Towards the Sato–Tate groups of trinomial hyperelliptic curves

We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form [Formula: see text] where [Formula: see text] is the genus of the curve and [Formula: see text] is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for [Formula: see text] curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus [Formula: see text] curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components [Formula: see text] for these families of curves.

On the closure of the Hodge locus of positive period dimension

Given $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/2ℤ ⊕ ℤ/2ℤ, ℤ/2ℤ ⊕ ℤ/4ℤ or ℤ/2ℤ ⊕ ℤ/6ℤ

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text], [Formula: see text] or [Formula: see text].

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text].

The Griffiths bundle is generated by groups

First the Griffiths line bundle of a $$\mathbf {Q}$$ Q -VHS $${\mathscr {V}}$$ V is generalized to a Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu ,r)$$ grif ( G , μ , r ) associated to any triple $$(\mathbf {G}, \mu , r)$$ ( G , μ , r ) , where $$\mathbf {G}$$ G is a connected reductive group over an arbitrary field F, $$\mu \in X_*(\mathbf {G})$$ μ ∈ X ∗ ( G ) is a cocharacter (over $$\overline{F}$$ F ¯ ) and $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) is an F-representation; the classical bundle studied by Griffiths is recovered by taking $$F=\mathbf {Q}$$ F = Q , $$\mathbf {G}$$ G the Mumford–Tate group of $${\mathscr {V}}$$ V , $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu , r)$$ grif ( G , μ , r ) . The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of $$\mathbf {G}$$ G -Zips. When $$\mathbf {G}$$ G is F-simple, we show that, up to positive multiples, the Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G},\mu ,r)$$ grif ( G , μ , r ) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by $$-\mu $$ - μ . As an application, we show that the Griffiths line bundle of a projective $${{\mathbf {G}{\text{- }}{} \mathtt{Zip}}}^{\mu }$$ G - Zip μ -scheme is nef.

On Deformations of 1-motives

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

Normalization of Closed Ekedahl—Oort Strata

We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

A note on the Fourier coefficients of a Cohen–Eisenstein series

We prove a formula for the coefficients of a weight [Formula: see text] Cohen–Eisenstein series of square-free level [Formula: see text]. This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let [Formula: see text] be an odd prime number. For any elliptic curve [Formula: see text] defined over [Formula: see text] of rank zero and square-free conductor [Formula: see text], if [Formula: see text], under certain conditions on the Shafarevich–Tate group [Formula: see text], we show that [Formula: see text] divides [Formula: see text] if and only if [Formula: see text] divides the class number [Formula: see text] of [Formula: see text]