Mathematical proceedings of the Cambridge Philosophical Society

Vol. 167, num. 1, p. 35-60

DOI: 10.1017/S0305004118000166

Date of publication: 2019-07

Abstract:

Let S be a smooth cubic surface over a finite field q. It is known that #S( q) = 1 + aq + q2 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.]]>

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 163, num. 3, p. 423-452

DOI: 10.1017/S0305004117000044

Date of publication: 2017-11-01

Abstract:

We are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which \begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath} when dim FS, dim FT ¿ 2 and dim FST ¿ [L : F] - 2.]]>

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 142, num. 3, p. 407-428

Date of publication: 2007-05

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 141, num. 3, p. 383-408

DOI: 10.1017/S0305004106009613

Date of publication: 2006-11

Abstract:

It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL(2)-type over Q by giving a moduli interpretation which translates the question into the diophantine, arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.]]>

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 138, num. 1, p. 117-128

Date of publication: 2005-01

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 135, num. 1, p. 123-131

DOI: 10.1017/S0305004102006618

Date of publication: 2003

Abstract:

A $genus$ (in the sense of Hirzebruch [4]) is a multiplicative invariant of cobordism classes of manifolds. Classical examples include the numerical invariants given by the signature and the $\widehat{A}$- and Todd genera. More recently genera were introduced which take as values modular forms on the upper half-plane, $\frak{h}=\{\,\tau\;|\;\mathrm{Im}(\tau)>0\,\}$. The main examples are the elliptic genus $\phi_{ell}$ and the Witten genus $\phi_W$; we refer the reader to the texts [7] or [9] for details]]>

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 124, num. 2, p. 215-229

Date of publication: 1998-09

Mathematical proceedings of the Cambridge Philosophical Society

Vol. 120, num. 0, p. 499-519

Date of publication: 1996-03