Bipartite graphs are widely used to model relationships between two types of entities. Community search retrieves densely connected subgraphs containing a query vertex, which has been extensively studied on unipartite graphs. However, community search on bipartite graphs remains largely unexplored. Moreover, all existing cohesive subgraph models on bipartite graphs can only be applied to measure the structure cohesiveness between two sets of vertices while overlooking the edge weight in forming the community. In this paper, we study the significant (alpha, beta)-community search problem on weighted bipartite graphs. Given a query vertex q, we aim to find the significant (alpha, beta)-community R of q which adopts (alpha, beta)-core to characterize the engagement level of vertices, and maximizes the minimum edge weight (significance) within R. To support fast retrieval of R, we first retrieve the maximal connected subgraph of (alpha, beta)-core containing the query vertex (the (alpha, beta)-community), and the search space is limited to this subgraph with a much smaller size than the original graph. A novel index structure is presented which can be built in O(delta * m) time and takes O(delta * m) space where m is the number of edges in G, delta is bounded by the square root of m and is much smaller in practice. Utilizing the index, the (alpha, beta)-community can be retrieved in optimal time. To further obtain R, we develop peeling and expansion algorithms to conduct searches by shrinking from the (alpha, beta)-community and expanding from the query vertex, respectively. The experimental results on real graphs not only demonstrate the effectiveness of the significant (alpha, beta)-community model but also validate the efficiency of our query processing and indexing techniques.