In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.

A new family of Hamiltonian-minimal Lagrangian tori in the complex Euclidean plane is constructed. They are the first known unstable ones and are characterized in terms of being the only Hamiltonian-minimal Lagrangian tori (with non-parallel mean curvature vector) in $C^2$ admitting a one-parameter group of isometries.

Consider the plane cubic curves over an algebraically closed field of characteristic 2. By blowing up the parameter space $P^9$ twice we obtain a variety B of complete cubics. We then compute the characteristic numbers for various families of cubics by intersecting cycles on B.

In the paper we find a further generalization of congruences of the K. Hardy and K. S. Williams [5] type which seems to be a full generalization of congruences of G. Gras [4]. Moreover we extend results of [5], [7], [8], [9] and in part of [6]. We apply ideas and methods of [.

Let $a, b$ and $n$ be integers with $n ⩾ 3$. We show that, in the sense of natural density, almost all integers represented by the binary form $ax^n − by^n$ are thus represented essentially uniquely. By exploiting this conclusion, we derive an asymptotic formula for the total number of integers represented by such a form. These conclusions augment earlier work of Hooley concerning binary cubic and quartic forms, and generalise or sharpen work of Hooley, Greaves, and Skinner and Wooley concerning sums and differences of two nth powers.

Let p be a rational prime number. We refine Brauer‘s elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds $(rd^2)^2^(d-1)$. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.

A Q-curve is an elliptic curve, defined over a number field, that is isogenous to each of its Galois conjugates. Ribet showed that Serre's conjectures imply that such curves should be modular. Let E be an elliptic curve defined over a quadratic field such that E is 3-isogenous to its Galois conjugate. We give an algorithm for proving any such E is modular and give an explicit example involving a quotient of $J_o$ (169). As a by-product, we obtain a pair of 19-isogenous elliptic curves, and relate this to the existence of a rational point of order 19 on $J_1$ (13).

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions $F(t_1, …, t_n)$ of WDVV equations of associativity polynomial in $t_1, …, t_n-1$, exp $t_n$.

We give a new proof of Shiota's theorem on Novikov's consecture, which states that the K.P. equation characters Jacobians among all indecomposable principally polarized abelian varieties.

Let $G$ be an inner anisotropic form of an unitary group of 3 variables over Q, such that $G_R≃U(2,1)$, and π be an automorphic representation of G(A) whose archimedean component π∞ is a degenerate limit of discrete series; such a π never occurs in the cohomology (coherent or étale) of a Shimura variety. We show that however it does ‘appear’ in the coherent cohomology of some line bundle over an associated Griffiths-Schmid variety. Moreover we study cup products between such cohomology classes and some other automorphic cohomology classes and we prove some non-vanishing results.

In this paper we will give necessary conditions for a Borel-fixed monomial ideal to be the generic initial ideal of a reduced, irreducible, non-degenerate curve in $P^3$.

In this paper we take into consideration a conjecture of Bloch equating in a suitable range and under some standard conjectures, the rank of the motivic cohomology of the special fiber of a semistable degeneration over the ring of integers of a number field with the order of zero of the local Euler factor of the L-function over the semistable prime corresponding to the special fiber. We then develop, following the philosophy that the fiber ‘at infinity’ of an arithmetic variety should be considered as ‘maximally degenerate’, a construction that goes in parallel to the one we use for the non-archimedean fiber. Namely, a definition of a double complex and a weighted operator N on it that plays the role of the logarithm of the (local) monodromy map at infinity. We like to see this archimedean theory as the analogue of the limiting mixed Hodge structure theory of a degeneration of projective varieties over a disc. In particular, this yields to a description of the (motivic) Deligne cohomology as homology of the mapping cone of N. The main result arising from this construction is the proof of a conjecture of Deninger. Namely we show that the archimedean Γ-factor of the Zeta-function of the archimedean fiber can be seen as the characteristic polynomial of an archimedean Frobenius acting on the subgroup of the invariants of N into the hypercohomology of our double complex.

Motivated by a result of Bost, we use the relationship between Faltings' heights of abelian varieties with complex multiplication and logarithmic derivatives of Artin L-functions at $s=0$ to investigate these heights. In particular, we prove that the height of an elliptic curve with complex multiplication by $Q√-d$ is bounded from below by an effective affine function of log d.

We consider polynomial mappings which have atypical fibres due to the asymptotic behavior at infinity. Fixing some proper extension of the polynomial mapping, we study the localizability at infinity of the variation of topology of fibres and the possibility of interpreting local results at infinity into global results. We prove local and global Bertini–Sard–Lefschetz type statements for noncompact spaces and nonproper mappings and we deduce results on the homotopy type or the connectivity of the fibres of polynomial mappings.

The goal of this paper is to determine Göbner bases of powers of determinantal ideals and to show that the Rees algebras of (products of) determinantal ideals are normal and Cohen–Macaulay if the characteristic of the base field is non-exceptional. Our main combinatorial result is a generalization of Schensted's Theorem on the Knuth–Robinson–Schensted correspondence.