Exponential sums, hypersurfaces with many symmetries and Galois representations
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Bibliographic record
Abstract
The main theme of this thesis is the study of compatible systems of $\\ell$-adic Galois representations provided by the étale cohomology of arithmetic varieties with a large group of symmetries. A canonical decomposition of these systems into isotypic components is proven (Section 3.1). The isotypic components are realized as the cohomology of the quotient with values in a certain sheaf, thus providing a geometrical interpretation for the rationality of the corresponding $L$-functions. A particular family of singular hypersurfaces $W_\\ell^{m,n}$ of degree $\\ell$ and dimension $m + n - 3$, admitting an action by a product of symmetric groups $S_m \\times S_n$, arises naturally when considering the average moments of certain exponential sums (Chapter 4); asymptotics for these moments are obtained by relating them to the trace of the Frobenius morphism on the cohomology of the desingularization of the corresponding varieties, following the approach of Livné. Two other closely related classes of smooth hypersurfaces admitting an $S_n$-action are introduced in Chapter 3, and the character of the representation of the symmetric group on their primitive cohomology is computed. In particular, a certain smooth cubic hypersurface of dimension 4 is shown to carry a compatible system of 2-dimensional Galois representations. A variant of the Faltings-Serre method is developed in Chapter 5 in order to explicitly determine the corresponding modular form, whose existence is predicted by Serre's conjecture. We provide a systematic treatment of the Faltings-Serre method in a form amenable to generalization to Galois representations of other fields and to other groups besides $\\GL_2$.
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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 | 0.000 | 0.000 |
| 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) | 0.000 | 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