On the torsion-freeness of the symmetric powers of an ideal

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

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Coarse equivalences of functorial constructions

Proceedings of the International Geometry Center ◽

10.15673/tmgc.v12i3.1609 ◽

2020 ◽

Vol 12 (3) ◽

pp. 69-77

Author(s):

Mykhailo Romanskyi

Keyword(s):

Metric Spaces ◽

Symmetric Powers

We consider the question of coarse equivalence of some functorial constructions (in particular, symmetric powers, hypersymmetric powers) in the category of metric spaces.

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Symmetric Powers of Motives

The Norm Residue Theorem in Motivic Cohomology ◽

10.23943/princeton/9780691191041.003.0014 ◽

2019 ◽

pp. 232-252

Author(s):

Christian Haesemeyer ◽

Charles A. Weibel

Keyword(s):

Topological Space ◽

Symmetric Group ◽

Homotopy Group ◽

Orbit Space ◽

Symmetric Power ◽

Homotopy Groups ◽

Integral Homology ◽

Symmetric Powers ◽

Mac Lane ◽

Abelian Monoid

This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎ *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.

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Geometrical symmetric powers in the motivic homotopy category

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2019.04.004 ◽

2019 ◽

Vol 223 (12) ◽

pp. 5396-5408

Author(s):

Joe Palacios

Keyword(s):

Homotopy Category ◽

Symmetric Powers ◽

Motivic Homotopy

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SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748018000476 ◽

2018 ◽

Vol 19 (5) ◽

pp. 1521-1572

Author(s):

Haruzo Hida ◽

Jacques Tilouine

Keyword(s):

Power Series ◽

Galois Representation ◽

Symmetric Power ◽

Selmer Groups ◽

Abelian Surfaces ◽

Symmetric Powers ◽

Branching Laws ◽

Hida Family

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

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