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 Linear estimate for the number of zeros for Abelian integrals in a kind quadratic reversible center of genus one Lijun Hong,Xiaochun Hong,Junliang Lu Keywords:Abelian integral, quadratic reversible center, weakened Hilbert's 16th problem, limit cycle. Abstract: In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we consider linear estimate for the number of zeros for Abelian integrals in a kind quadratic reversible center of genus one, and obtain that its upper bound of the associated number of zeros for Abelian integrals under arbitrary polynomial perturbations of degree n is 2[(n+1)/2]+[n/2]+2 (n≥1), the upper bound linearly depends on n.