An official website of the United States government
We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y^2 = x^p−1 and y2 = x^{2p}−1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves.
more »
« less
Towards the Sato–Tate groups of trinomial hyperelliptic curves
We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1 : y2 = x2g+2 + c, C2 : y2 = x2g+1 + cx, C3 : y2 = x2g+1 + c, where g is the genus of the curve and c in Q* is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus C1 curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form C1, C2 and C3. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components ST^0(C) for these families of curves.
more »
« less
- Award ID(s):
- 2002085
- PAR ID:
- 10544955
- Publisher / Repository:
- World Scientific
- Date Published:
- Journal Name:
- International Journal of Number Theory
- Volume:
- 17
- Issue:
- 10
- ISSN:
- 1793-0421
- Page Range / eLocation ID:
- 2175 to 2206
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $$m$$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.more » « less
-
Seven doubly 13 C-labeled isotopomers of methyl β- d -glucopyranoside, methyl β- d -xylopyranoside, methyl β- d -galactopyranoside, methyl β- d -galactopyranosyl-(1→4)-β- d -glucopyranoside and methyl β- d -galactopyranosyl-(1→4)-β- d -xylopyranoside were prepared, crystallized, and studied by single-crystal X-ray crystallography and solid-state 13 C NMR spectroscopy to determine experimentally the dependence of 2 J C1,C3 values in aldopyranosyl rings on the C1–C2–O2–H torsion angle, θ 2 , involving the C2 carbon of the C1–C2–C3 coupling pathway. Using X-ray crystal structures to determine θ 2 in crystalline samples and by selecting compounds that exhibit a relatively wide range of θ 2 values in the crystalline state, 2 J C1,C3 values measured in crystalline samples were plotted against θ 2 and the resulting plot compared to that obtained from density functional theory (DFT) calculations. For θ 2 values ranging from ∼90° to ∼240°, very good agreement was observed between the experimental and theoretical plots, providing strong validation of DFT-calculated spin-coupling dependencies on exocyclic C–O bond conformation involving the central carbon of geminal C–C–C coupling pathways. These findings provide new experimental evidence supporting the use of 2 J CCC values as non-conventional spin-coupling constraints in MA′AT conformational modeling of saccharides in solution, and the use of NMR spin-couplings not involving coupled hydroxyl hydrogens as indirect probes of C–O bond conformation. Solvomorphism was observed in crystalline βGal-(1→4)-βGlcOCH 3 wherein the previously-reported methanol solvate form was found to spontaneously convert to a monohydrate upon air-drying, leading to small but discernible conformational changes in, and a new crystalline form of, this disaccharide.more » « less
-
Water soluble ferrocene (Fc) derivatives are promising cathode materials for aqueous organic redox flow batteries (AORFBs) towards scalable energy storage. However, their structure–performance relationship and degradation mechanism in aqueous electrolytes remain unclear. Herein, physicochemical and electrochemical properties, battery performance, and degradation mechanisms of three Fc catholytes, (ferrocenylmethyl)trimethylammonium chloride (C1-FcNCl), (2-ferrocenyl-ethyl)trimethylammonium chloride (C2-FcNCl), and (3-ferrocenyl-propyl)trimethylammonium chloride (C3-FcNCl) in pH neutral aqueous electrolytes were systemically investigated. UV-Vis and gas chromatography (GC) studies confirmed the thermal and photolytic C x -Cp − ligand dissociation decomposition pathways of both discharged and charged states of C1-FcNCl and C2-FcNCl catholytes. In contrast, in the case of the C3-FcNCl catholyte, the electron-donating 3-(trimethylammonium)propyl group strengthens the coordination between the C 3 -Cp − ligand and the Fe 3+ or Fe 2+ center and thus mitigates the ligand-dissociation degradation. Consistently, the Fc electrolytes displayed cycling stability in both half-cell and full-cell flow batteries in the order of C1-FcNCl < C2-FcNCl < C3-FcNCl.more » « less
-
Abstract Methods that can simultaneously install multiple different functional groups to heteroarenes via C−H functionalizations are valuable for complex molecule synthesis, which, however, remain challenging to realize. Here we report the development of vicinal di‐carbo‐functionalization of indoles in a site‐ and regioselective manner, enabled by the palladium/norbornene (Pd/NBE) cooperative catalysis. The reaction is initiated by the Pd(II)‐mediated C3‐metalation and specifically promoted by the C1‐substituted NBEs. The mild, scalable, and robust reaction conditions allow for a good substrate scope and excellent functional group tolerance. The resulting C2‐arylated C3‐alkenylated indoles can be converted to diverse synthetically useful scaffolds. The combined experimental and computational mechanistic study reveals the unique role of the C1‐substituted NBE in accelerating the turnover‐limiting oxidative addition step.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S1793042121500822
