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Register a new userOn the u-Invariant of Function Fields of Curves Over Complete Discretely Valued Fields
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Let K be a complete discretely valued field with residue field k. If char(K) = 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable over K is bounded by N. The method of proof is via introducing the notion of uniform boundedness for the p-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over K are uniformly bounded.
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Let K be a complete discretely valued field and F the function field of a curve over K. If the characteristic of the residue field k of K is p > 0, then we give a bound for the Brauer p-simension of F in terms of the p-rank of k. If k is a perfect field of characteristic 2, we show that the u-invaraint of F is at most 8.
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an isogeny estimate, providing an explicit bound on the degree of a minimal isogeny between two isogenous elliptic curves. We also give several corollaries of these two results.
Let $C$ be a smooth projective curve over $mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $phi:mathcal{E}to C$ its minimal regular model. For each $Pin C$ such that $E$ has good reduction at $P$, i.e., the fiber $mathcal{E}_P=phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $mathcal{E}_P$ over the residue field $kappa_P$ of $P$ are of the form $q_P^{1/2}e^{itheta_P},q_{P}e^{-itheta_P}$, where $q_P=q^{deg(P)}$ and $0letheta_Plepi$. The goal of this note is to determine given an integer $Bge 1$, $alpha,betain[0,pi]$ the number of $Pin C$ where the reduction of $E$ is good and such that $deg(P)le B$ and $alphaletheta_Plebeta$.
We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $mathbb{F}_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^ u +1$, and $K_d := mathbb{F}_p(mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the new part of $J$ is isogenous over $overline{mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $phi(r)/2$ with endomorphism algebra $mathbb{Z}[mu_r]^+$. For a prime $ell$ with $ell mid pr$, we prove that $J[ell](L)={0}$ for any abelian extension $L$ of $overline{mathbb{F}}_p(t)$.
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