Related papers: A mod p variant of the Andr\'e-Oort conjecture

We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that…

We show that the reduction mod p of an orthogonal linear representation is orthogonal, and we generalize this fact to representations of algebras with involution.The proofs make an essential use of the notion of " middle lattices ".

We give an a geometric interpretation of the Hasse-Arf theorem for function fields using the recently proved Oort conjecture.

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using…

We study the induced Ekedahl-Oort stratification on the moduli of curves of genus 4 in positive characteristic.

Using linear forms in logarithms we prove an explicit result of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C}) \times…

We formulate a version of the Breuil--Mezard conjecture for quaternion algebras, and show that it follows from the Breuil--Mezard conjecture for GL_2. In the course of the proof we establish a mod p analogue of the Jacquet--Langlands…

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime $p\equiv 1\pmod 4$ and integer $a\not\equiv0\pmod p$, we prove that \begin{align*}&(-1)^{|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}\prod_{1\le…

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes…

For quadratic curves over $F_p$, the number of solutions, which is governed by an analogue of the Mordell-Weil group, is expressed with the Legendre symbol of a coefficient of quadratic curves. Focusing on the number of solutions, a…

We prove the level 1 case of Serre's conjecture.

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the…

We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this…

We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the…

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one…

We prove modularity of certain residually reducible ordinary 2-dimensional $p$-adic Galois representations with determinant a finite order odd character $\chi$. For certain non-quadratic $\chi$ we prove an $R=T$ result for $T$ the weight 1…

We discuss Base Change functoriality for mod p eigenforms for GL(2) over number fields. We carry out systematic computer experiments and collect data supporting its existence in cases of field extensions K/F where F is imaginary quadratic…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

In this paper we prove, assuming the Generalized Riemann Hypothesis, the Andr?e-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional…