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My question concerns holonomic sequences of rational numbers, meaning assignments $a(n) : \mathbb{N} \to \mathbb{Q}$ fulfilling a linear recurrent relation of the form $$ a(n+k) = \sum_{i=0}^{k-1} p_i …

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added t …

The elements of $S$ are conjectured to be $\mathbb{Q}$-linearly independent, and so a basis for the $\mathbb{Q}$-linear span of the multiple zeta values. This is what Francis Brown accomplished at t …

In Arakelov geometry, the conventional wisdom is that the ``closed fibre at $\infty$'' should be viewed as totally degenerate. This is the extreme opposite of smoothness. A visualization in the case o …

That will depend on your more specific interests. One point of entrance into arithmetic geometry could be to have a good understanding of Szpiro's discriminant-conductor inequality for non-isotrivial …

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real …

Erdos and Selfridge have proved that the product of two or more consecutive non-zero integers is never a power (Illinois J. Math, vol. 19, no. 2, 1975). This implies in particular that for $n > 1$ a p …

Will Sawin and Michael Stoll have noted that, as a consequence of Faltings's "Big Theorem," a hyperelliptic equation $y^2 = f(x)$ with $\deg{f} > 6$ (genus $> 2$) and not admitting a degree $2$ non-co …

This is motivated by a basic number theory question I asked the previous day: Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the corresp …

Let me add a different perspective on Stefan Keil's question: why is the isogeny class of an elliptic curve $E/\mathbb{Q}$ determined by the sequence of numbers $|E(\mathbb{F}_p)|$ for (almost) all pr …

It is all about the $g$-cuspidal singularity $y^2 = x^{2g+1}$ of the special fibre, and how this singularity affects the Picard group in terms of the Picard group of the normalization (which is an abe …

One way of seeing this is by appealing to Rumely's general local-global principle over $\bar{\mathbb{Z}}$, applied here to the moduli stack: an algebraic scheme over the algebraic integers $\bar{\mat …