Formal Groups and Atkin and Swinnerton-Dyer Congruences
Open Access
- Author:
- Kibelbek, Jonas
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- March 03, 2011
- Committee Members:
- Wen Ching W Li, Dissertation Advisor/Co-Advisor
Wen Ching W Li, Committee Chair/Co-Chair
Woodrow Dale Brownawell, Committee Member
Robert Charles Vaughan, Committee Member
Ling Rothrock, Committee Member - Keywords:
- congruences
modular forms
formal groups - Abstract:
- We examine the arithmetic structure of the Fourier coefficients of cusp forms and explore their relationship to the formal logarithms of integral formal group laws. Specifically, we use formal group theory to extend the Atkin and Swinnerton-Dyer congruences to certain cases of noncongruence subgroups Γ of SL<sub>2</sub>(<b>Z</b>) whose modular curves X<sub>Γ</sub> are defined over some finite extension <b>K</b> of <b>Q</b>. A theorem of Ditters provides the crucial link between integral formal group laws and congruences of Atkin and Swinnerton-Dyer type. <p>We associate to any space S<sub>k</sub>(Γ) of cusp forms a formal group law <b><i>F</i></b><sub>Γ,k</sub>. In the case that Γ is a congruence subgroup of SL<sub>2</sub>(<b>Z</b>), we prove that this formal group law is integral, using the Atkin-Lehner-Li theory of newforms. We also prove that <b><i>F</i></b><sub>Γ,2</sub> is integral at almost all places in the case that the modular curve X<sub>Γ</sub> is an elliptic or hyperelliptic curve, using C̆ech cohomology. From this, we prove a more general form of Atkin and Swinnerton-Dyer congruences at a strongly ordinary place ℘ for weight 2 cusp forms for such Γ; the resulting congruences are twisted by the frobenius automorphism at ℘.