Sharedwww / job / Short.texOpen in CoCalc
Author: William A. Stein
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3\newcommand{\doc}{Research Summary}
4\newcommand{\myname}{William A.\ Stein}
5\newcommand{\phone}{(510) 883-9938}
6\newcommand{\email}{{\tt was@math.berkeley.edu}}
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9Berkeley, CA  94709\\
10USA}
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63\begin{document}
64\begin{center}
65\LARGE\bf \doc
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70\mysection{Introduction}
71My research program reflects the essential interplay between abstract
72theory and explicit machine computation during
73the latter half of the twentieth century;
74it sits at the intersection of recent work of
75B.~Mazur, K.~Ribet, R.~Taylor, and A.~Wiles
76on Galois representations
77with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre
78on explicit computations involving modular abelian varieties.
79My work on the Birch and Swinnerton-Dyer
80conjecture for modular abelian varieties and search for new
81examples of modular icosahedral Galois representations
82has led me to discover and implement algorithms for
83explicitly computing with modular forms.
84
85\mysection{Objectives}
86The main outstanding problem in my field
87is the conjecture of Birch and Swinnerton-Dyer (\nobreak{BSD} conjecture),
88which ties together the constellation of
89arithmetic invariants of an
90elliptic curve.
91There is still no general class of elliptic curves for which
92the full BSD conjecture is known to hold.
93Approaches to the BSD conjecture that rely on congruences
94between modular forms are likely to
95require a deeper understanding of the analogous
96conjecture for modular abelian varieties, which are higher
97dimensional analogues of elliptic curves.
98
99As a first step, I have obtained theorems that make possible
100computation of some of the arithmetic invariants of
101modular abelian varieties.
102My objective is to find ways to explicitly compute all of
103the arithmetic invariants.
104Cremona has enumerated these invariants for the first
105few thousand elliptic curves, and I am working to do the same
106for abelian varieties.
107It is hoped that this work will continue to
108yield theoretical results.
109I am also writing modular forms software that I hope will be used
110by many mathematicians and have practical applications in
111the development of elliptic curve cryptosystems.
112
113My long-range goal is to give a general hypothesis, valid for
114infinitely many abelian varieties,
115under which the full BSD conjecture holds.
116My approach involves combining Euler system techniques of K.~Kato and
117K.~Rubin with visibility and congruence ideas of Mazur and Ribet.
118
119
120\mysection{Modular abelian varieties}
121My primary objective is to verify the BSD conjecture for
122specific modular abelian varieties, by using the
123rich theory of their arithmetic.
124
125The BSD conjecture asserts that if~$A$ is a modular abelian
126variety with $L(A,1)\neq 0$, then
127$$\frac{L(A,1)}{\Omega_A} = 128\frac{\#\Sha(A)\cdot\prod c_p} 129{\# A(\Q)_{\tor}\cdot\#\Adual(\Q)_{\tor}}.$$
130Here $A(\Q)_{\tor}$ is the group of rational torsion points on~$A$;
131the Shafarevich-Tate group $\Sha(A)$ is a measure of the failure
132of the local-to-global principle; the Tamagawa numbers~$c_p$ are the
133orders of certain component groups associated to~$A$;
134the real number~$\Omega_A$ is
135the volume of~$A(\R)$ with respect to a basis of differentials having
136everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.
137
138\mysubsection{The ratio $L(A,1)/\Omega_A$}
139Extending Manin's work on elliptic curves, A.~Agash\'{e} and I
140found a computable formula for the rational number $L(A,1)/\Omega_A$.
141Using similar techniques, I hope to find computable formulas
142for rational parts of special values of twists, and of $L$-functions
143attached to forms of weight greater than two.   I have
144already computed $L(A,1)/\Omega_A$ for several thousand abelian
145varieties, and hope to extend these computations.
146
147\mysubsection{The Tamagawa numbers $c_p$}
148When~$A$ has semistable reduction at~$p$, I have
149found a way to explicitly compute
150the number~$c_p$, up to a power of~$2$.
151I hope to find a way to compute~$c_p$ in the remaining cases.
152
153\mysubsection{Bounding $\#\BigSha$}
154V.~Kolyvagin and K.~Kato
155obtained upper bounds on~$\#\Sha(A)$.
156To verify the full BSD conjecture for certain abelian
157varieties, it is necessary is to make these bounds explicit.
158Kolyvagin's bounds involve computations with Heegner points,
159and Kato's involve a study of the Galois representations
160associated to~$A$.  I plan to carry out such
161computations in many specific cases.
162
163One approach to showing that~$\Sha(A)$ is as large as predicted
164by the BSD conjecture is suggested by Mazur's notion of
165the visible part of~$\Sha(A)$.
166Consider an abelian variety~$A$ that sits naturally
167in the Jacobian $J_0(N)$ of the modular curve $X_0(N)$.
168The {\em visible part} of $\Sha(A)$ is the collection of those
169elements of $\Sha(A)$
170that go to~$0$ under the natural map to $\Sha(J_0(N))$.
171Cremona and Mazur observed that if an element of order~$p$
172in~$\Sha(A)$ is visible,
173then it is explained by a jump in the rank of Mordell-Weil,
174in the sense that there is another abelian subvariety $B\subset J_0(N)$
175such that $p \mid \#(A\intersect B)$ and~$B$ has many rational points.
176I am trying to find the precise degree  to which this observation
177can be turned around: if there is
178another abelian variety~$B$ with many rational points and
179$p\mid \#( A\intersect B)$, then under what hypotheses
180is there an element of~$\Sha(A)$ of order~$p$?
181
182
183\mysection{Icosahedral Galois representations}
184E.~Artin conjectured
185that the $L$-series associated to any continuous irreducible
186representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.
187Recent exciting work of Taylor and others suggests that a complete proof
188of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,
189is on the horizon.
190
191By combining the main result of
192a recent paper of K.~Buzzard and Taylor
193with a computer computation,
194Buzzard and I recently
195proved that the icosahedral Artin representations of
196conductor~$1376=2^5\cdot 43$ are modular.
197If I can extend a congruence result of J.~Sturm, then
198 our method will yield several more examples.
199These ongoing computations are laying a part of the
200foundation necessary for a full proof of the Artin conjecture
201for odd two-dimensional~$\rho$, as well as stimulating
202the development of new algorithms for computing with modular forms
203in characteristic~$\ell$.
204
205\end{document}
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