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Register a new userAlgebraic maps constant on isomorphism classes of unpolarized abelian varieties are constant
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We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results are motivated by and shed light on a proposed construction of a cryptographic protocol for multiparty non-interactive key exchange.
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We prove formulas for the p-adic logarithm of quaternionic Darmon points on p-adic tori and modular abelian varieties over Q having purely multiplicative reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties.
Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(tau,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/tau B(k)$ varies when we take an unramified abelian cover $pi:Cto C$ defined over $k$.
Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $Delta(v)$ of $End(E(v))$ is divisible by $N$ and the ratio $Delta(v)/N$ is relatively prime to $NM$. We also discuss similar questions for reductions of abelian varieties. The subject of this paper was inspired by an exercise in Serres Abelian $ell$-adic representations and elliptic curves and questions of Mihran Papikian and Alina Cojocaru.
In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g geq 3$, there are infinitely many abelian varieties over $mathbb Q$ with adelic Galois representation having image equal to all of $operatorname{GSp}_{2g}(widehat{mathbb Z})$.
Let $k$ be a field of characteristic $q$, $cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p e q$ be a prime number and $cactocac$ a finite geometrically textsc{Galois} and etale cover defined over $k$ with function field $K:=k(cac)$. Let $(tau,B)$ be the $K/k$-trace of $A/K$. We give an upper bound for the $bbz_p$-corank of the textsc{Selmer} group $text{Sel}_p(Atimes_KK)$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $bbz$-rank of the textsc{Lang-Neron} group $A(K)/tauB(k)$. In the case of a geometric tower of curves whose textsc{Galois} group is isomorphic to $bbz_p$, we give sufficient conditions for the textsc{Lang-Neron} group of $A$ to be uniformly bounded along the tower.
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