Related papers: Congruences of modular forms and modularity of Tat…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…

Conditionally on a conjecture on the \'etale cohomology of Hilbert modular surfaces and some minor technical assumptions, we establish new instances of the equivariant BSD-conjecture in rank $0$ with applications to the arithmetic of…

Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D^b(X) as in D^b(Y).…

Let $p\neq{2,3}$ be a prime number and let $\Gamma \subset \mathrm{SL}_{2}(\mathbb{Z})$ be a congruence subgroup with modular curve $X_{\Gamma}/K$ and Jacobian $J(X_{\Gamma})$. In this paper we give an explicit group-theoretic description…

A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate…

Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p.

Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$ of characteristic $p>0$ with $\mathrm{Gal}(Y/X)\cong\mathbb{Z}/p\mathbb{Z}$, and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We consider the…

We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2,3,4,5. Our construction makes use…

Under some assumptions, we compute the Picard group of moduli of stable sheaves on Abelian surfaces. Considering relative moduli spaces, it is sufficient to consider the moduli of stable sheaves on the product of elliptic curves. By using…

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over…

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different…

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological…

This is an updated version of ANT-0166. Generalizing results of Stroeker and Top we show that the 2-ranks of the Tate-Shafarevich groups of the elliptic curves $y^2 = (x+k)(x^2+k^2)$ can become arbitrarily large. We also present a…

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $\beta_C$ denote…

In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the…

Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if…

We apply Tate's conjecture on algebraic cycles to study the N\'eron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the…

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic…