Matching Tag: gross-conjecture
Generalized Gross conjecture. The coinvariant group X ′ ( k ∞ c ) Gal ( k ∞ c / k ) X'(k^\mathrm{c}_\infty)_{\operatorname{Gal}(k^\mathrm{c}_\infty/k)} X ′ ( k ∞ c ) Gal ( k ∞ c / k )
Fix an odd prime p p p ι : C ≃ C p \iota:\mathbb{C}\simeq\mathbb{C}_p ι : C ≃ C p ψ ∈ Irr C p ( G ) \psi\in\operatorname{Irr}_{\mathbb{C}_p}(\mathcal{G}) ψ ∈ Irr C p ( G ) S S S
Let K ′ K' K ′ H ′ H' H ′ I ( H ′ ) I(H') I ( H ′ ) r K ′ r_{K'} r K ′ S ( K ′ ) S(K') S ( K ′ ) θ H ′ \theta_{H'} θ H ′
Let ψ ∈ Irr C p ( G ) \psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G) ψ ∈ Irr C p ( G ) ι : C ≃ C p \iota:\mathbb C\simeq\mathbb C_p ι : C ≃ C p r S ( ψ ) r_S(\psi) r S ( ψ )
Let ψ ∈ Irr C p ( G ) \psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G) ψ ∈ Irr C p ( G ) r S ( ψ ) r_S(\psi) r S ( ψ ) s = 0 s=0 s = 0 S S S L L L
With K / F K/F K / F G G G V V V D ′ = ∏ v ∈ V I G v , G ⊂ Z [ G ] D'=\prod_{v\in V}I_{G_v,G}\subset\mathbb{Z}[G] D ′ = ∏ v ∈ V I G v , G ⊂ Z [ G ] Θ K , S , T \Theta_{K,S,T} Θ K , S , T
Let ( H / F , S , T , ( v i ) i , ( w i ) i ) (H/F,S,T,(v_i)_i,(w_i)_i) ( H / F , S , T , ( v i ) i , ( w i ) i ) K / F K/F K / F S S S H ⊂ K H\subset K H ⊂ K O K , S , T × \mathcal{O}_{K,S,T}^{\times} O K , S , T ×
