Split abelian surfaces over finite fields and reductions of genus-2 curves.

*(English)*Zbl 1357.14059It is conjectured in [V. K. Murty and V. M. Patankar, Int. Math. Res. Not. 2008, Article ID rnn033, 27 p. (2008; Zbl 1152.14043)] that if the absolute endomorphism ring of an absolutely simple abelian variety \(A\) over a number field \(K\) is commutative then the reduction \(A_{\mathfrak p}\) should be simple for almost all primes \({\mathfrak p}\). As this conjecture is widely believed and given the importance of computation, even speculative, in this area, the authors have judged that it is not too soon to try to estimate the number of exceptions.

They define (anticipating that the conjecture of Murthy and Pantakar is correct) \[ \pi_{\mathrm{split}}(A/K,z)=\#\{\mathfrak p\mid N(\mathfrak p)\leq z \text{ and }A_{\mathfrak p}\text{ is split}\} \] and conjecture that if \(A\) is defined over \(K\), has a principal polarisation and has \(\mathrm{End}_{\bar K}A={\mathbb Z}\), then \[ \pi_{\mathrm{split}}(A/K,z)\sim C_A{{\sqrt{z}}\over{\log z}} \] as \(z\to\infty\), for some \(C_A>0\). They cautiously use the word “hope” as well as “conjecture”. Their aim is to provide some evidence.

Some much weaker results are known, but the first evidence is that there is an analogy with the Lang-Trotter conjecture estimating the number of rational primes \(p<z\) such that a given elliptic curve over \({\mathbb Q}\) has \(p+1+a\) \({\mathbb F}_p\)-rational points for some given \(a\). This is also supposed to grow like \({\sqrt{z}}\over{\log z}\). They give a heuristic that suggests both their conjecture and the Lang-Trotter conjecture (without, however, offering any estimate of the constants involved).

This heuristic indicates that the conjecture made here should hold for \(K={\mathbb Q}\) if the probability that a randomly chosen principally polarised abelian surface over \({\mathbb F}_q\) splits is about \(q^{-1/2}\). The main technical theorem (and it is a theorem, despite the conjectural starting points and the presence nearby of GRH) states that this is indeed the actual probability. Precisely, they show that \[ (\log q)^{-3}(\log\log q)^{-4}\ll q^{-5/2}\# {\mathcal A}_{2,\mathrm{split}}({\mathbb F}_q) \ll (\log q)^4(\log\log q)^2 \] for all \(q\), where “split” means not simple over the field of definition \({\mathbb F}_q\). Slightly sharper bounds are obtained under GRH. At the end of the paper there is a suggestion about what should be expected if by split one means instead geometrically split, i.e. split over \(\overline{{\mathbb F}}_q\).

The proof of the estimate above occupied most of the paper. In outline it is quite straightforward. One divides principally polarised abelian surfaces over \({\mathbb F}_q\) that are isogenous to products \(E_1\times E_2\) into four classes and estimates the number in each class separately. The classes are: \(E_1\) and \(E_2\) are not isogenous; \(E_1=E_2\) is ordinary; \(E_1\) is supersingular and \(E_2\) is not; and \(E_1=E_2\) is supersingular. The number of surfaces over \({\mathbb F}_q\) in these classes is called \(W_q\), \(X_q\), \(Y_q\) and \(Z_q\) respectively, and one estimates all of these from above and \(W_q\) from below. As one would naively expect, \(W_q\) is the dominant term, of approximate order \(q^{5/2}\) (i.e. up to powers of \(\log q\) and \(\log \log q\)) whereas the others are of approximate order \(q^2\).

The technically hard part of the paper is in obtaining these estimates. The results depend on good estimates for class numbers, due to Littlewood with subsequent refinements and improvements. This is where the different bounds under GRH arise. Further estimates of multiplicative functions, coming from very classical analytic number theory techniques, are also needed, as well, of course, as an understanding of the surfaces, which depends on [G. Frey and E. Kani, Prog. Math. 89, 153–176 (1991; Zbl 0757.14015)]. Each of the classes requires separate treatment.

At the end of the paper there is some computational evidence for the conjecture and for some other estimates, obtained with help from A. Sutherland.

A crucial ingredient of the proof earlier is a bound on the sum \(f(q)\) of the relative conductors of all ordinary elliptic curves over \({\mathbb F}_q\), which is shown to be \(\sim q(\log q)^2\). For small \(q\) this number can be computed explicitly: in fact the number \(f(q)/q)\) appears to decrease, suggesting that in fact \(f(q)\sim q\).

Similarly, with a suitable weighted count (as usual, one should divide by the size of the automorphism group, as in Cohen-Lenstra) one finds experimentally that \(\sqrt{q}\times P\), where \(P\) is the probability of a surface being split, appears to be actually less than \(\log\log q\) and perhaps even bounded.

They define (anticipating that the conjecture of Murthy and Pantakar is correct) \[ \pi_{\mathrm{split}}(A/K,z)=\#\{\mathfrak p\mid N(\mathfrak p)\leq z \text{ and }A_{\mathfrak p}\text{ is split}\} \] and conjecture that if \(A\) is defined over \(K\), has a principal polarisation and has \(\mathrm{End}_{\bar K}A={\mathbb Z}\), then \[ \pi_{\mathrm{split}}(A/K,z)\sim C_A{{\sqrt{z}}\over{\log z}} \] as \(z\to\infty\), for some \(C_A>0\). They cautiously use the word “hope” as well as “conjecture”. Their aim is to provide some evidence.

Some much weaker results are known, but the first evidence is that there is an analogy with the Lang-Trotter conjecture estimating the number of rational primes \(p<z\) such that a given elliptic curve over \({\mathbb Q}\) has \(p+1+a\) \({\mathbb F}_p\)-rational points for some given \(a\). This is also supposed to grow like \({\sqrt{z}}\over{\log z}\). They give a heuristic that suggests both their conjecture and the Lang-Trotter conjecture (without, however, offering any estimate of the constants involved).

This heuristic indicates that the conjecture made here should hold for \(K={\mathbb Q}\) if the probability that a randomly chosen principally polarised abelian surface over \({\mathbb F}_q\) splits is about \(q^{-1/2}\). The main technical theorem (and it is a theorem, despite the conjectural starting points and the presence nearby of GRH) states that this is indeed the actual probability. Precisely, they show that \[ (\log q)^{-3}(\log\log q)^{-4}\ll q^{-5/2}\# {\mathcal A}_{2,\mathrm{split}}({\mathbb F}_q) \ll (\log q)^4(\log\log q)^2 \] for all \(q\), where “split” means not simple over the field of definition \({\mathbb F}_q\). Slightly sharper bounds are obtained under GRH. At the end of the paper there is a suggestion about what should be expected if by split one means instead geometrically split, i.e. split over \(\overline{{\mathbb F}}_q\).

The proof of the estimate above occupied most of the paper. In outline it is quite straightforward. One divides principally polarised abelian surfaces over \({\mathbb F}_q\) that are isogenous to products \(E_1\times E_2\) into four classes and estimates the number in each class separately. The classes are: \(E_1\) and \(E_2\) are not isogenous; \(E_1=E_2\) is ordinary; \(E_1\) is supersingular and \(E_2\) is not; and \(E_1=E_2\) is supersingular. The number of surfaces over \({\mathbb F}_q\) in these classes is called \(W_q\), \(X_q\), \(Y_q\) and \(Z_q\) respectively, and one estimates all of these from above and \(W_q\) from below. As one would naively expect, \(W_q\) is the dominant term, of approximate order \(q^{5/2}\) (i.e. up to powers of \(\log q\) and \(\log \log q\)) whereas the others are of approximate order \(q^2\).

The technically hard part of the paper is in obtaining these estimates. The results depend on good estimates for class numbers, due to Littlewood with subsequent refinements and improvements. This is where the different bounds under GRH arise. Further estimates of multiplicative functions, coming from very classical analytic number theory techniques, are also needed, as well, of course, as an understanding of the surfaces, which depends on [G. Frey and E. Kani, Prog. Math. 89, 153–176 (1991; Zbl 0757.14015)]. Each of the classes requires separate treatment.

At the end of the paper there is some computational evidence for the conjecture and for some other estimates, obtained with help from A. Sutherland.

A crucial ingredient of the proof earlier is a bound on the sum \(f(q)\) of the relative conductors of all ordinary elliptic curves over \({\mathbb F}_q\), which is shown to be \(\sim q(\log q)^2\). For small \(q\) this number can be computed explicitly: in fact the number \(f(q)/q)\) appears to decrease, suggesting that in fact \(f(q)\sim q\).

Similarly, with a suitable weighted count (as usual, one should divide by the size of the automorphism group, as in Cohen-Lenstra) one finds experimentally that \(\sqrt{q}\times P\), where \(P\) is the probability of a surface being split, appears to be actually less than \(\log\log q\) and perhaps even bounded.

Reviewer: G. K. Sankaran (Bath)