Modular <i>p</i>-adic <i>L</i>-functions attached to real quadratic fieldsand arithmetic applications
Bibliographic record
Abstract
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mi>Γ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo></m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> ${f\in S_{k_{0}+2}(\Gamma_{0}(Np))}$ be a normalized N -new eigenform with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mpadded> <m:mi>p</m:mi> </m:mpadded> <m:mo>∤</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> ${p\,{\nmid}\,N}$ and such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msubsup> <m:mo>≠</m:mo> <m:msup> <m:mi>p</m:mi> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:math> ${a_{p}^{2}\neq p^{k_{0}+1}}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo>ord</m:mo> <m:mi>p</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>a</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo><</m:mo> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> ${\operatorname{ord}_{p}(a_{p})<k_{0}+1}$ . By Coleman’s theory, there is a p -adic family <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝐅</m:mi> </m:math> ${\mathbf{F}}$ of eigenforms whose weight <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> ${k_{0}+2}$ specialization is f . Let K be a real quadratic field and let ψ be an unramified character of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>Gal</m:mo> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mover> <m:mi>K</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${\operatorname{Gal}(\overline{K}/K)}$ . Under mild hypotheses on the discriminant of K and the factorization of N , we construct a p -adic L -function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℒ</m:mi> <m:mrow> <m:mrow> <m:mi>𝐅</m:mi> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>ψ</m:mi> </m:mrow> </m:msub> </m:math> ${\mathcal{L}_{\mathbf{F}/K,\psi}}$ interpolating the central critical values of the Rankin L -functions associated to the base change to K of the specializations of <jats:inline-formula id="j_crelle-2014-0088_ineq_9991_w2aab3b7e4374b1b6b1aab1c13
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How this classification was reachedexpand
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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".