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Texas A&M University

1. Barquero Sanchez, Adrian Alberto. The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture.

Degree: 2016, Texas A&M University

URL: http://hdl.handle.net/1969.1/157725

In this thesis we start by giving a quick review of the classical Chowla-Selberg formula. We
then recall a conjecture of Colmez which relates the Faltings height of an abelian variety with
complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of certain Artin L?functions at s = 0. It turns out that in the case in which the abelian variety is a CM elliptic curve, the conjecture of Colmez can be seen as a geometric reformulation of the classical Chowla-Selberg formula.
Then we will focus our attention on establishing a generalization of the classical Chowla-
Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler?s gamma function ? and an analogous function ?2 at rational numbers.
Finally, we will study the above mentioned conjecture of Colmez. We will prove that if F
is any fixed totally real number field of degree [F : Q] ? 3, then there are infinitely many CM
extensions E ? F such that E ? Q is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at ?arbitrary? prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be seen as an explicit non-abelian Chowla-Selberg formula.
*Advisors/Committee Members: Masri, Riad (advisor), Papanikolas, Matthew (committee member), Young, Matthew P (committee member), Zhou, Lan (committee member).*

Subjects/Keywords: Chowla-Selberg formula; Colmez conjecture; Faltings height; non-abelian CM fields

Record Details Similar Records

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

APA (6^{th} Edition):

Barquero Sanchez, A. A. (2016). The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture. (Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/157725

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):

Barquero Sanchez, Adrian Alberto. “The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture.” 2016. Thesis, Texas A&M University. Accessed November 14, 2019. http://hdl.handle.net/1969.1/157725.

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

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Barquero Sanchez, Adrian Alberto. “The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture.” 2016. Web. 14 Nov 2019.

Vancouver:

Barquero Sanchez AA. The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture. [Internet] [Thesis]. Texas A&M University; 2016. [cited 2019 Nov 14]. Available from: http://hdl.handle.net/1969.1/157725.

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

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Barquero Sanchez AA. The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture. [Thesis]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/157725

Not specified: Masters Thesis or Doctoral Dissertation

Texas A&M University

2. Karadag, Tekin. Modular Forms and L-functions.

Degree: 2016, Texas A&M University

URL: http://hdl.handle.net/1969.1/157819

In this thesis, our main aims are expressing some strong relations between modular forms, Hecke operators, and L-functions. We start with background information for modular forms and give some information about the linear space of modular forms. Next, we introduce Hecke operators and their properties. Also we find a basis of Hecke eigenforms. Then, we explain how to construct L-functions from modular forms. Finally, we give a nice functional equation for completed L-function.
*Advisors/Committee Members: Young, Matthew P (advisor), Masri, Riad Mohamed (committee member), Cline, Daren H B (committee member).*

Subjects/Keywords: modular forms; L-functions; Hecke operators

Record Details Similar Records

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

APA (6^{th} Edition):

Karadag, T. (2016). Modular Forms and L-functions. (Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/157819

Not specified: Masters Thesis or Doctoral Dissertation

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

Karadag, Tekin. “Modular Forms and L-functions.” 2016. Thesis, Texas A&M University. Accessed November 14, 2019. http://hdl.handle.net/1969.1/157819.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Karadag, Tekin. “Modular Forms and L-functions.” 2016. Web. 14 Nov 2019.

Vancouver:

Karadag T. Modular Forms and L-functions. [Internet] [Thesis]. Texas A&M University; 2016. [cited 2019 Nov 14]. Available from: http://hdl.handle.net/1969.1/157819.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Karadag T. Modular Forms and L-functions. [Thesis]. Texas A&M University; 2016. Available from: http://hdl.handle.net/1969.1/157819

Not specified: Masters Thesis or Doctoral Dissertation