specified comparison maps to de Rham filtration quotients, defined modulo $E^\times$. For a critical integer $n$, neither archimedean factor for $M$ at $n$ nor for $M^\vee$ at $1-n$ has a pole. Put $s=(-1)^n$. The assertion is \[ \frac{L(M,n)}{(2\pi i)^{n d_s}c^s(M)}\in E \ \subseteq\ E\otimes_{{\mathbb{Q}}}{\mathbb{C}}\simeq\prod_{\sigma:E\hookrightarrow{\mathbb{C}}}{\mathbb{C}}. \] The membership is diagonal across all embeddings. The finite $L$-values and compatible realizations are prerequisites; continuation and nonvanishing are not additional conclusions of this period law. The recorded scope fixes the period and sign conventions.

Formulation and concept references: [S1].

Short English statement

After normalization by the predicted periods, do all coefficient-field embeddings of a critical motivic $L$-value arise from one element of that field?

Research attempt

Current frontier and normalization audit.

Deligne's original Sections 1.7 and 2.6 were inspected, including the comparison determinants and the coefficient-field formulation [E1]. The supplied sign and finite-$L$ conventions are retained. Recent factorization results for automorphic motives over CM fields have regularity and archimedean rationality hypotheses; they do not establish the statement for every pure motive over ${\mathbb{Q}}$ [E2]. The present attempt makes no new analytic-continuation or nonvanishing assertion: the selected finite values and compatible realizations remain prerequisites as in the original record.

Attack route.

Restriction of coefficients can turn an $E$-valued assertion into scalar statements, but may keep only a norm and lose the compatibility between embeddings. Componentwise algebraicity has the same weakness. The proposed repair is to use all coefficient-weighted traces, or a finite set of coefficient-weighted determinant probes. The linear-algebraic reduction will be proved exactly; the missing motivic step is to realize those probes by comparison identities, rather than treating arbitrary scalar operations on $L$-values as new motives.

Attempt: trace-dual recovery of a diagonal element.

Let $d=[E:{\mathbb{Q}}]$ and write $B=E\otimes_{{\mathbb{Q}}}{\mathbb{C}}$. In its product description, $z=(z_\sigma)_{\sigma:E\hookrightarrow{\mathbb{C}}}$ is an arbitrary vector; in the application it is the period-normalized vector in the original statement. Define the ${\mathbb{C}}$-linear functional $\operatorname{Tr}_B(z)=\sum_\sigma z_\sigma$. Choose a ${\mathbb{Q}}$-basis $e_1,\ldots,e_d$ of $E$ and its trace-dual basis $e_1^*,\ldots,e_d^*$, characterized by $\operatorname{Tr}_{E/{\mathbb{Q}}}(e_i e_j^*)=\delta_{ij}$. This basis exists because the trace pairing of a number field is nondegenerate. Extending that pairing to ${\mathbb{C}}$ gives \begin{equation}\label{wsref-0-65-71-37-31-3a-72-65-63-6f-6e-73-74-72-75-63-74} z=\sum_{i=1}^d\operatorname{Tr}_B(e_i z)e_i^*. \end{equation} Indeed, both sides have the same pairing with every $e_j$, which is a ${\mathbb{C}}$-basis of $B$. Nondegeneracy then gives equality. In particular, \begin{equation}\label{wsref-0-65-71-37-31-3a-63-72-69-74-65-72-69-6f-6e} z\in E\quad\Longleftrightarrow\quad \operatorname{Tr}_B(e_i z)\in{\mathbb{Q}}\quad(1\leq i\leq d). \end{equation} No prior algebraicity of the individual $z_\sigma$ is needed. The converse direction is not a heuristic use of all embeddings: the displayed formula constructs the one element of $E$ they must represent.

Applied to \[ z_\sigma=\frac{L(M,\sigma,n)} {(2\pi i)^{n d_s}c^s(M)_\sigma},\qquad s=(-1)^n, \] this gives a finite family of scalar rationality targets. It preserves the source's sign and comparison determinants; it is not a replacement of them by their absolute values. A change of allowed $E$-rational bases multiplies $z$ by a diagonal element of $E^\times$. Since multiplication by such an element maps $E$ bijectively onto itself, the criterion is independent of that choice, exactly as the original period law requires [E1].

Attempt: recovering the traces from finite determinant probes.

For each basis element form the degree-at-most-$d$ polynomial \[ Q_i(T)=\operatorname{Norm}_{B/{\mathbb{C}}}(1+T e_i z) =\prod_\sigma(1+T\sigma(e_i)z_\sigma). \] Its linear coefficient is $\operatorname{Tr}_B(e_i z)$. Fix $d+1$ distinct rational numbers $t_0,\ldots,t_d$. If \begin{equation}\label{wsref-0-65-71-37-31-3a-70-72-6f-62-65-73} Q_i(t_j)\in{\mathbb{Q}}\qquad(1\leq i\leq d, \ 0\leq j\leq d), \end{equation} then all coefficients of $Q_i$ are rational. To see this, the coefficient vector is obtained from its value vector by inverting the rational Vandermonde matrix $(t_j^k)_{0\leq j,k\leq d}$. Its determinant is $\prod_{a<b}(t_b-t_a)\ne0$, so its inverse also has rational entries. Thus (eq71:probes) implies (eq71:criterion) and hence $z\in E$. Conversely, $z\in E$ makes every $Q_i$ a rational polynomial and implies all these probe identities. This is a finite exact equivalence, not an approximation by numerically recognized algebraic numbers.

The useful conceptual difference from a lone norm test is that multiplication by the $e_i$ remembers which coefficient embedding is which. A hypothetical coefficient-compatible comparison construction for these probes would settle the period membership assertion without requiring separate nonvanishing. At present no such general construction is supplied. In particular, tensor products multiply local Euler factors in prescribed ways; they do not make $1+t e_i z$ into the normalized critical value of an automatically available auxiliary motive.

Stress test: a counterexample to scalar-only descent.

Take $E={\mathbb{Q}}(\sqrt2)$ and order the two embeddings by $\sqrt2\mapsto\sqrt2,-\sqrt2$. Set $z=(2,3)$. Both coordinates are rational, its trace is $5$, and its norm is $6$. Even the whole characteristic polynomial \[ (T-2)(T-3)=T^2-5T+6 \] is rational. Nevertheless $z\notin E$: if $(a+b\sqrt2,a-b\sqrt2)=(2,3)$ with $a,b\in{\mathbb{Q}}$, then $a=5/2$ and $b=-1/(2\sqrt2)\notin{\mathbb{Q}}$. The omitted probe detects the defect: \[ \operatorname{Tr}_B(\sqrt2 z)=-\sqrt2\notin{\mathbb{Q}}. \] Thus all characteristic coefficients of the unweighted vector being rational is still weaker than the target. Numerical matches of a norm, individual minimal polynomials, or independent algebraicity in each embedding do not supply the common coefficient-field element.

Likewise $(0,1)$ cannot be a diagonal element of any quadratic field: an embedding of a field is injective, so an element with one zero embedding is zero in every embedding. In contrast, the zero vector satisfies (eq71:criterion) and (eq71:probes); the argument introduces no extraneous nonvanishing demand. These examples concern arbitrary vectors in $B$, not actual motivic $L$-values, and are therefore not counterexamples to Deligne's conjecture. The exact verification script checks the quadratic basis and probes symbolically.

Outcome.

Outcome: Conditional reduction.

The coefficient-equivariant conclusion is reduced exactly to finitely many weighted trace rationalities, equivalently to a finite rational determinant probe test. The reconstruction and the failure of unweighted scalar information are proved. These are elementary descent lemmas developed for the notebook, without a priority claim.

The decisive missing input is arithmetic: establish those weighted identities for the actual normalized comparison vector of every motive and every critical integer in the given scope. Scalar special-value results for restricted automorphic families do not supply that input automatically. No new general period theorem, analytic continuation, or nonvanishing statement is claimed.

Sources

• S1 Kazuki Morimoto, On special values of L-functions for quaternion unitary groups of degree2 and GL$2$, Oberwolfach Report22/2014; with Deligne's definitions and twist convention. https://ems.press/content/serial-article-files/46514?nt=1. Catalogue formulation source.
• S2 Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields. https://arxiv.org/abs/2509.02303. Catalogue research/status reference; not a proof endorsement.
• S3 Motivic pieces of curves: L-functions and periods. https://arxiv.org/abs/2601.21934. Catalogue research/status reference; not a proof endorsement.
• E1 Pierre Deligne, Valeurs de fonctions $L$ et périodes d'intégrales, Proceedings of Symposia in Pure Mathematics 33, part 2 $1979$, 313--346, Sections 1.7 and 2.1--2.6. https://publications.ias.edu/sites/default/files/33_Valeursde.pdf. Original comparison and coefficient conventions inspected 27 September 2026; supplements the incorporated definitions in [S1].
• E2 Harald Grobner, Michael Harris and Jie Lin, Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields, arXiv:2509.02303 $2025$, abstract and stated automorphic hypotheses. https://arxiv.org/abs/2509.02303. Inspected 27 September 2026; regularity and archimedean rationality assumptions are not removed.