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Penn State University

1. Yelton, Jeffrey Samuel. Hyperelliptic Jacobians and their associated $\ell$-adic Galois representations.

Degree: 2015, Penn State University

URL: https://submit-etda.libraries.psu.edu/catalog/25868

Let k be a field of characteristic different from 2, and let K be the extension of k obtained by adjoining the symmetric functions of the independent transcendental elements α_{1}, α_{2}, ... , α_{d} for some d ≥ 3. For each prime ℓ, we examine the natural ℓ-adic representation of the absolute Galois group of K associated to the ``generic" Jacobian J of the hyperelliptic curve over K whose Weierstrass roots are the α_{i}'s. In particular, we show that the image of the absolute Galois group in the group of automorphisms of the ℓ-adic Tate module T_{ℓ}(J) contains the entire symplectic group \Sp(T_{ℓ}(J)) when ℓ \neq 2, and that it contains the level-2 congruence subgroup Γ(2) \lhd \Sp(T_{2}(J)) when ℓ = 2. We also derive formulas for generators for the field extension K(J[4]) / K, as well as formulas for generators of K(J[8]) / K in the case that J is an elliptic curve.
We then give a full desription of the infinite algebraic extension of K generated by the coordinates of all 2-power torsion points of J when d is 3, 5, or 6, by giving recursive formulas for the generators. This is done in all of the above cases by associating to J a regular tree and assigning elements of the algebraic closure of K to the vertices of the tree. We also use these constructions to describe several other algebraic extensions of K related to the subgroup of dyadic torsion of J.
*Advisors/Committee Members: Yuri Zarhin, Dissertation Advisor/Co-Advisor, Mihran Papikian, Committee Member, Wen Ching Li, Committee Member, Donald Richards, Special Member.*

Subjects/Keywords: abelian variety; elliptic curve; Galois theory

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Yelton, J. S. (2015). Hyperelliptic Jacobians and their associated $\ell$-adic Galois representations. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/25868

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Yelton, Jeffrey Samuel. “Hyperelliptic Jacobians and their associated $\ell$-adic Galois representations.” 2015. Thesis, Penn State University. Accessed October 31, 2020. https://submit-etda.libraries.psu.edu/catalog/25868.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Yelton, Jeffrey Samuel. “Hyperelliptic Jacobians and their associated $\ell$-adic Galois representations.” 2015. Web. 31 Oct 2020.

Vancouver:

Yelton JS. Hyperelliptic Jacobians and their associated $\ell$-adic Galois representations. [Internet] [Thesis]. Penn State University; 2015. [cited 2020 Oct 31]. Available from: https://submit-etda.libraries.psu.edu/catalog/25868.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Yelton JS. Hyperelliptic Jacobians and their associated $\ell$-adic Galois representations. [Thesis]. Penn State University; 2015. Available from: https://submit-etda.libraries.psu.edu/catalog/25868

Not specified: Masters Thesis or Doctoral Dissertation

Penn State University

2. Mayanskiy, Evgeny S. An asymptotic Mukai model of M6.

Degree: 2013, Penn State University

URL: https://submit-etda.libraries.psu.edu/catalog/18733

We study the Mukai construction of a general curve of genus 6 as a complete intersection of the Grassmannian of lines in P4 with a codimension 5 quadric in the Pl{ücker space. We formulate the relevant GIT problem in general and then solve it for the large values of the GIT parameter. This allows us to conclude that asymptotically Mukai compact model of M6 parametrizes double anticanonical curves on the smooth del Pezzo surface of degree 5. As a byproduct of our study we obtain an explicit geometric interpretation of Ozeki classification of orbits of a certain prehomogeneous space. This complements earlier results of J.A. Todd [17].
*Advisors/Committee Members: Yuri Zarhin, Dissertation Advisor/Co-Advisor, Yuri Zarhin, Committee Chair/Co-Chair, Robert Charles Vaughan, Committee Member, Dale Brownawell, Committee Member, Karl Schwede, Committee Member, Murat Gunaydin, Committee Member.*

Subjects/Keywords: Genus 6 curves; Mukai models; Hassett-Keel program; geometric invariant theory

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Mayanskiy, E. S. (2013). An asymptotic Mukai model of M6. (Thesis). Penn State University. Retrieved from https://submit-etda.libraries.psu.edu/catalog/18733

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Mayanskiy, Evgeny S. “An asymptotic Mukai model of M6.” 2013. Thesis, Penn State University. Accessed October 31, 2020. https://submit-etda.libraries.psu.edu/catalog/18733.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Mayanskiy, Evgeny S. “An asymptotic Mukai model of M6.” 2013. Web. 31 Oct 2020.

Vancouver:

Mayanskiy ES. An asymptotic Mukai model of M6. [Internet] [Thesis]. Penn State University; 2013. [cited 2020 Oct 31]. Available from: https://submit-etda.libraries.psu.edu/catalog/18733.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Mayanskiy ES. An asymptotic Mukai model of M6. [Thesis]. Penn State University; 2013. Available from: https://submit-etda.libraries.psu.edu/catalog/18733

Not specified: Masters Thesis or Doctoral Dissertation