## Iwasawa Invariants of Elliptic Curves and p_ADIC Measures

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2010

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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..

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Supervisor: Anupam Saikia

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MATHEMATICS