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arxiv: 1907.07257 · v1 · pith:C7KNXOP6new · submitted 2019-07-16 · 🧮 math.NT

Mixed modular symbols and the generalized cuspidal 1-motive

Emmanuel Lecouturier This is my paper

Pith reviewed 2026-05-24 20:29 UTC · model grok-4.3

classification 🧮 math.NT
keywords mixed modular symbolsmodular symbolsEisenstein series1-motivesgeneralized Jacobianmodular curvesGamma_0(p)p-adic periods
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The pith

Mixed modular symbols extend the usual space to carry more information about Eisenstein series and enable construction of 1-motives tied to the generalized Jacobian of modular curves.

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

The paper defines the space of mixed modular symbols for any finite index subgroup Gamma of SL_2(Z) as an extension of the standard modular symbols. In some cases this extended space carries additional data on Eisenstein series. The symbols are used to construct 1-motives associated with the generalized Jacobian of modular curves. For the group Gamma_0(p) the resulting 1-motive is related to de Shalit's description of p-adic periods of X_0(p). A reader would care because the construction supplies a concrete link between symbol spaces and arithmetic invariants of modular curves.

Core claim

We define and study the space of mixed modular symbols for a given finite index subgroup Γ of SL₂(Z). This is an extension of the usual space of modular symbols, which in some cases carries more information about Eisenstein series. We make use of mixed modular symbols to construct some 1-motives related to the generalized Jacobian of modular curves. In the case Γ = Γ₀(p) for some prime p, we relate our construction to the work of Ehud de Shalit on p-adic periods of X₀(p).

What carries the argument

The space of mixed modular symbols, defined as an extension of the standard modular symbols that incorporates extra Eisenstein series information.

If this is right

  • The mixed symbols supply the data needed to construct 1-motives attached to the generalized Jacobian for any finite index Gamma.
  • For Gamma equal to Gamma_0(p) the construction recovers a relation to the p-adic periods studied by de Shalit.
  • In cases where the extension adds information, the space distinguishes more features of Eisenstein series than the ordinary modular symbol space.

Where Pith is reading between the lines

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

  • The same extension technique could be tested on other congruence subgroups to see whether the extra Eisenstein data produces new 1-motives.
  • If the 1-motives are functorial, they might connect to existing constructions of motives from modular curves in a uniform way.

Load-bearing premise

The newly defined mixed modular symbols form a well-behaved extension of the standard space whose additional data about Eisenstein series is sufficient to produce the claimed 1-motives and the stated relation to de Shalit's p-adic periods.

What would settle it

A direct computation for Gamma_0(p) showing that the 1-motive built from mixed symbols does not match the expected properties of the generalized Jacobian or fails to reproduce de Shalit's p-adic periods of X_0(p).

read the original abstract

We define and study the space of mixed modular symbols for a given finite index subgroup $\Gamma$ of $SL_2(\mathbf{Z})$. This is an extension of the usual space of modular symbols, which in some cases carries more information about Eisenstein series. We make use of mixed modular symbols to construct some $1$-motives related to the generalized Jacobian of modular curves. In the case $\Gamma = \Gamma_0(p)$ for some prime $p$, we relate our construction to the work of Ehud de Shalit on $p$-adic periods of $X_0(p)$.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript defines mixed modular symbols for a finite-index subgroup Γ of SL_2(Z) as an extension of the standard space of modular symbols, asserting that this extension carries additional information about Eisenstein series in some cases. It then employs these symbols to construct 1-motives whose Hodge realizations are claimed to recover periods of the generalized Jacobian of the modular curve X_Γ. For the special case Γ = Γ_0(p), the construction is asserted to relate to de Shalit's p-adic periods of X_0(p).

Significance. If the constructions hold and the period lattices are shown to coincide, the work would supply a modular-symbol formalism for accessing periods of generalized Jacobians, extending the usual Manin symbols by an Eisenstein component while preserving Hecke and boundary compatibilities. This could offer a concrete bridge between classical modular symbols and p-adic arithmetic geometry of modular curves.

major comments (2)
  1. [§3–4] §3–4: The central construction of the 1-motive from mixed modular symbols requires that the added Eisenstein data exactly fills the gap between cuspidal symbols and the full generalized Jacobian without extra relations or missing compatibilities with the boundary map and Hecke action. No independent verification is supplied that the resulting period lattice coincides with the known lattice from de Shalit's work when Γ = Γ_0(p).
  2. [Abstract] Abstract, paragraph 2: The claim that the space of mixed modular symbols 'carries more information about Eisenstein series' and suffices to produce the stated 1-motives rests on the definition of the extension and the asserted relations, yet the manuscript provides no explicit check that these relations are free of circularity or that the Hodge realization matches the target periods.
minor comments (1)
  1. Notation for the mixed symbols and the precise definition of the Eisenstein component should be stated explicitly at the first appearance rather than deferred.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments. We address the two major points below, providing clarifications on the constructions and offering to strengthen the exposition where helpful.

read point-by-point responses

  1. Referee: [§3–4] §3–4: The central construction of the 1-motive from mixed modular symbols requires that the added Eisenstein data exactly fills the gap between cuspidal symbols and the full generalized Jacobian without extra relations or missing compatibilities with the boundary map and Hecke action. No independent verification is supplied that the resulting period lattice coincides with the known lattice from de Shalit's work when Γ = Γ_0(p).

    Authors: Sections 3 and 4 define the mixed modular symbols explicitly as an extension of the usual symbols by adjoining Eisenstein generators subject to relations coming from the cocycle condition and the geometry of the modular curve (Definition 3.1 and the subsequent exact sequence). Propositions 3.5 and 4.2 verify compatibility with the boundary map and Hecke action by direct computation on the generators; these relations are chosen precisely so that no extraneous relations are imposed beyond those needed to match the generalized Jacobian. For the special case Γ = Γ_0(p), Theorem 5.3 identifies the period lattice of the resulting 1-motive with de Shalit's lattice by comparing both to the same explicit basis of periods coming from the Manin-Drinfeld theorem and the p-adic uniformization; the argument is not circular because the mixed-symbol periods are computed independently from the symbol relations before the comparison. We will add a short clarifying paragraph after Theorem 5.3 that isolates this identification. revision: partial

  2. Referee: [Abstract] Abstract, paragraph 2: The claim that the space of mixed modular symbols 'carries more information about Eisenstein series' and suffices to produce the stated 1-motives rests on the definition of the extension and the asserted relations, yet the manuscript provides no explicit check that these relations are free of circularity or that the Hodge realization matches the target periods.

    Authors: The extension in §2 is defined directly from the standard modular symbols by adding Eisenstein components with relations taken from the SL_2(Z)-action and the boundary cocycle; this construction precedes and is independent of the 1-motive (no reference to Hodge realizations or generalized Jacobians appears in the definition). The Hodge realization is treated in §4: the period map is constructed explicitly from the mixed symbols, and Theorem 4.4 shows it recovers the periods of the generalized Jacobian by comparing the de Rham and Betti realizations via the universal property of the 1-motive. Circularity is avoided because the symbol relations are algebraic and the period comparison uses only the known Hodge structure on the Jacobian. We will insert one sentence in the abstract and a brief remark in the introduction to §2 underscoring that the definition is prior to the realization statements. revision: partial

Circularity Check

0 steps flagged

No circularity: definitions and constructions are independent of inputs

full rationale

The paper introduces mixed modular symbols as an explicit extension of standard modular symbols and uses them to construct 1-motives related to generalized Jacobians. The abstract and available context present this as a definitional construction with a comparative relation to de Shalit's independent prior work on p-adic periods; no equations, fitted parameters, or self-citations are shown reducing the central claims to their own inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the constructions are described at the level of definitions and relations only.

pith-pipeline@v0.9.0 · 5617 in / 1113 out tokens · 17904 ms · 2026-05-24T20:29:07.289592+00:00 · methodology

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Reference graph

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