By Dan Abramovich, Jonathan Lubin (auth.), Dorian Goldfeld, Jay Jorgenson, Peter Jones, Dinakar Ramakrishnan, Kenneth Ribet, John Tate (eds.)

Serge Lang was once an iconic determine in arithmetic, either for his personal vital paintings and for the indelible effect he left at the box of arithmetic, on his scholars, and on his colleagues. Over the process his profession, Lang traversed a huge quantity of mathematical floor. As he moved from topic to topic, he came across analogies that resulted in very important questions in such parts as quantity thought, mathematics geometry, and the speculation of negatively curved areas. Lang's conjectures will continue many mathematicians occupied a ways into the longer term. within the spirit of Lang’s immense contribution to arithmetic, this memorial quantity includes articles through well known mathematicians in quite a few parts of the sphere, specifically quantity conception, research, and Geometry, representing Lang’s personal breadth of curiosity and impression. a distinct creation by way of John Tate encompasses a short and interesting account of the Serge Lang’s lifestyles. This volume's workforce of 6 editors also are hugely famous mathematicians and have been just about Serge Lang, either academically and in my opinion. the quantity is acceptable to analyze mathematicians within the components of quantity conception, research, and Geometry.

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**Sample text**

In view of the following lemma, A0 and B 0 provide an example of A and B respectively as above. J / we mean the ring of endomorphisms of J defined over Q. 1. J / preserves A0 and B 0 . Proof. J / does not preserve A0 (the case of B 0 is symmetric). Then since the Jf ’s are simple, that means that some abelian subvariety Jg of A0 is isogenous to some abelian subvariety Jh of B 0 , where g ¤ h. Pick a prime `. If f is a newform, then let f denote the canonical absolutely irreducible `-adic representation attached to f .

Stein Key words elliptic curves • abelian varieties • modular degree • congruence primes • multiplicity one Mathematics Subject Classification (2010): 11G05, 11610, 11G18, 11F33 1 Introduction Let E be an elliptic curve over Q. N /, where N is the conductor of E. N / ! N / (this can always be done by replacing E by an isogenous curve if needed). N / ! N / ! N /. N /; C/ be the newform attached to E. mod rE / for all n). Section 2 is about relations between rE and mE . For example, mE j rE . In [FM99, Q.

EndT M . Suppose m is a maximal ideal of T that satisfies the hypotheses of the lemma. To prove that Tm D T0m it suffices to prove the following claim: t u Claim: The map jT is surjective locally at m. Proof. It suffices to show that M is generated by a single element over T locally at m, and in turn, by Nakayama’s lemma, it suffices to check that the dimension of the T=m -vector space M=mM is at most one. N /=Fp /Œm. M=mM / Ä 1, which proves the claim. 12. , see [Dia97]). 8, all we needed was (locally) a non-zero free T-module (of finite rank, say) that is attached functorially to J .