arXiv subjects
Publications and source records attributed to Armand Brumer.
We extend methods of Fontaine, Abrashkin and Schoof to obtain criteria determining number fields K over which no non-zero abelian variety with everywhere good reduction exists. As an application, under the GRH, we find 24744 such fields of various degrees up to 16.
We say that an abelian variety $A_{/\mathbf Q}$ of dimension $g$ is {\em prosaic} if it is semistable, with good reduction at 2 and its points of order $2$ generate a $2$-extension of ${\mathbf Q}$. For $p \equiv 1 \bmod{8}$, let $M_u$ be the maximal 2-primary unramified abelian extension of $K = {\mathbf Q}(\sqrt{-p})$ and let $h_2 =[M_u:K]$. We construct an indecomposable group scheme $\Xi_p$ over ${\mathbf Z}[\frac{1}{p}]$ of exponent 2 with field of points $M_u$. Assume that $A$ is prosaic, with bad reduction at only one prime $p$. Then $p \equiv 1 \bmod{8}$ and $A$ is totally toroidal at $p$. We prove that if End $A={\mathbf Z}$, then there is a ${\mathbf Q}$-isogenous abelian variety $B$ such that $B[2]$ is a subquotient of $\Xi_p$. We thereby show that $2g+2 \le h_2$ and $p$ has the form $a^2+16b^2$, with $a+4b \equiv \pm 1 \bmod{8}$. Moreover, if $2g + 4 \le h_2$, then $p$ has the form $a^2+64b^2$, with $a \equiv \pm 1 \bmod{8}$.
For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action on $\ell$-torsion points of $A$ is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of $\mathrm{PSp_4}(\mathbb{F}_\ell)$, sampling of the characteristic polynomials of Frobenius, and the Khare--Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus~$2$ curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.
Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves. In the appendix, Serre proves a result extending his work on the reduction of G-invariant bilinear forms modulo primes to the case of G-covariant forms.
An abelian threefold $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_7$-extension over ${\mathbb Q}$ with ramification index 7 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be a semistable abelian variety of dimension $3d$ and conductor $N^d$ with $B[2]$ filtered by copies of $A[2]$. We give a sufficient and computable class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$.
A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on ell-division fields of semistable abelian varieties A, mainly when A[ell] is reducible, by considering extension problems for groups schemes of small rank. Our general results imply, for instance, that the least prime conductor of an abelian surface is 277.
An abelian surface $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_5$-extension with ramification index 5 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be any semistable abelian variety of dimension $2d$ and conductor $N^d$ such that $B[2]$ is filtered by copies of $A[2]$. We give a sufficient class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$. As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in $\{277, 349,461,797,971\}$. The general applicability of our criterion is discussed in the data section.
Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let $\mathcal E$ be an absolutely irreducible group scheme of rank $p^4$ over $\mathbb Z_p$. We provide a complete description of the Honda systems of $p$-divisible groups $\mathcal G$ such that $\mathcal G[p^{n+1}]/\mathcal G[p^n] \simeq \mathcal E$ for all $n$. Then we find a bound for the abelian conductor of the second layer $\mathbb Q_p(\mathcal G[p^2])/\mathbb Q_p(\mathcal G[p])$, stronger in our case than can be deduced from Fontaine's bound. Let $π\!: \, {\rm Sp}_{2g}(\mathbb Z_p) \to {\rm Sp}_{2g}(\mathbb F_p)$ be the reduction map and let $G$ be a closed subgroup of ${\rm Sp}_{2g}(\mathbb Z_p)$ with $\overline{G} = π(G)$ irreducible and generated by transvections. We fill a gap in the literature by showing that if $p=2$ and $G$ contains a transvection, then $G$ is as large as possible in ${\rm Sp}_{2g}(\mathbb Z_p)$ with given reduction $\overline{G}$, i.e. $G = π^{-1}(\overline{G})$. One simple application arises when $A = J(C)$ is the Jacobian of a hyperelliptic curve $C\!: \, y^2 + Q(x)y = P(x)$, where $Q(x)^2 + 4P(x)$ is irreducible in $\mathbb Z[x]$ of degree $m=2g+1$ or $2g+2$, with Galois group $\mathcal S_m \subset {\rm Sp}_{2g}(\mathbb F_2)$. If the Igusa discriminant $I_{10}$ of $C$ is odd and some prime $q$ exactly divides $I_{10}$, then $G = {\operatorname{Gal}}(\mathbb Q(A[2^\infty])/\mathbb Q)$ is $\tildeπ^{-1}(\mathcal S_m)$, where $\tildeπ\!: \, {\rm GSp}_{2g}(\mathbb Z_p) \to {\rm Sp}_{2g}(\mathbb F_p)$. When $m = 5$, $Q(x) = 1$ and $I_{10} = N$ is a prime, $A = J(C)$ is an example of a $\textit{favorable}$ abelian surface. We use the machinery above to obtain non-existence results for certain favorable abelian surfaces, even for large $N$.
We study the arithmetic of division fields of semistable abelian varieties A over the rationals. The Galois group of the 2-division field of A is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in "Paramodular abelian varieties of odd conductor," arXiv:1004.4699.
Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(μ_\ell)$ be the cyclotomic field generated by the group of $\ell^{th}$-roots of unity. We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is "small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent. We show that if $\ell=2$ or 3 and $H$ is nilpotent then the reduction of $A$ at $p$ is totally toroidal, so its conductor is $p^{\dim A}$. The Jacobian of the modular curve $X_0(41)$ is a simple semistable abelian variety of dimension 3, with bad reduction only at $p=41$ and the Galois group of its 2-division field is a 2-group. For $\ell=2$, 3 or 5, there exist elliptic curves $E$ of prime conductor such that ${\bf Q}(E[\ell]) = {\bf Q}(μ_{2 \ell})$. We characterize the abelian varieties that are isogenous to products $E^d$.
We show that there do not exist semistable varietes defined over the rationals with good reduction outside one prime p if p = 2, 3, 5 or 7.