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Genus bounds for curves with fixed Frobenius eigenvalues. (English) Zbl 1286.14035
Let \(A\) be a \(d\)-dimensional abelian variety over a finite field \(\mathbb{F}_q\). The Weil polynomial of \(A\) is a monic polynomial in \(\mathbb{Z}[x]\) of degree \(2d\) whose complex roots can be written as \(\sqrt{q}\exp(i\theta_j)\) for real numbers \(-\pi<\theta_j\leq\pi\), which are called the Frobenius angles of \(A\). If \(C\) is a curve over \(\mathbb{F}_q\), the Frobenius angles of \(C\) are defined to be the Frobenius angles of its Jacobian.
The aim of the paper is to find upper bounds for the genus \(g\) of curves \(C\) whose non-negative Frobenius angles all lie in a given finite set \(S\subset [0,\pi]\). If \(S=\{0\}\) set \(r=1/2\), and otherwise take \(r=\#(S\cap\{\pi\})+2\sum_{\theta\in S\setminus\{0,\pi\}}\lceil \pi/2\theta\rceil\). If \(s=\#S\), the bounds are: \[ g\leq 23s^2q^{2s}\log q,\quad g<(\sqrt{q}+1)^{2r}(1+q^{-r})/2. \] When applied to the set \(S\) of all non-negative Frobenius angles of elliptic curves over \(\mathbb{F}_q\), this result yields the bound \(g\leq 510q^{8\sqrt{q}+3}\log q\) for the genus of a curve \(C\) whose Jacobian is isogenous over \(\mathbb{F}_q\) to a product of elliptic curves. These upper bounds are far from being sharp. The authors use a simple linear programming argument to show that a curve over \(\mathbb{F}_2\) with totally split Jacobian has genus \(g\leq 26\), and this bound is sharp because it is attained by the modular curve \(X(11)\). This had been proved by I. Duursma and J.-Y. Enjalbert [in: Finite fields with applications to coding theory, cryptography and related areas. Proceedings of the 6th international conference on finite fields and applications, Oaxaca, México, May 21–25, 2001. Mullen, Gary L. (ed.) et al., Berlin: Springer 86–93 (2002; Zbl 1058.14052)].
However, the most striking application of the above upper bounds is the derivation of a lower bound, \(d>\sqrt{\log\log q/6\log q}\), for the dimension \(d\) of the largest simple isogeny factor of a curve of genus \(g>2\). This gives an effective version of a result by Serre: if the genus of a sequence \(C_n/\mathbb{F}_q\) of curves tends to infinity with \(n\), then the dimension of the largest \(\mathbb{F}_q\)-simple isogeny factor tends to infinity too.

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G20 Curves over finite and local fields
14G15 Finite ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
Magma; SageMath
Full Text: DOI arXiv
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