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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.
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Compactified Jacobians as Mumford models
We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.
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- PAR ID:
- 10462234
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- Volume:
- 376
- Issue:
- 1070
- ISSN:
- 0002-9947
- Page Range / eLocation ID:
- 4605 to 4630
- 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/10.1090/tran/8875
