We show that the naive adaptation of Malle’s conjecture to fair counting functions is not true in general.
We show that the naive adaptation of Malle’s conjecture to fair counting functions is not true in general.
We give simple geometric proofs of Aprodu, Farkas, Papadima, Raicu and Weyman’s theorem on syzygies of tangent-developable surfaces of rational normal curves and Raicu and Sam’s result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green’s conjecture for general curves of genus over an algebraically closed field with or . Our approach provides a new way to study tangent-developable surfaces in general. Along the way, we show the arithmetic normality of tangent-developable surfaces of arbitrary smooth projective curves of large degree.
Let be the germ of a complex hypersurface isolated singularity of equation , with at least of dimension . We consider the family of analytic -modules generated by the powers of and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of in the neighbourhood of the singularity.
We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of -term complexes categorifying the classical concatenation/near-concatenation identity for symmetric functions, and one of our main results is that the exactness of these -term complexes is equivalent to the Koszul property of the underlying algebra . We further generalize these ribbon Schur functors to the notion of a multi-Schur functor and construct a canonical filtration of these objects whose associated graded pieces are described explicitly; one consequence of this filtration is a complete equivariant description of the syzygies of arbitrary Segre products of Koszul modules over the Segre product of Koszul algebras. Further applications to the equivariant structure of derived invariants, symmetric function identities, and Koszulness of certain classes of modules are explored at the end, along with a characteristic-free computation of the regularity of the Schur functor applied to the tautological subbundle on projective space.
We classify the automorphism groups of del Pezzo surfaces of degrees 1 and 2 over an algebraically closed field of characteristic 2. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.
A Thue–Mahler equation is a Diophantine equation of the form
where is an irreducible binary form of degree at least with integer coefficients, is a nonzero integer and are rational primes. Existing algorithms for resolving such equations require computations in the field , where , , are distinct roots of . We give a new algorithm that requires computations only in making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell–Weil sieve that makes it practical to tackle Thue–Mahler equations of higher degree and with larger sets of primes than was previously possible. We give several examples including one of degree
In this paper, we begin the study of the Fermat equation over real biquadratic fields. In particular, we prove that there are no nontrivial solutions to the Fermat equation over for .
Over a finite field , abelian varieties with commutative endomorphism rings can be described by using modules over orders in étale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when is locally Gorenstein, we show that the group structure of is determined by . In fact, the same conclusion is attained if has local Cohen–Macaulay type at most , under the additional assumption that is ordinary or is prime, although the conclusion is not true in general. Second, the description in the Gorenstein case is used to characterize cyclic isogeny classes in terms of conductor ideals. Third, going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with rational points in which every abelian group of order is realized as a group of rational points. Finally, we study when an abelian variety over and its dual satisfy or fail to satisfy several interrelated properties, namely
We develop obstruction theory for lifting characteristic- local Galois representations valued in reductive groups of type , , or . An application of the Emerton–Gee stack then reduces the existence of crystalline lifts to a purely combinatorial problem when is not too small.
As a toy example, we show for all local fields , with , all representations admit a crystalline lift , where is the exceptional Chevalley group of type .
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