Let $K$ be a global function field of characteristic $p>0$ and $\ell\neq p$ be a prime number. We study Selmer groups over a $\mathbb{Z}_\ell$-extension $K_\infty/K$. For a lisse $\mathbb Z_\ell$-sheaf we prove that the Pontryagin dual of the associated Selmer group is a finitely generated torsion module over the Iwasawa algebra and has $\mu$-invariant equal to zero. This gives a positive-characteristic, prime to $p$, analogue of Greenberg's $\mu=0$ conjecture. Our result applies in particular to abelian varieties, fine Selmer groups, and adjoint representations. We also prove an analogue of the weak Leopoldt conjecture in this context over $K_\infty$, and deduce that the framed deformation ring of a residual representation is a formal power series ring. The same conclusion holds for the unframed deformation ring if the residual representation has no non-scalar endomorphisms.
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\Q$ containing $\infty$ but not $p$, any continuous odd representation $G_{\Q,S} \to \GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where...
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
Let $p\geq 5$ be a prime, and let $f$ be a $p$-ordinary cuspidal newform of even weight $k = 2r \geq 2$ and level $N$, with $p \nmid N$ and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for $f$, formulated in terms of the $p$-adic $L$-function of $f$, under the sole assumptions that the as...
For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\...
We prove Mond's conjecture for $\mathcal A$-finite corank-one map germs $(\mathbb C^3,0)\to(\mathbb C^4,0)$: the $\mathcal A_e$-codimension is at most the image Milnor number, with equality for quasihomogeneous germs. For a quasihomogeneous germ we determine the derivations of its image modulo the conductor fields, in...
Rich'ard Rim'anyi· 0 citations
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