7 research outputs found
Numerical modular symbols for elliptic curves
We present a detailed analysis of how to implement the computation of modular symbols for a given elliptic curve by using numerical approximations. This method turns out to be more effcient than current implementations as the conductor of the curve increases
Numerical modular symbols for elliptic curves
We present a detailed analysis of how to implement the computation of modular symbols for a given elliptic curve by using numerical approximations. This method turns out to be more effcient than current implementations as the conductor of the curve increases
Integrality of twisted L-values of elliptic curves
Under suitable, fairly weak hypotheses on an elliptic curve
and a primitive non-trivial Dirichlet character , we show that the
algebraic -value at is an algebraic integer. For
instance, for semistable curves is integral whenever
admits no isogenies defined over . Moreover we give examples
illustrating that our hypotheses are necessary for integrality to hold
Mordell-Weil group as Galois modules
We study the action of the Galois group of a finite extension of
number fields on the points on an elliptic curve . For an odd prime , we
aim to determine the structure of the -adic completion of the Mordell-Weil
group as a -module only using information of over
and the completions of
Complete verification of strong BSD for many modular abelian surfaces over
We develop the theory and algorithms necessary to be able to verify the
strong Birch--Swinnerton-Dyer Conjecture for absolutely simple modular abelian
varieties over . We apply our methods to all 28 Atkin--Lehner
quotients of of genus , all 97 genus curves from the LMFDB
whose Jacobian is of this type and six further curves originally found by Wang.
We are able to verify the strong BSD Conjecture unconditionally and exactly in
all these cases; this is the first time that strong BSD has been confirmed for
absolutely simple abelian varieties of dimension at least . We also give an
example where we verify that the order of the Tate--Shafarevich group is
and agrees with the order predicted by the BSD Conjecture.Comment: 94 page