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CM cycles on Kuga-Sato varieties over Shimura curves and Selmer groups

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Yara Elias, Carlos de Vera-Piquero

Number Theory · Submitted 30 Oct 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a suitable cohomology class does not vanish.

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arXiv ID1610.09627 (version 1)
Pith IDMPADDBW7, Pith's permanent record of this paper and its review
CitationCarlos de Vera-Piquero, Yara Elias. "CM cycles on Kuga-Sato varieties over Shimura curves and Selmer groups." arXiv:1610.09627 (2016). https://pith.science/paper/MPADDBW7
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,504 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.92 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.21 · compares with prior results 0.17 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a suitable cohomology class does not vanish.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. contribution 0.96 0.95 0.16 asserted 1.00
s1 Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a suitable cohomology class does not vanish. result 0.72 0.93 0.66 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

Given a modular form f of even weight larger than two and an imaginary quadratic field K satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga-Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to f enjoying the compatibility properties of an Euler system. Then we use Kolyvagin's method, as adapted by Nekovar to higher weight modular forms, to bound the size of the relevant Selmer group associated to f and K and prove the finiteness of the (primary part) of the Shafarevich-Tate group, provided that a suitable cohomology class does not vanish.

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