We present a Mordell–Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb {Q}(\sqrt {d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and ${\lvert d \rvert < 100}$.

Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

We show that geodesics in $\mathbf {H}$ attached to a maximal split torus or a real quadratic torus in $GL_{2, \mathbf {Q}}$ are the only irreducible algebraic curves in $\mathbf {H}$ whose image in $\mathbf {R}^2$ via the j-invariant is contained in an algebraic curve.

For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

In this paper we construct a $\mathbb{Q}$-linear tannakian category $\mathsf{MEM}_{1}$ of universal mixed elliptic motives over the moduli space ${\mathcal{M}}_{1,1}$ of elliptic curves. It contains $\mathsf{MTM}$, the category of mixed Tate motives unramified over the integers. Each object of $\mathsf{MEM}_{1}$ is an object of $\mathsf{MTM}$ endowed with an action of $\text{SL}_{2}(\mathbb{Z})$ that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over ${\mathcal{M}}_{1,1}$ whose fiber over the tangential base point $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$ at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of $\mathsf{MEM}$ is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes the arithmetic relations, which describe the ‘infinitesimal Galois action’. We use the presentation to give a new and more conceptual proof of the Ihara–Takao congruences.

In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

In this paper, we prove that the modular curve $X(11)$ over a field of characteristic 3 admits the Mathieu group $M_{11}$ as an automorphism group. We also examine some aspects of the geometry of the curve $X(11)$ in characteristic 3. In particular, we show that every point of the curve is a point of inflection, the curve has 110 hyperflexes and there are no inflectional triangles and 11232 inflectional pentagons, of which 144 are self-conjugate. The hyperflexes correspond to the supersingular elliptic curves. We comment on the relationship of Ward's quadrilinear invariant for $M_{12}$ to our work and announce for the first time the equations for Klein's A-curve of level 11. We also comment on the relation of our work to some unpublished work of Bott and Tate.

1991 Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10, 14N10, 20B25, 20C34.