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

• Published in: The Ramanujan journal 2022-03, Vol.57 (3), p.1001-1019
• Main Authors: Amir, Malik, Hong, Letong
• Format: Article
• Language: English
• Subjects: Combinatorics; Conductors; Curves; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-021-00388-w

Inspired by Lehmer’s conjecture on the non-vanishing 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 ≥ 4 . We use these methods for weight 2 and 3 newforms and apply our results to L -functions of modular elliptic curves and certain K 3 surfaces with Picard number ≥ 19 . In particular, for the complete list of weight 3 newforms f λ ( z ) = ∑ 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 , 2 N λ ) = 1 , then 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 a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 } .