We study the diagonalizability of the Atkin Ut-operator acting on Drinfeld cusp forms for Γ0(t): starting with the slopes of eigenvalues and then moving to the space of cusp forms for Γ1(t) to use Teitelbaum's interpretation as harmonic cocycles which makes computations more explicit. We prove Ut is diagonalizable in odd characteristic for (relatively) small weights and explicitly compute the eigenvalues. In even characteristic we show that it is not diagonalizable when the weight is odd (except for the trivial cases) and prove some cases of non diagonalizability in even weight as well. We also formulate a few conjectures, supported by numerical search, about diagonalizability of Ut and the slopes of its eigenforms.
We define oldforms and newforms for Drinfeld cusp forms of level t and conjecture that their direct sum is the whole space of cusp forms. Moreover we describe explicitly the matrix U associated to the action of the Atkin operator Ut on cusp forms of level t and use it to compute tables of slopes of eigenforms. Building on such data, we formulate conjectures on bounds for slopes, on the diagonalizability of Ut and on various other issues. Via the explicit form of the matrix U we are then able to verify our conjectures in various cases (mainly in small weights).Date:
Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K /F be an -adic Lie extension ( = p) unramified outside a finite set of primes S and such that Gal(K /F) has no elements of order . We shall prove that, under certain conditions, Sel A (K ) ∨ has no nontrivial pseudo-null submodule.
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