On p-adic L-functions for symplectic representations of GL(N) over number fields
Pith reviewed 2026-05-24 08:18 UTC · model grok-4.3
The pith
A distribution on Gal_p interpolates all critical L-values at p for symplectic cuspidal representations of GL_N over number fields when a regular non-critical spin p-refinement exists.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The author constructs a distribution L_p(tilde π) on Gal_p that interpolates all the standard critical L-values of π at p, for π a regular algebraic cuspidal automorphic representation of GL_N(A_F) of symplectic type that is spherical at primes dividing p and admits a regular non-critical spin p-refinement to Q-parahoric level, where Q is the (n,n)-parabolic. The distribution satisfies a natural growth condition, becoming a bounded measure when the refinement is ordinary. This leads to the non-vanishing of L(π × (χ ∘ N_{F/Q}), 1/2) for all but finitely many Dirichlet characters χ of p-power conductor when π is unitary with very regular weight and Q-ordinary at all primes above p.
What carries the argument
The distribution L_p(tilde π) on Gal_p, built from the regular non-critical spin p-refinement of π to Q-parahoric level and shown to interpolate the critical L-values.
Load-bearing premise
π must be spherical at all primes dividing p and admit a regular non-critical spin p-refinement to the (n,n)-parabolic level.
What would settle it
A concrete symplectic representation π meeting the hypotheses for which the constructed distribution at a specific point of Gal_p fails to equal the corresponding critical L-value of π.
read the original abstract
Let $F$ be a number field, and $\pi$ a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_N(\mathbb{A}_F)$ of symplectic type. When $\pi$ is spherical at all primes $\mathfrak{p}|p$, we construct a $p$-adic $L$-function attached to any regular non-critical spin $p$-refinement $\tilde\pi$ of $\pi$ to $Q$-parahoric level, where $Q$ is the $(n,n)$-parabolic. More precisely, we construct a distribution $L_p(\tilde\pi)$ on the Galois group $\mathrm{Gal}_p$ of the maximal abelian extension of $F$ unramified outside $p\infty$, and show that it interpolates all the standard critical $L$-values of $\pi$ at $p$ (including, for example, cyclotomic and anticyclotomic variation when $F$ is imaginary quadratic). We show that $L_p(\tilde\pi)$ satisfies a natural growth condition; in particular, when $\tilde\pi$ is ordinary, $L_p(\tilde\pi)$ is a (bounded) measure on $\mathrm{Gal}_p$. As a corollary, when $\pi$ is unitary, has very regular weight, and is $Q$-ordinary at all $\mathfrak{p}|p$, we deduce non-vanishing $L(\pi\times(\chi\circ N_{F/\mathbb{Q}}),1/2) \neq 0$ of the twisted central value for all but finitely many Dirichlet characters $\chi$ of $p$-power conductor.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript constructs a p-adic distribution L_p(tilde π) attached to a regular non-critical spin p-refinement tilde π of a regular algebraic cuspidal automorphic representation π of GL_N(A_F) of symplectic type. Under the hypotheses that π is spherical at all primes above p, the distribution is defined on Gal_p (the Galois group of the maximal abelian extension of F unramified outside p and infinity) and is shown to interpolate all standard critical L-values of π at p, including cyclotomic and anticyclotomic twists when F is imaginary quadratic. The distribution satisfies a natural growth condition (reducing to a bounded measure when tilde π is ordinary), and a non-vanishing corollary is deduced for twisted central values L(π × (χ ∘ N_{F/Q}), 1/2) when π is unitary, very regular, and Q-ordinary at primes above p.
Significance. If the construction and interpolation properties hold, the result supplies a new family of p-adic L-functions for symplectic representations of GL(N) over general number fields, extending existing constructions in the literature on p-adic L-functions and Iwasawa theory. The explicit growth condition and the non-vanishing corollary under ordinary hypotheses are concrete strengths; the conditional hypotheses on sphericity and the existence of a regular non-critical spin refinement are clearly stated, which limits the scope but strengthens the precision of the claims.
minor comments (3)
- The definition of the (n,n)-parabolic Q and the notion of 'spin p-refinement' are used from the outset; a brief reminder of these notions in the introduction would improve accessibility for readers outside the immediate subfield.
- The growth condition is stated in the abstract but the precise statement (e.g., the precise bound or the space of distributions) should be recalled explicitly when the non-vanishing corollary is deduced, to make the logical dependence transparent.
- Notation for Gal_p and the precise meaning of 'standard critical L-values at p' could be clarified with a short sentence or reference to the relevant L-function normalization early in the paper.
Simulated Author's Rebuttal
We thank the referee for their detailed summary, positive assessment of significance, and recommendation of minor revision. No specific major comments were provided in the report, so we have no individual points requiring point-by-point rebuttal or revision at this stage. The manuscript already states its hypotheses clearly (sphericity at p, existence of regular non-critical spin refinement), and we believe the construction and interpolation results stand as described.
Circularity Check
No significant circularity; construction and interpolation are the stated theorem
full rationale
The paper's central claim is the existence of a distribution L_p(tilde pi) under explicit hypotheses on pi (spherical at p, regular non-critical spin p-refinement to Q-parahoric level) together with a proof that this object interpolates the critical L-values and satisfies a growth condition. This is a standard constructive result in p-adic L-functions; the interpolation property is the content of the theorem rather than a self-referential fit or renaming. No load-bearing self-citation chains, ansatze smuggled via prior work, or fitted parameters relabeled as independent predictions appear in the abstract or stated argument structure. The non-vanishing corollary is derived from the growth condition under further ordinary/very regular hypotheses and is therefore independent of the core construction. The derivation is self-contained against external benchmarks.
discussion (0)
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