Degeneration of Abelian Varieties

The Iwasawa Main Conjectures for GL2

“…Such a component will be called a rational component of genus 2r. The local structure of the minimal compactification at the boundary component S [g] is given by the following theorem whose proof is similar to [FC,V.2.7] (see also [Lan08, 7.2.3.11]); the r = 0 case has already been noted.…”

Siegel Modular Forms and Their Applications

“…We now give Satake's so-called spherical map of the Hecke algebra H p (Γ, G) to the Hecke algebras H p (M ) and H p (T), cf., [84,16,28,39]. The images will land in the W M -invariant (resp.…”

“…In [28] the authors define a 'twisted' version of these spherical maps where they put |e 2ρn(m) | and |e 2ρM (t) | instead of the multipliers above. In this way one avoids square roots of p. If one uses this twisted version one should also twist the action of the Weyl group on the co-character group Y of T by e ρ too: in the usual action S g permutes the a i and d i and the i-th generator τ i of (Z/2Z) g interchanges a i and d i .…”

“…Under the twisted action τ i sends (u i , v i ) to (p g+1−i v i , p i−g−1 u i ), while the permutation (i i + 1) ∈ S g sends (u i /v i ) to pu i+1 /v i+1 . The formula is w · φ(t) = |e ρ(w −1 t)−ρ(t) |φ(w −1 t) for w ∈ W and t ∈ T, cf., [28].…”

“…exist over Z as was shown by Faltings. We refer to an extensive, but very condensed survey of this theory in [28]. In particular, Faltings constructed the Satake compactification over Z as the image of a toroidal compactificationà g by the sections of a sufficiently big power of det(E), the determinant of the Hodge bundle.…”