Modular Symbols with Values in Beilinson-Kato Distributions

Cecilia Busuioc; Jeehoon Park; Owen Patashnick; Glenn Stevens

doi:10.48550/arXiv.2311.14620

Modular Symbols with Values in Beilinson-Kato Distributions

Cecilia Busuioc, Jeehoon Park, Owen Patashnick, Glenn Stevens

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

For each integer n ≥ 1, we construct a GL_n(Q)-invariant modular symbol ξ_n with coefficients in a space of distributions that takes values in the Milnor K_n-group of the modular function field. The Siegel distribution μ on Q², with values in the modular function field, serves as the building block for ξ_n; we define ξ_n essentially by taking the n-Steinberg product of μ. The most non-trivial part of this construction is the cocycle property of ξ_n; we prove it by using an induction on n based on the first two cases ξ₁ and ξ₂; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor K₂-group modulo torsion satisfy the Manin relations.

Original language	English
Number of pages	29
Journal	Transactions of the American Mathematical Society
Early online date	23 Dec 2025
DOIs	https://doi.org/10.48550/arXiv.2311.14620 https://doi.org/10.1090/tran/9590
Publication status	E-pub ahead of print - 23 Dec 2025

Bibliographical note

Accepted in the Transactions of the American Mathematical Society

Keywords

math.NT
math.AG
math.KT
14F67 (Primary) 19D45 (Secondary)

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title = "Modular Symbols with Values in Beilinson-Kato Distributions",

abstract = " For each integer n ≥ 1, we construct a GLn(Q)-invariant modular symbol ξn with coefficients in a space of distributions that takes values in the Milnor Kn-group of the modular function field. The Siegel distribution μ on Q2, with values in the modular function field, serves as the building block for ξn; we define ξn essentially by taking the n-Steinberg product of μ. The most non-trivial part of this construction is the cocycle property of ξn; we prove it by using an induction on n based on the first two cases ξ1 and ξ2; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor K2-group modulo torsion satisfy the Manin relations. ",

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author = "Cecilia Busuioc and Jeehoon Park and Owen Patashnick and Glenn Stevens",

note = "Accepted in the Transactions of the American Mathematical Society",

year = "2025",

month = dec,

day = "23",

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language = "English",

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T1 - Modular Symbols with Values in Beilinson-Kato Distributions

AU - Busuioc, Cecilia

AU - Park, Jeehoon

AU - Patashnick, Owen

AU - Stevens, Glenn

N1 - Accepted in the Transactions of the American Mathematical Society

PY - 2025/12/23

Y1 - 2025/12/23

N2 - For each integer n ≥ 1, we construct a GLn(Q)-invariant modular symbol ξn with coefficients in a space of distributions that takes values in the Milnor Kn-group of the modular function field. The Siegel distribution μ on Q2, with values in the modular function field, serves as the building block for ξn; we define ξn essentially by taking the n-Steinberg product of μ. The most non-trivial part of this construction is the cocycle property of ξn; we prove it by using an induction on n based on the first two cases ξ1 and ξ2; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor K2-group modulo torsion satisfy the Manin relations.

AB - For each integer n ≥ 1, we construct a GLn(Q)-invariant modular symbol ξn with coefficients in a space of distributions that takes values in the Milnor Kn-group of the modular function field. The Siegel distribution μ on Q2, with values in the modular function field, serves as the building block for ξn; we define ξn essentially by taking the n-Steinberg product of μ. The most non-trivial part of this construction is the cocycle property of ξn; we prove it by using an induction on n based on the first two cases ξ1 and ξ2; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor K2-group modulo torsion satisfy the Manin relations.

KW - math.NT

KW - math.AG

KW - math.KT

KW - 14F67 (Primary) 19D45 (Secondary)

U2 - 10.48550/arXiv.2311.14620

DO - 10.48550/arXiv.2311.14620

M3 - Article (Academic Journal)

SN - 0002-9947

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

ER -

Modular Symbols with Values in Beilinson-Kato Distributions

Abstract

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Keywords

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