1. January 21, 2010 2 1.1. Why define a curve to be f rather than V (f) ⊂ P(k)? 3 1.2. Cubic plane curves 3 2. January 26, 2010 4 2.1. A little bit about smoothness 4 2.2. Weierstrass form 5 3. January 28, 2010 6 3.1. An algebro-geometric description of the group law in terms of divisors 6 3.2. Why are the two group laws the same? 7 4. February 2, 2010 7 4.1. Overview 7 4.2. Uniqueness of Weier...

τ(n)e(nz). The two equations were proven for τ(n) by Mordell, using what are now known as the Hecke operators. The inequality was proven by Deligne as a consequence of his proof of the Weil conjectures. Those results determine everything about af (n) except for the distribution of the af (p) ∈ [−2, 2]. Define θf (p) ∈ [0, π] by af (p) = 2 cos θf (p). It is conjectured that for each f the θf (p)...

In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify th...

In this talk the definition of polarization of abelian variety and Weil paring will be given. After introducing some proposition of Weil pairing, we will show the Zarhin’s trick about principal polarization.

In this paper, we define the Schrödinger-Weil representation for the Jacobi group and construct covariant maps for the Schrödinger-Weil representation. Using these covariant maps, we construct Jacobi forms with respect to an arithmetic subgroup of the Jacobi group.

Let M(n, d) denote the moduli space of stable holomorphic vector bundles of coprime rank n and degree d over a fixed Riemann surface Σ of genus g ≥ 2. Let Λ be a fixed line bundle over Σ of degree d and let MΛ(n, d) ⊂ M(n, d) denote the space consisting of those bundles with determinant Λ. The spaces M(n, d) and MΛ(n, d) are nonsingular complex projective varieties whose geometry has been much ...

Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average ran...

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J . Denote the Mordell–Weil rank of J(K) by r. We give an explicit and practical Chabauty-style criterion for showing that a given subset K ⊆ C(K) is in fact equal to C(K). This criterion is likely to be successful if r ≤ d(g − 1). We also show that the only sol...

We describe the large scale geometry of the mapping class group and of the pants graph (or equivalently the Teichmüller space in the Weil-Petersson metric) of a compact orientable surface, from the point of view of coarse median spaces. We derive various results about coarse rank and quasi-isometric rigidity of such spaces. In particular, we show that a quasi-isometric embedding of a mapping cl...

Given an elliptic curve E1 over a number field K and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2Selmer groups of E1, E2 and A, we can obtain information about X(E1/K)[2] and we give examples where...