Expositiones Mathematicae最新文献

Using our recent results on the algebraicity of the Hodge locus for variations of Hodge structures of level at least 3, we improve the results of Lawrence–Venkatesh in direction of the refined Bombieri–Lang conjecture.

We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form.

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves $X_0(D,N)$. In this paper we compute the $Q$-rational points on the Atkin–Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.

We compute the rational points on the Atkin–Lehner quotient $X_0^{+}\left(125\right)$ using the quadratic Chabauty method. Our work completes the study of exceptional rational points on the curves $X_0^{+}\left(N\right)$ of genus between 2 and 6. Together with the work of several authors, this completes the proof of a conjecture of Galbraith.

Two continued fractions $U\left(q\right)$ and $V\left(q\right)$ of order twenty-four are obtained from a general continued fraction identity of Ramanujan. Some theta-function and modular identities for $U\left(q\right)$ and $V\left(q\right)$ are established to prove general theorems for the explicit evaluations of $U(\pm q)$ and $V(\pm q)$. From the theta-function identities of $U\left(q\right)$ and $V\left(q\right)$, three colour partition identities are derived as application to partition theory of integer. Further, $2$-, $4$- and $8$-dissection formulas are established for the continued fractions $U^{\ast }\left(q\right)≔q^{-5/2}U\left(q\right)$ and $V^{\ast }\left(q\right)≔q^{-1/2}V\left(q\right)$, and their reciprocals.

In this paper, the composition of Bhargava’s cubes is generalized to the ring of integers of a number field of narrow class number one, excluding the case of totally imaginary number fields. The exclusion of the latter case arises from the nonexistence of a bijection between (classes of) binary quadratic forms and an ideal class group. This problem, together with a related mistake in another paper of the author, is addressed in the appendix.

We prove the classical Riemann–Roch theorems for the Adams operations ${\psi }^j$ on $K$-theory: a statement with coefficients on $Z\left[j^{-1}\right]$, that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

We generalize Mattei’s result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.

We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.