# \$2^k\$-Selmer groups and Goldfeld's conjecture, II

Posted in
Speaker:
Alexander Smith
Affiliation:
Harvard University
Date:
Tue, 2019-06-04 15:00 - 16:00
Location:
MPIM Lecture Hall

Choose some positive \$k\$ and a rational elliptic curve \$E\$, and choose \$k\$ pairs of primes \$(p_i, p'_i)\$. Take \$d_0\$ to be \$p_1 p_2 \dots p_k\$, and consider the family of \$d\$ given by replacing \$p_i\$ with \$p'_i\$ for some set of \$i\$. Under special circumstances, we show that \$2^k\$-Selmer elements of a twist \$E^{d_0}\$ can be constructed from \$2^k\$-Selmer elements of the remaining twists \$E^d\$. By elaborating on this strategy we show that, in a grid of twists of \$E\$, some information about the distribution of \$2^k\$-Selmer groups over this grid can be found from symbols whose values are subject to analytic control.

 © MPI f. Mathematik, Bonn Impressum & Datenschutz
{"ft_lang_label":"__label__en","ft_lang_prob":0.7822841,"math_prob":0.9593411,"size":721,"snap":"2019-35-2019-39","text_gpt3_token_len":213,"char_repetition_ratio":0.10041841,"word_repetition_ratio":0.0,"special_character_ratio":0.29819694,"punctuation_ratio":0.11034483,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97598004,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-18T07:32:20Z\",\"WARC-Record-ID\":\"<urn:uuid:2e2ac204-3503-4695-8d4b-bbc028d5fede>\",\"Content-Length\":\"19330\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f2031ad2-13ae-4f53-9d63-014bbeefe95b>\",\"WARC-Concurrent-To\":\"<urn:uuid:6f334642-f52e-4fef-ad21-778ccb275511>\",\"WARC-IP-Address\":\"195.37.209.187\",\"WARC-Target-URI\":\"http://www.mpim-bonn.mpg.de/node/9489\",\"WARC-Payload-Digest\":\"sha1:QQSW6WYVJGUFLQF33DLTNFLFAODEHB7B\",\"WARC-Block-Digest\":\"sha1:AS2R3LCVDNTX24RFVWSB62EB57K747IA\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573258.74_warc_CC-MAIN-20190918065330-20190918091330-00552.warc.gz\"}"}