# Anomalous Subvarieties—Structure Theorems and Applications

@article{Bombieri2007AnomalousST, title={Anomalous Subvarieties—Structure Theorems and Applications}, author={Enrico Bombieri and David Masser and Umberto Zannier}, journal={International Mathematics Research Notices}, year={2007}, volume={2007} }

When a fixed algebraic variety in a multiplicative group variety is intersected with the union of all algebraic subgroups of fixed dimension, a key role is played by what we call the anomalous subvarieties. These arise when the algebraic variety meets translates of subgroups in sets larger than expected. We prove a Structure Theorem for the anomalous subvarieties, and we give some applications, emphasizing in particular the case of codimension two. We also state some related conjectures about…

## 104 Citations

### On abelian points of varieties intersecting subgroups in a torus

- MathematicsJournal de théorie des nombres de Bordeaux
- 2022

We show, under some natural conditions, that the set of abelian points on the non-anomalous dense subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic…

### On the Torsion Anomalous Conjecture in CM abelian varieties

- Mathematics
- 2014

The Torsion Anomalous Conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties. In this work we prove, with an effective…

### Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension

- Mathematics
- 2009

In fact both authors stated stronger conjectures. Zilber conjectured a finiteness statement for subvarieties that are contained in an improper intersection of X with an algebraic subgroup of S. Pink…

### Unlikely Intersections of Curves with Algebraic Subgroups in Semiabelian Varieties

- Mathematics
- 2021

. Let G be a semiabelian variety and C a curve in G that is not contained in a proper algebraic subgroup of G . In this situation, conjectures of Pink and Zilber imply that there are at most ﬁnitely…

### An Addendum to the elliptic torsion anomalous conjecture in codimension 2

- MathematicsRendiconti del Seminario Matematico della Università di Padova
- 2019

The torsion anomalous conjecture states that for any variety V in an abelian variety there are only finitely many maximal V-torsion anomalous varieties. We prove this conjecture for V of codimension…

### Unlikely, likely and impossible intersections without algebraic groups

- Mathematics
- 2013

We formulate function field analogues for the PinkZilber Conjecture and for the Bounded Height Conjecture. The “special” varieties in our formulation are varieties defined over the constant field. We…

### Unlikely intersections for curves in multiplicative groups over positive characteristic

- Mathematics
- 2014

The conjectures associated with the names of Zilber–Pink greatly generalize results associated with the names of Manin–Mumford and Mordell–Lang, but unlike the latter they are at present restricted…

### Unlikely intersections and multiple roots of sparse polynomials

- Mathematics
- 2014

We present a structure theorem for the multiple non-cyclotomic irreducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents.…

### Unlikely intersections for curves in additive groups over positive characteristic

- Mathematics
- 2017

The conjectures associated with the names of Zilber-Pink greatly generalize results associated with the names of Manin-Mumford and Mordell-Lang, but unlike the latter they are at present restricted…

### Homology torsion growth and Mahler measure

- Mathematics
- 2010

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the…

## References

SHOWING 1-10 OF 28 REFERENCES

### Intersecting a plane with algebraic subgroups of multiplicative groups

- Mathematics
- 1999

Consider an arbitrary algebraic curve defined over the field of all alge- braic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the…

### Finiteness results for multiplicatively dependent points on complex curves

- Mathematics
- 2003

Let n be a positive integer, and let C be a curve in G m that we suppose (for convenience) is absolutely irreducible. When H is a fixed algebraic subgroup of G m, the intersection of C with H by…

### A Common Generalization of the Conjectures of André-Oort

- Mathematics
- 2005

Consider any irreducible closed subvariety Z ⊂ S. Since any intersection of special subvarieties is a finite union of special subvarieties, there exists a unique smallest special subvariety…

### Intersection de sous-groupes et de sous-variétés I

- Mathematics
- 2005

We study the intersection of a subvariety X of an abelian variety A over with the union of all the algebraic subgroups of A of given dimension d. Our main result states that if we remove a suitable…

### A relative Dobrowolski lower bound over abelian extensions

- Mathematics
- 2000

Let a be a non-zero algebraic number, not a root of unity. A wellknown theorem by E. Dobrowolski provides a lower bound for the Weil height h(a) which, in simplified form, reads h(a) » D-l-’, where D…

### Intersecting curves and algebraic subgroups: conjectures and more results

- Mathematics
- 2005

This paper solves in the affirmative, up to dimension n = 5, a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the…

### Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe

- MathematicsCompositio Mathematica
- 1998

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…

### Fundamentals of Diophantine Geometry

- Mathematics
- 1983

1 Absolute Values.- 2 Proper Sets of Absolute Values. Divisors and Units.- 3 Heights.- 4 Geometric Properties of Heights.- 5 Heights on Abelian Varieties.- 6 The Mordell-Weil Theorem.- 7 The…

### Heights in Diophantine Geometry

- Mathematics
- 2006

I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil…

### Exponential Sums Equations and the Schanuel Conjecture

- Mathematics
- 2002

A uniform version of the Schanuel conjecture is discussed that has some model‐theoretical motivation. This conjecture is assumed, and it is proved that any ‘non‐obviously‐contradictory’ system of…