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 K 3 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 } .
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Variation of canonical height for\break Fatou points on ℙ 1
Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ( a t ) - h D ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number.
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- Award ID(s):
- 2050037
- NSF-PAR ID:
- 10454883
- Date Published:
- Journal Name:
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- Volume:
- 0
- Issue:
- 0
- ISSN:
- 0075-4102
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/crelle-2022-0078
