论文标题
在$ l $ - 模块化椭圆曲线和某些$ k3 $表面上
On $L$-Functions of Modular Elliptic Curves and Certain $K3$ Surfaces
论文作者
论文摘要
受莱默（Lehmer）对Ramanujan $τ$功能的非散步的猜想的启发，可能会问一个奇数整数$α$是否可以等于$τ（n）$或任何新形式$ f（z）$的系数。 Balakrishnan，Craig，Ono和Tsai使用Lucas序列和二芬太汀分析的理论来表征均匀重量$ k \ geq 4 $的新形式的不可接受的值。我们使用这些方法的重量$ 2 $和$ 3 $ newforms，并将我们的结果应用于模块化椭圆曲线的$ l $ - 符合条件，而Picard Number $ \ ge 19 $的某些$ k3 $表面。特别是，对于完整的重量列表，$ 3 $ newforms $f_λ（z）= \ suma_λ（n）q^n $是$η$ - 产品，对于$n_λ$，某些椭圆曲线$e_λ$的指挥\ begin {align*}a_λ（n）\ in \ in \ ,,＆\ { - 5,9，\ pm 11,25，\ pm41，\ pm 43，-45，-45，\ pm47,49，\ pm53,55，\ pm53,55，\ pm59，\ pm59，\ pm59，\ pm61，pm61，\ pm 67 \ \ \ \ \ \ \ \ \＆\＆\ \＆\＆\＆\＆\＆\＆\＆\＆\＆\＆\， \ { - 69，\ pm 71，\ pm 73,75，\ pm79，\ pm81，\ pm 83，\ pm89，\ pm 93 \ pm 97，99 \}。 \ end {align*}假设有概括的riemann假设，我们可以排除其他可能性的可能性，\ begin {align*}a_λ（n）\ in \ in \ { - 5,9，\ pm 11,25，-45,49,49,55，-69,75，-69,75，-69,75，-69,75，-69,75，\ pm 81，dm pm 81，93，93，93，999999999. \ end {align*}
Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $τ$-function, one may ask whether an odd integer $α$ can be equal to $τ(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $k\geq 4$. We use these methods for weight $2$ and $3$ newforms and apply our results to $L$-functions of modular elliptic curves and certain $K3$ surfaces with Picard number $\ge 19$. In particular, for the complete list of weight $3$ newforms $f_λ(z)=\sum a_λ(n)q^n$ that are $η$-products, and for $N_λ$ the conductor of some elliptic curve $E_λ$, we show that if $|a_λ(n)|<100$ is odd with $n>1$ and $(n,2N_λ)=1$, then \begin{align*} a_λ(n) \in \,& \{-5,9,\pm 11,25, \pm41, \pm 43, -45,\pm47,49, \pm53,55, \pm59, \pm61, \pm 67\}\\ & \,\,\, \cup \, \{-69,\pm 71, \pm 73,75, \pm79,\pm81, \pm 83, \pm89,\pm 93 \pm 97, 99\}. \end{align*} Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving \begin{align*} a_λ(n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{align*}
