Iwasawa Invariants of Elliptic Curves and p_ADIC Measures
| dc.contributor.author | Barman, Rupam | |
| dc.date.accessioned | 2015-09-16T10:40:24Z | |
| dc.date.accessioned | 2023-10-20T12:30:30Z | |
| dc.date.available | 2015-09-16T10:40:24Z | |
| dc.date.available | 2023-10-20T12:30:30Z | |
| dc.date.issued | 2010 | |
| dc.description | Supervisor: Anupam Saikia | en_US |
| dc.description.abstract | The central theme of our work is to investigate Iwasawa invariants associated with elliptic curves and p-adic measures. Iwasawa D- and D-invariants of an elliptic curve contain valuable information about the curve. On the other hand, p-adic L-functions over Q arise as D-transforms of certain p-adic measures, hence there is considerable interest in Iwasawa invariants of such measures and their D-transforms. Suppose that E1 and E2 are elliptic curves defined over Q. Let p be an odd prime where E1 and E2 have good ordinary reduction. Assume that E1[pi] D= E2[pi] as Galois modules for i = D(E1)+1. Also assume that both E1(Q)[p] and E2(Q)[p] are trivial. Under the above assumptions we prove that D(E1) = D(E2). Also, if E1[pi] D= E2[pi] as Galois modules for i = D(E1), then D(E1) D D(E2). This result is an extension of earlier works of Greenberg and Vatsal, who studied this problem for elliptic curves with D-invariants zero. We also illustrate our results through some numerical examples. We find a generalization of an existing result of Satoh, Kida and Childress that deals with p-adic measures on Zp to p-adic measures on Zn p for any n. Let O be the ring of integers of a finite extension of Qp, where p is a fixed odd prime. If D is a O-valued measure on Zn p , then it gives a power series in n variables with coefficients in O. Analogous to the case n = 1, we define Iwasawa invariants of such a power series for any n. Given a O-valued measure D on Zn p , one obtains a new measure D = P D12V D D D P Dn2V (D D (D1; D D D ; Dn))jUn on Un while defining the D-transform of D. Extending by 0, D gives a measure on Zn p . We obtain a relation between the Iwasawa invariants of the power series associated to D and the D-transform for any n D 1. We prove the relation by deriving certain p-adic properties of Mahler coefficients of the continuous functions fm(x) = Dux m D and fm1;DDD ;mn(x1; D D D ; xn) = Dux1 m1 D D D D Duxn mn D , where u is a topological generator of 1 + pZp.. | en_US |
| dc.identifier.other | ROLL NO. 06612301 | |
| dc.identifier.uri | https://gyan.iitg.ac.in/handle/123456789/275 | |
| dc.language.iso | en | en_US |
| dc.relation.ispartofseries | TH-0816; | |
| dc.subject | MATHEMATICS | en_US |
| dc.title | Iwasawa Invariants of Elliptic Curves and p_ADIC Measures | en_US |
| dc.type | Thesis | en_US |