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Register a new userFiniteness of reductions of Hecke orbits
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We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimension $g$ only finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga-Satake construction, we also show that only finitely many supersingular $K3$-surfaces admit CM lifts. Our tools include $p$-adic Hodge theory and group theoretic techniques.
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We formulate some properties of a conjectural object $X_{fun}(r,n)$ parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we give formulas for $goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its reductions at $goth p$ (where $goth p$ is a prime of $Bbb F_q[theta]$). Also, we describe their geometric interpretation. These results are analogs of the corresponding results of reductions of Shimura varieties. Finally, we give conjectural formulas for Hodge numbers (over the fields generated by Hecke correspondences) of middle cohomology submotives of $X_{fun}(r,n)$.
Let $q:=e^{2 pi iz}$, where $z in mathbb{H}$. For an even integer $k$, let $f(z):=q^hprod_{m=1}^{infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$ of a modular form and the points in the support of $T_m.D$. In this paper, we extend this result to general $Gamma_0(N)$ in terms of values of harmonic weak Maass forms of weight $0$. By the distribution of Hecke points, this applies to obtain an asymptotic behaviour of convolutions of sums of divisors of an integer and sums of exponents of a modular form.
Let $j(z)$ be the modular $j$-invariant function. Let $tau$ be an algebraic number in the complex upper half plane $mathbb{H}$. It was proved by Schneider and Siegel that if $tau$ is not a CM point, i.e., $[mathbb{Q}(tau):mathbb{Q}] eq2$, then $j(tau)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $Gamma_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $tau$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at the cusp $i infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to $i infty$. We prove, under a mild assumption on $f$, that for any fixed $tau$, if $N$ is a prime such that $ Ngeq 23 text{ and } N ot in {23, 29, 31, 41, 47, 59, 71},$ then $f(T_m.tau)$ are transcendental for infinitely many positive integers $m$ prime to $N$.
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. Using this, we extend Faltings finiteness criteria to determine the equivalence of two $ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.
Some new results concerning the equation $sigma(N)=aM, sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
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