On the -factors of motives. II.
On torsion in J₁(N)
On torsion in J₁(N), II
On torsion points on an elliptic curves via division polynomials.
On uniform lower bound of the Galois images associated to elliptic curves
Let be a prime and let be a number field. Let be the Galois representation given by the Galois action on the -adic Tate module of an elliptic curve over . Serre showed that the image of is open if has no complex multiplication. For an elliptic curve over whose -invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of .
On unlikely intersections of complex varieties with tori
On x³ + y³ + z³ = 3μxyz and Jacobi polynomials
Optimal curves differing by a 3-isogeny
Stein and Watkins conjectured that for a certain family of elliptic curves E, the X₀(N)-optimal curve and the X₁(N)-optimal curve of the isogeny class 𝓒 containing E of conductor N differ by a 3-isogeny. In this paper, we show that this conjecture is true.
Optimal curves differing by a 5-isogeny
For i = 0,1, let be the -optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational point of order...
Optimal levels for modular mod 2 representations over totally real fields.
Ordinariness in good reductions of Shimura varieties of PEL-type
Ordinary reduction of K3 surfaces
Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.