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This paper concerns towers of curves over a finite field with many rational points, following Garcia--Stichtenoth and Elkies. We present a new method to produce such towers. A key ingredient is the study of algebraic solutions to Fuchsian differential equations modulo $p$. We apply our results to towers of modular curves, and find new asymptotically good towers.
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It is shown that the real class field towers are always finite. The proof is based on Castelnuovos theory of the algebraic surfaces and a functor from such surfaces to the Etesi C*-algebras.
We fix a monic polynomial $f(x) in mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $cdots to C_m to C_{m-1} to cdots to C_0 =mathbb A^1$, with total Galois group $mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.
In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of Kuhne and Yuan and Zhang, we deduce the equidistribution of generic sequence of preperiodic parameters for families of polarized endomorphisms with marked points.
Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:mathbb{P}^1_Ktomathbb{P}^1_K$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism $mathbb{P}^1_{R_S}tomathbb{P}^1_{R_S}$. A finite Galois invariant subset $Xsubsetmathbb{P}^1_K(bar{K})$ has good reduction outside $S$ if its closure in $mathbb{P}^1_{R_S}$ is etale over $R_S$. We study triples $(f,Y,X)$ with $X=Ycup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many $text{PGL}_2(R_S)$-equivalence classes of triples with $text{deg}(f)=d$ and $sum_{Pin Y}e_f(P)ge2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Yto X$ is specified, and we give an exhaustive analysis for degree $2$ maps on $mathbb{P}^1$ when $Y=X$.
Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(tau,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/tau B(k)$ varies when we take an unramified abelian cover $pi:Cto C$ defined over $k$.
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