On L-functions of modular elliptic curves and certain K3 surfaces

Abstract

Abstract Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan

$$\tau $$

τ

-function, one may ask whether an odd integer

$$\alpha $$

α

can be equal to

$$\tau (n)$$

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

$$k\ge 4$$

k ≥ 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

$$\ge 19$$

≥ 19

. In particular, for the complete list of weight 3 newforms

$$f_\lambda (z)=\sum a_\lambda (n)q^n$$

f λ

( z )

= ∑

a λ

( n )

q n

that are

$$\eta $$

η

-products, and for

$$N_\lambda $$

N λ

the conductor of some elliptic curve

$$E_\lambda $$

E λ

, we show that if

$$|a_\lambda (n)|<100$$

|

a λ

( n )

| < 100

is odd with

$$n>1$$

n > 1

and

$$(n,2N_\lambda )=1$$

( n , 2

N λ

) = 1

, then

$$\begin{aligned} a_\lambda (n) \in&\{-5,9,\pm 11,25, \pm 41, \pm 43, -45,\pm 47,49, \pm 53,55, \pm 59, \pm 61,\\&\pm 67, -69,\pm 71,\pm 73,75, \pm 79,\pm 81, \pm 83, \pm 89,\pm 93 \pm 97, 99\}. \end{aligned}$$

a λ

( n )

∈

{ - 5 , 9 , ± 11 , 25 , ± 41 , ± 43 , - 45 , ± 47 , 49 , ± 53 , 55 , ± 59 , ± 61 ,

± 67 , - 69 , ± 71 , ± 73 , 75 , ± 79 , ± 81 , ± 83 , ± 89 , ± 93 ± 97 , 99 } .

Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving

$$\begin{aligned} a_\lambda (n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{aligned}$$

a λ

( n )

∈

{ - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 }

.