We study the failure of a local-global principle for the existence of l-isogenies for elliptic curves over number fields K. Sutherland has shown that over ‫ޑ‬ there is just one failure, which occurs for l D 7 and a unique j -invariant, and has given a classification of such failures when K does not contain the quadratic subfield of the l-th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new "exceptional" source of such failures arising from the exceptional subgroups of PGL 2 ‫ކ.‬ l /. By constructing models of two modular curves, X s .5/ and X S 4 .13/, we find two new families of elliptic curves for which the principle fails, and we show that, for quadratic fields, there can be no other exceptional failures.

Let S be a smooth cubic surface over a finite field F q . It is known that #S(F q ) = 1 + aq + q 2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.Note that a smooth cubic surface S with a(S) = 7 is split, i.e. all its lines are defined over F q . It has been known for a long time, prior to the work of Swinnerton-Dyer [37], that such a surface exists over F q if and only if q = 2, 3, 5; this result appears to be first due to Hirschfeld [16, Thm. 20.1.7].We briefly explain the proof of Theorem 1.1. Any smooth cubic surface over an algebraically closed field is the blow-up of P 2 in 6 rational points in general position. Whilst this does not hold over other fields in general, the trace values 1, 2, 3, 4, 5 can be obtained from cubic surfaces which are blow-ups of P 2 in collections of closed points in general position. We therefore show that such collections exist over every finite field F q , via combinatorial arguments. Trace 0 can be obtained by blowing-up certain collections of closed points of total degree 7, and contracting a line. The existence of the remaining traces −1, −2 can be deduced from work of Rybakov [29] and Swinnerton-Dyer [37], respectively. 1.2. Corrections to Manin's and Swinnerton-Dyer's tables. Let S be a smooth cubic surface over a finite field F q . Building on work of Frame [12] and Swinnerton-Dyer [36], Manin constucted a table (Table 1 of [27, p. 176]) of the conjugacy classes of W (E 6 ) and their properties, such as the trace a(S).Urabe [39,40] was the first to notice that Manin's table contains mistakes regarding the calculation of H 1 (F q , PicS) (the issue being that H 1 (F q , PicS) must have square order). In our investigation we found some new mistakes. These concern the Galois orbit on the lines, where the mistake can be traced back to , and the index [27, §28.2] of the surface. The index is the size of the largest Galois invariant collection of pairwise skew lines overF q .Manin's table has a surface of index 2, which led him to state [27, Thm. 28.5(i)] that the index can only take one of the values 0, 1, 2, 3, 6. Our investigations reveal, however, that index 2 does not occur, and that index 5 can occur, hence the correct statement is the following. Theorem 1.2. Let S be a smooth cubic surface over a finite field. Then the index of S can only take one of the values 0, 1, 3, 5, 6.

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin–Najman, Özman–Siksek, and most recently Box, we determine the above set of primes for the three quadratic fields, ${\mathbb {Q}}(\sqrt {-10})$, ${\mathbb {Q}}(\sqrt {5})$, and ${\mathbb {Q}}(\sqrt {7})$, providing the first such examples after Mazur’s 1978 determination for $K = {\mathbb {Q}}$. The termination of the algorithm relies on the Generalised Riemann Hypothesis.