arXiv subjects
Publications and source records attributed to Antonio Rojas-Leon.
Given a prime $p$ and an integer $d>1$, we give a numerical criterion to decide whether the $\ell$-adic sheaf associated to the one-parameter exponential sums $t\mapsto \sum_xψ(x^d+tx)$ over ${\mathbb F}_p$ has finite monodromy or not, and work out some explicit cases where this is computable.
For Artin-Schreier curve y^q -y = f(x) defined over a finite field F_q of q elements, we show that the Weil bound for the number of the rational points over extension fields of F_q can often be greatly improved, essentially removing an extra factor of size about the square root of q in the error term.
For ψa nontrivial additive character on the finite field F_q, the map t \mapsto \sum_{x \in F_q} ψ(f(x)+tx) is the Fourier transform of the map t \mapsto ψ(f(t))$. As is well-known, this has a cohomological interpretation, producing a continuous ell-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both ell-adic and p-adic methods. Using ell-adic techniques, we give an explicit formula for the degree of this L-function and determine the complex absolute values of its roots. Using p-adic techniques, we study the p-adic absolute values of the roots.
This paper looks at the L-function of the k-th symmetric power of the ell-adic sheaf over the affine line associated to the generalized Airy family of exponential sums. We compute the degree of this rational function and the local factors at infinity.
Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.
We study in detail the family of Calabi-Yau hypersurfaces defined by x_1+...+x_n+1/(x_1...x_n)=t over a finite field k. We determine its local and global monodromy and the trivial factors of its moment zeta function and Dwork's unit root zeta function.