Person: GEZER, BETÜL
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GEZER
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BETÜL
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Publication Sequences associated to elliptic curves(Editura Acad Romane, 2022-01-01) Gezer, Betül; GEZER, BETÜL; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; AAH-1547-2021Let E be an elliptic curve defined over a field K (with char(K) 2) given by a Weierstrass equation and let P = (x, y) is an element of E(K) be a point. Then for each n >= 1 and some gamma is an element of K* we can write the x- and y-coordinates of the point [n]P as [n]P = (phi(n)(P)/psi(2)(n)(P), omega(n)(P)/psi(3)(n)(P)) = (gamma(2)G(n)(P)/F-n(2)(P), gamma H-3(n)(P)/F-n(3)(P))where phi(n), psi(n), omega n is an element of K[x, y], gcd(phi(n), psi(n)) = 1 andF-n(P) =gamma(1-n2)psi(n)(P), G(n)(P) = gamma(-2n2) phi(n)(P), H-n(P) = gamma(-3n2)omega(n)(P)are suitably normalized division polynomials of E. In this work we show the coefficients of the elliptic curve E can be defined in terms of the sequences of values (G(n)(P))(n >= 0) and (H-n(P))(n >= 0) of the suitably normalized division polynomials of E evaluated at a point P is an element of E(K). Then we give the general terms of the sequences (G(n)(P))(n >= 0) and (H-n(P))(n >= 0) associated to Tate normal form of an elliptic curve. As an application of this we determine square and cube terms in these sequences.Publication On the product of translated division polynomials and somos sequences(Wydawnictwo Naukowe Uam, 2023-09-01) Gezer, Betül; Bizim, Osman; GEZER, BETÜL; BİZİM, OSMAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; AAH-1547-2021; AAH-1468-2021We consider the product sequences of the sequences (psi n(P)), (phi n(P)), and (omega n(P)) (n is an element of N) of values of the translated division polynomials of an elliptic curve E/K evaluated at a point P is an element of E(K)2. We prove that these sequences are purely periodic when K is a finite field. Then we use p-adic properties of these sequences to obtain p-adic convergence of product of the Somos 4 and Somos 5 sequences.Publication Sequences generated by elliptic curves(Polish Acad Sciences Inst Mathematics-IMPAN, 2019-01-01) Gezer, Betül; Bizim, Osman; GEZER, BETÜL; BİZİM, OSMAN; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; AAH-1468-2021; AAH-1547-2021Publication Representations of positive integers by positive quadratic forms(Southeast Asian Mathematical Soc-seams, 2011-01-01) TEKCAN, AHMET; Gezer, Betül; GEZER, BETÜL; Bizim, Osman; BİZİM, OSMAN; Özkoç, Arzu; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; AAH-1468-2021; AAH-8518-2021; AAH-1547-2021In this work we consider the representations of positive integers by quadratic forms F-1 = x(1)(2) + x(1)x(2) + 8x(2)(2) and G(1) = 2x(1)(2) + x(1)x(2) + 4x(2)(2) of discriminant 31 and we obtain some results concerning the modular forms (sci) (T; F, phi(tau s)). Moreover we construct a basis for the cusp form space S-4 (Gamma(0) (31), 1), and then we give some formulas for the number of representations of positive integer n by positive definite quadratic forms.Publication Sequences associated to elliptic curves with non-cyclic torsion subgroup(Hacettepe Üniversitesi, 2020-01-01) Gezer, Betül; GEZER, BETÜL; Bursa Uludağ Üniversitesi/Fen Fakültesi/Matematik Bölümü; AAH-1547-2021Let E be an elliptic curve defined over K given by a Weierstrass equation and let P = (x, y) is an element of E(K) be a point. Then for each n >= 1 we can write the x- and y-coordinates of the point [n]P as[n]P = (G(n)(P)/F-n(2)(P), H-n(P)/F-n(3)(P))where F-n, G(n), and H-n is an element of K[x, y] are division polynomials of E. In this work we give explicit formulas for sequences(F-n(P))(n >= 0), (G(n)(P))(n >= 0), and (H-n(P))(n >= 0)associated to an elliptic curve E defined over Q with non-cyclic torsion subgroup. As applications we give similar formulas for elliptic divisibility sequences associated to elliptic curves with non-cyclic torsion subgroup and determine square terms in these sequences.