Related papers: A Chabauty-Coleman bound for surfaces

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what…

We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give…

We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of…

Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22]) states that a curve of genus greater than one over any number field has only finitely many points. Again a natural question is how many points can such a curve have.…

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

We prove a lower bound that agrees with Manin's prediction for the number of rational points of bounded height on the Fermat cubic surface. As an application we provide a simple counterexample to Manin's conjecture over the rationals.

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs…

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties $A_f$ attached by Shimura to normalized newforms $f \in S_2( \Gamma_0(N))$. We present all the curves corresponding to principally…

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational…

Based on computational evidence, we formulate a number of conjectures on the distribution of rational points on curves of genus 2 over the rational numbers, in terms of the size of the coefficients of an equation of the form y^2 = f(x) >.

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields…

The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of…

Given number fields $L \supset K$, smooth projective curves $C$ defined over $L$ and $B$ defined over $K$, and a non-constant $L$-morphism $h \colon C \to B_L$,we consider the curve $C_h$ defined over $K$ whose $K$-rational points…

We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves.

In this paper I prove that for any prime $p$ there is a constant $C_p>0$ such that for any $n>0$ and for any $p$-power $q$ there is a smooth, projective, absolutely irreducible curve over $\mathbb{F}_q$ of genus $g\leq C_p q^n$ without…

In this paper, we prove that for each number field $F$ there exists a uniform bound on the prime levels $p$ of elliptic curves $E/F$ for which $F(E[p])=F(\zeta_p)$. Under the Generalized Riemann Hypothesis, we also give uniform bounds on…

We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.

For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$…

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is…