Consider two families of hyperelliptic curves (over ℚ), $C^{n,a}:y²=xⁿ+a$ and $C_{n,a}:y²=x(xⁿ+a)$, and their respective Jacobians $J^{n,a}$, $J_{n,a}$. We give a partial characterization of the torsion part of $J^{n,a}\left(\mathbb{Q}\right)$ and $J_{n,a}\left(\mathbb{Q}\right)$. More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8,a}\left(\mathbb{Q}\right)$. Namely, we show that $J_{8,a}{\left(\mathbb{Q}\right)}_{tors}=J_{8,a}\left(\mathbb{Q}\right)\left[2\right]$. In addition, we characterize the torsion parts of $J_{p,a}\left(\mathbb{Q}\right)$, where p is an odd prime, and...

We explicitly determine the elliptic $K3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$, for each of the two possible maximal fibre types, $I_{19}$ and $I_{14}^{*}$, the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I_{13}^{*}$, and there exist two (resp. one) one-parameter families of such surfaces.

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for all monic polynomials...

Let L be a finite Galois CM-extension of a totally real field K. We show that the validity of an appropriate special case of the Equivariant Tamagawa Number Conjecture leads to a natural construction for each odd prime p of explicit elements in the (non-commutative) Fitting invariants over ${\mathbb{Z}}_p\left[G\right]$ of a certain tame ray class group, and hence also in the analogous Fitting invariants of the p-primary part of the ideal class group of L. These elements involve the values at s=1 of the Artin L-series of characters...

Let $p$ be a prime and let $K$ be a number field. Let ${\rho }_{E,p}:{\mathrm{G}}_K\to \mathrm{Aut}\left(T_{p}E\right)\cong {\mathrm{GL}}_2\left({\mathbb{Z}}_p\right)$ be the Galois representation given by the Galois action on the $p$-adic Tate module of an elliptic curve $E$ over $K$. Serre showed that the image of ${\rho }_{E,p}$ is open if $E$ has no complex multiplication. For an elliptic curve $E$ over $K$ whose $j$-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 ${\rho }_{E,p}$.

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.

For i = 0,1, let $E_i$ be the $X_i\left(N\right)$-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...