scholarly journals Motivic invariants of symmetric powers of curves

2021 ◽  
pp. 106913
Author(s):  
Rahul Gupta
Keyword(s):  
Symmetric Powers

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

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).

scholarly journals Coarse equivalences of functorial constructions

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.

Symmetric Powers of Motives

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.

scholarly journals SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

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.

scholarly journals MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES

2019 ◽  
Vol 236 ◽  
pp. 251-310 ◽  
Author(s):  
MARC LEVINE
Keyword(s):  
Euler Characteristic ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Vector Bundles ◽  
Reductive Group ◽  
Characteristic Classes ◽  
Euler Characteristics ◽  
Symmetric Powers ◽  
Maximal Tori ◽  
General Calculus

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

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