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Full-Text Articles in Physical Sciences and Mathematics
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
Doctoral Dissertations
We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …
Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen
Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen
Doctoral Dissertations
Let E1 x E2 over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E1 x E2, denoted as ap(E1) + ap(E2), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical …
Modular Forms And Elliptic Curves Over The Cubic Field Of Discriminant - 23, Paul E. Gunnells, Dan Yasaki
Modular Forms And Elliptic Curves Over The Cubic Field Of Discriminant - 23, Paul E. Gunnells, Dan Yasaki
Paul Gunnells
Let F be the cubic field of discriminant –23 and let O Ϲ F be its ring of integers. By explicitly computing cohomology of congruence subgroups of 〖GL〗_2(O) , we computationally investigate modularity of elliptic curves over F.
Class Numbers Of Ray Class Fields Of Imaginary Quadratic Fields, Omer Kucuksakalli
Class Numbers Of Ray Class Fields Of Imaginary Quadratic Fields, Omer Kucuksakalli
Open Access Dissertations
Let K be an imaginary quadratic field with class number one and let [Special characters omitted.] be a degree one prime ideal of norm p not dividing 6 d K . In this thesis we generalize an algorithm of Schoof to compute the class number of ray class fields [Special characters omitted.] heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura's reciprocity law. We have discovered a very interesting phenomena where p divides the class number of [Special characters omitted.] . This is a counterexample to the elliptic …
Power Residues Of Fourier Coefficients Of Elliptic Curves With Complex Multiplication, T Weston, E Zaurova
Power Residues Of Fourier Coefficients Of Elliptic Curves With Complex Multiplication, T Weston, E Zaurova
Mathematics and Statistics Department Faculty Publication Series
Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).