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The study of rational points on an abelian variety A over a number field K gives rise to its Tate-Shafarevich group X(A/K), which plays an important role in understanding the arithmetic of A/K. The Tate-Shafarevich group is conjectured to be finite, and if it is, then for elliptic curves E/K the Cassels-Tate pairing forces the order of X(E/K) to be a perfect square. This implication is false for abelian varieties of dimension 2 or higher. It is known, that if A/K is principally polarised and the order of X(A/K) is finite, then this order is a square or twice a square. Poonen and Stoll showed that both cases occur. When A/K is not principally polarised, the situation remains unclear. William Stein constructed for all odd primes p < 25 000, p ̸ = 37, an abelian variety A p /Q of dimension p − 1, such that X(A p /Q) has order p times a square. He conjectures that for any given square-free positive integer k there is an abelian variety A/Q, such that #X(A/Q) = k ·. However, it is an open question what to expect if the dimension of A/Q is bounded. Restricting to abelian surfaces B/Q, then the results of Poonen, Stoll and Stein imply that there are examples such that #X(B/Q) = k · , for k ∈ {1, 2, 3}. In this thesis we focus in depth on abelian surfaces B/Q, such that there are elliptic curves E 1 /Q and E 2 /Q and an isogeny ϕ : E 1 × E 2 → B. We want to compute the order of X(B/Q) with respect to the order of the Tate-Shafarevich group of E 1 × E 2 , which has square order. To achieve this goal, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture. The main tool used is the Cassels-Tate equation. we construct a non-simple non-principally po-larised abelian surface B/Q whose Tate-Shafarevich group has order k times a square. All these examples do not assume the finiteness of the Tate-Shafarevich group. Furthermore, we compute numerically how often the order of X(B/Q) equals five times a square, for cyclic isogenies ϕ : E 1 × E 2 → B of degree 5. It turns out that this happens to be the case in approx. 50% of the first 20 million examples we have checked. Finally, we address the question …

On non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals