Consequences of functional equations for pairs of p-adic L-functions

C\'edric Dion; Florian Ito Sprung

arxiv: 1906.08377 · v2 · pith:YWQ5AHPInew · submitted 2019-06-19 · 🧮 math.NT

Consequences of functional equations for pairs of p-adic L-functions

C\'edric Dion , Florian Ito Sprung This is my paper

Pith reviewed 2026-05-25 19:48 UTC · model grok-4.3

classification 🧮 math.NT

keywords p-adic L-functionselliptic curvessupersingular primesfunctional equationsIwasawa invariantsorders of vanishing

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The pith

Functional equations for pairs of p-adic L-functions imply coefficient relations, vanishing parity, and Iwasawa invariant invariance at supersingular primes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the functional equations satisfied by pairs of p-adic L-functions attached to elliptic curves at supersingular primes lead to three main consequences. First, the leading term of one function in the pair determines the sub-leading term of the other. Second, the orders of vanishing of the two functions have the same parity. Third, the Iwasawa invariants are the same for a p-adic L-function and its conjugate twist. These relations follow directly from the form of the functional equations and provide concrete links between analytic and arithmetic data in Iwasawa theory.

Core claim

The functional equations of the pairs of p-adic L-functions at supersingular primes imply a relationship between the leading and sub-leading terms, that the orders of vanishing have the same parity, and that the Iwasawa invariants are invariant under conjugate twists.

What carries the argument

The functional equations relating the pair of p-adic L-functions, which interchange the leading and sub-leading coefficients while preserving the Iwasawa invariants.

If this is right

The leading coefficient of one p-adic L-function is determined by the sub-leading coefficient of its pair member.
The orders of vanishing of the paired p-adic L-functions have the same parity.
The mu and lambda Iwasawa invariants remain unchanged under conjugate twists of the p-adic L-functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These relations could be used to compute Iwasawa invariants for one function using data from its pair without separate calculation.
Verification of the coefficient relations could be done by direct computation for specific elliptic curves with small conductors.

Load-bearing premise

The functional equations hold in the specific form that directly relates the coefficients and preserves the invariants under the twist.

What would settle it

A counterexample consisting of an elliptic curve at a supersingular prime where the leading term does not match the value predicted from the sub-leading term of the paired function would disprove the consequences.

read the original abstract

We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper takes known functional equations for supersingular p-adic L-functions and derives three explicit consequences: a leading-term relation, a parity statement on vanishing orders, and invariance of Iwasawa invariants under conjugate twists.

read the letter

The paper's main point is straightforward. It assumes the functional equations for the pair of p-adic L-functions attached to an elliptic curve at a supersingular prime, then pulls out three relations that follow from them. One links the leading and sub-leading coefficients using ideas from Wuthrich and Bianchi. Another gives a parity result on the orders of vanishing. The third shows that the Iwasawa invariants are unchanged under conjugate twists of the functions. These are presented as new statements built on earlier work rather than re-proving the equations themselves. The stress-test finds no internal contradictions once the equations are granted in the stated form, which matches the abstract's framing. The contribution is mostly organizational: it makes certain analytic-algebraic links explicit in a way that could support computations tied to the main conjecture. That is real but narrow progress inside Iwasawa theory for supersingular primes. The soft spot is dependence on the prior constructions of the functional equations. If those hold without extra normalizations or adjustments, the consequences are fine; the paper does not supply independent checks. The derivations appear mechanical rather than introducing new techniques, so the strength tracks the strength of the cited functional equations. This is for readers already working in this specialized corner of elliptic curves and Iwasawa theory. Someone outside that area will not find much to use. A referee can check the algebra without needing broad new ideas, so the paper deserves peer review even if the results stay incremental.

Referee Report

0 major / 3 minor

Summary. The manuscript derives consequences of the functional equations satisfied by pairs of p-adic L-functions attached to elliptic curves at supersingular primes p. The three main results are a relation between the leading and sub-leading coefficients (drawing on Wuthrich–Bianchi ideas), a parity statement on the orders of vanishing of the two p-adic L-functions, and the invariance of the associated Iwasawa invariants under conjugate twists.

Significance. If the functional equations are applied in the stated form, the results supply concrete arithmetic relations among coefficients, vanishing orders, and Iwasawa invariants that are not immediate from the existing literature on supersingular p-adic L-functions. These relations may be useful for studying the p-adic BSD conjecture or main conjectures in the supersingular setting, and the derivations appear to follow directly once the functional equations are granted.

minor comments (3)

The precise normalization and sign conventions in the functional equations (presumably stated in §2 or §3) should be recalled explicitly when deriving the coefficient relation in the leading-term section, to make the application of the Wuthrich–Bianchi argument fully self-contained.
Notation for the two p-adic L-functions (e.g., L_p^+ and L_p^-) and for the Iwasawa invariants μ^±, λ^± should be introduced once at the beginning and used consistently; occasional switches between subscript and superscript notation appear in the parity and invariance statements.
The manuscript cites the constructions of the p-adic L-functions but does not list the exact references for the functional equations themselves; adding these citations at the first use of each equation would improve traceability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the significance of the derived relations for p-adic BSD and main conjectures in the supersingular case, and for recommending minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No circularity: consequences derived from external functional equations

full rationale

The paper states it proves consequences (leading/sub-leading term relations via Wuthrich-Bianchi ideas, parity of vanishing orders, Iwasawa invariant invariance under twists) of functional equations for pairs of p-adic L-functions at supersingular primes. These functional equations are taken as given from prior literature rather than constructed, fitted, or defined inside the paper. The derivation chain therefore starts from an external assumption and produces independent consequences; no step reduces by construction to the paper's own inputs, self-citations, or ansatzes. This matches the default case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that functional equations exist and take a usable form for these p-adic L-functions; no free parameters or invented entities are indicated.

axioms (1)

domain assumption Functional equations for pairs of p-adic L-functions attached to elliptic curves at supersingular primes hold in the form required to derive the stated consequences.
The paper proves consequences of these equations rather than establishing the equations.

pith-pipeline@v0.9.0 · 5575 in / 1255 out tokens · 32691 ms · 2026-05-25T19:48:35.448373+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Distributions p-adiques associées aux séries de Hecke

Yvette Amice and Jacques Vélu. Distributions p-adiques associées aux séries de Hecke. pages 119–131. Astérisque, Nos. 24–25, 1975

work page 1975
[2]

Consequences of the functional equa tion of the p-adic L-function of an elliptic curve

Francesca Bianchi. Consequences of the functional equa tion of the p-adic L-function of an elliptic curve. Functiones et Approximatio, Commentarii Mathematici

work page
[3]

Mazur, J

B. Mazur, J. Tate, and J. Teitelbaum. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. , 84(1):1–48, 1986

work page 1986
[4]

On the p-adic L-function of a modular form at a supersingular prime

Robert Pollack. On the p-adic L-function of a modular form at a supersingular prime. Duke Math. J. , 118(3):523–558, 2003

work page 2003
[5]

The Šafarevič-Tate group in cyclotomic Zp-extensions at supersingular primes

Florian Sprung. The Šafarevič-Tate group in cyclotomic Zp-extensions at supersingular primes. J. Reine Angew. Math. , 681:199–218, 2013

work page 2013
[6]

On pairs of p-adic L-functions for weight-two modular forms

Florian Sprung. On pairs of p-adic L-functions for weight-two modular forms. Algebra Number Theory, 11(4):885–928, 2017

work page 2017
[7]

Ito Sprung

Florian E. Ito Sprung. Iwasawa theory for elliptic curve s at supersingular primes: a pair of main conjectures. J. Number Theory , 132(7):1483–1506, 2012

work page 2012
[8]

M. M. Višik. Nonarchimedean measures associated with Di richlet series. Mat. Sb. (N.S.) , 99(141)(2):248–260, 296, 1976

work page 1976
[9]

The sub-leading coeﬃcient of the L-function of an elliptic curve

Christian W uthrich. The sub-leading coeﬃcient of the L-function of an elliptic curve. In Pub- lications mathématiques de Besançon. Algèbre et théorie de s nombres, 2016 , volume 2016 of Publ. Math. Besançon Algèbre Théorie Nr. , pages 95–96. Presses Univ. Franche-Comté, Besançon, 2017. Cédric Dion, Département de mathématiques et de statistiqu e, Univers...

work page 2016

[1] [1]

Distributions p-adiques associées aux séries de Hecke

Yvette Amice and Jacques Vélu. Distributions p-adiques associées aux séries de Hecke. pages 119–131. Astérisque, Nos. 24–25, 1975

work page 1975

[2] [2]

Consequences of the functional equa tion of the p-adic L-function of an elliptic curve

Francesca Bianchi. Consequences of the functional equa tion of the p-adic L-function of an elliptic curve. Functiones et Approximatio, Commentarii Mathematici

work page

[3] [3]

Mazur, J

B. Mazur, J. Tate, and J. Teitelbaum. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. , 84(1):1–48, 1986

work page 1986

[4] [4]

On the p-adic L-function of a modular form at a supersingular prime

Robert Pollack. On the p-adic L-function of a modular form at a supersingular prime. Duke Math. J. , 118(3):523–558, 2003

work page 2003

[5] [5]

The Šafarevič-Tate group in cyclotomic Zp-extensions at supersingular primes

Florian Sprung. The Šafarevič-Tate group in cyclotomic Zp-extensions at supersingular primes. J. Reine Angew. Math. , 681:199–218, 2013

work page 2013

[6] [6]

On pairs of p-adic L-functions for weight-two modular forms

Florian Sprung. On pairs of p-adic L-functions for weight-two modular forms. Algebra Number Theory, 11(4):885–928, 2017

work page 2017

[7] [7]

Ito Sprung

Florian E. Ito Sprung. Iwasawa theory for elliptic curve s at supersingular primes: a pair of main conjectures. J. Number Theory , 132(7):1483–1506, 2012

work page 2012

[8] [8]

M. M. Višik. Nonarchimedean measures associated with Di richlet series. Mat. Sb. (N.S.) , 99(141)(2):248–260, 296, 1976

work page 1976

[9] [9]

The sub-leading coeﬃcient of the L-function of an elliptic curve

Christian W uthrich. The sub-leading coeﬃcient of the L-function of an elliptic curve. In Pub- lications mathématiques de Besançon. Algèbre et théorie de s nombres, 2016 , volume 2016 of Publ. Math. Besançon Algèbre Théorie Nr. , pages 95–96. Presses Univ. Franche-Comté, Besançon, 2017. Cédric Dion, Département de mathématiques et de statistiqu e, Univers...

work page 2016