Related papers: On the section conjecture over function fields and…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the…

We construct \'etale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized…

In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in an arbitrary fixed arithmetic progression. Contrary to the classical…

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…

We prove that Grothendieck's Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes,…

We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for…

We prove the following result which was conjectured by Stichtenoth and Xing: let $g$ be the genus of a projective, irreducible non-singular curve over the finite field $\Bbb F_{q^2}$ and whose number of $\Bbb F_{q^2}$-rational points…

We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang's conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many…

We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We…

We prove that a refinement of Stark's Conjecture formulated by Rubin is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of…

In this paper, we prove the Shafarevich conjecture for certain complete intersections of hypersurfaces in abelian varieties defined over a number field $K$ using the Lawrence-Venkatesh method. The main new inputs we need are computation of…

Let $k$ be a field that is finitely generated over the field of rational numbers and $Br(k)$ the Brauer group of $k$. Let $X$ be an absolutely irreducible smooth projective variety over $k$, let $Br(X)$ be the cohomological…

In three preprints [Pan1], [Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here $G$ is a reductive group scheme). The preprint…

Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate-Shafarevich group of a $K$-torus $T$ is $Sha(T , V) = \ker\left(H^1(K , T) \to \prod_{v \in V} H^1(K_v , T)\right)$. We prove that if $K = k(X)$ is…

By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic $\chi$-volume along semiample divisors.…

We compute the $3$-class groups $A_n$ of the fields $F_n$ in the cyclotomic $\mathbf{Z}_3$-extensions of the real quadratic fields of discriminant $f<100,000$. In all cases the orders of $A_n$ remain bounded as $n$ goes to infinity. This is…

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.

K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite…

We prove Vojta's generalized abc conjecture for algebraic tori over function fields with exceptional sets that can be determined effectively. Additionally, we establish a version of the conjecture for toric varieties. As an application, we…