On L-functions of modular elliptic curves and certain K3 surfaces
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -function, one may ask whether an odd integer $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> can be equal to $$\tau (n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$k\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . 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 $$\ge 19$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>19</mml:mn> </mml:mrow> </mml:math> . In particular, for the complete list of weight 3 newforms $$f_\lambda (z)=\sum a_\lambda (n)q^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>∑</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> that are $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> -products, and for $$N_\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> the conductor of some elliptic curve $$E_\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> , we show that if $$|a_\lambda (n)|<100$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> <mml:mo><</mml:mo> <mml:mn>100</mml:mn> </mml:mrow> </mml:mrow> </mml:math> is odd with $$n>1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$(n,2N_\lambda )=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , then $$\begin{aligned} a_\lambda (n) \in&\{-5,9,\pm 11,25, \pm 41, \pm 43, -45,\pm 47,49, \pm 53,55, \pm 59, \pm 61,\\&\pm 67, -69,\pm 71,\pm 73,75, \pm 79,\pm 81, \pm 83, \pm 89,\pm 93 \pm 97, 99\}. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>9</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>11</mml:mn> <mml:mo>,</mml:mo> <mml:mn>25</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>41</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>43</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>45</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>47</mml:mn> <mml:mo>,</mml:mo> <mml:mn>49</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>53</mml:mn> <mml:mo>,</mml:mo> <mml:mn>55</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>59</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mn>61</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow/> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>67</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>69</mml:mn> <mml:mo>,</mml:mo>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it