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If in a given rank r, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number \ell, there is an absolutely irreducible \ell-adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r if once there is an irreducible rank r complex linear representation with determinant isomorphic to this torsion character, then for any \ell, there is an absolutely irreducible one of rank r and determinant this given character, which is defined over \bar { \mathbb {Z}}_\ell. We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne's companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong's conjecture (Gaitsgory for \ell \ge 3). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple. [ABSTRACT FROM AUTHOR]

Let \sigma and \tau denote a pair of absolutely irreducible p-ordinary and p-distinguished Galois representations into \operatorname {GL}_2(\overline {\mathbb {F}}_p). Given two primitive forms (f,g) such that \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where \overline {\rho }_f\cong \sigma and \overline {\rho }_g\cong \tau, we show that the Iwasawa Main Conjecture for the double product \rho _f\otimes \rho _g depends only on the residual Galois representation \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p). More precisely, if IMC(f\otimes g) is true for one pair (f,g) with \overline {\rho }_f \cong \sigma and \overline {\rho }_g\cong \tau and whose \mu-invariant equals zero, then it is true for every congruent pair too. [ABSTRACT FROM AUTHOR]

Let F \in \mathbb {Z}[x_0, \ldots, x_n] be homogeneous of degree d and assume that F is not a 'nullform', i.e., there is an invariant I of forms of degree d in n+1 variables such that I(F) \neq 0. Equivalently, F is semistable in the sense of Geometric Invariant Theory. Minimizing F at a prime p means to produce T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e \in \mathbb {Z}_{\ge 0} such that F_1 = p^{-e} F([x_0, \ldots, x_n] \cdot T) has integral coefficients and v_p(I(F_1)) is minimal among all such F_1. Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w \in \mathbb {Z}_{\ge 0}^{n+1} to F. We show that for any dimension n and degree d, there is a complete set of weight vectors consisting of [0,w_1,w_2,\dots,w_n] with 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1}. When n = 2, we improve the bound to d. This answers a question raised by Kollár. These results are valid in a more general context, replacing \mathbb {Z} and p by a PID R and a prime element of R. Based on this result and a further study of the minimization process in the planar case n = 2, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma. [ABSTRACT FROM AUTHOR]

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy. [ABSTRACT FROM AUTHOR]

We describe a Kolyvagin system-theoretic refinement of Gross–Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve E over an imaginary quadratic field K is -1. When the root number of E over K is 1, we first establish the structure theorem of the p^\infty-Selmer group of E over K. The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various "Kolyvagin systems" and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of p^\infty-Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one p-converse to the theorem of Gross–Zagier and Kolyvagin. [ABSTRACT FROM AUTHOR]

This note proves two theorems regarding Fermat-type equation x^r + y^r = dz^p where r \geq 5 is a prime. Our main result shows that, for infinitely many integers d, the previous equation has no non-trivial primitive solutions such that 2 \mid x+y or r \mid x+y, for a set of exponents p of positive density. We use the modular method with a symplectic argument to prove this result. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]