# Consani–Scholten quintic explained

In the mathematical fields of algebraic geometry and arithmetic geometry, the Consani–Scholten quintic is an algebraic hypersurface (the set of solutions to a single polynomial equation in multiple variables) studied in 2001 by Caterina Consani and Jasper Scholten. It has been used as a test case for the Langlands program.

## Definition

Consani and Scholten define their hypersurface from the (projectivized) set of solutions to the equation

P(x,y)=P(z,w)

in four complex variables, where

P(x,y)=x5+y5-(5xy-5)(x2+y2-x-y).

In this form the resulting hypersurface is singular: it has 120 double points. Its Hodge diamond is

The Consani–Scholton quintic itself is the non-singular hypersurface obtained by blowing up these singularities. As a non-singular quintic threefold, it is a Calabi–Yau manifold.

## Modularity

According to the Langlands program, for any Calabi–Yau threefold

X

over

Q

, the Galois representations giving the action of the absolute Galois group on the

\ell
{"ft_lang_label":"__label__en","ft_lang_prob":0.90984017,"math_prob":0.96109384,"size":1580,"snap":"2023-14-2023-23","text_gpt3_token_len":395,"char_repetition_ratio":0.10786802,"word_repetition_ratio":0.0,"special_character_ratio":0.19683544,"punctuation_ratio":0.08243728,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99360764,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-02T21:02:15Z\",\"WARC-Record-ID\":\"<urn:uuid:2386a927-0e84-4bae-a94e-0da80326714a>\",\"Content-Length\":\"9996\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e2e81c9a-149a-4786-9772-bdc283f8ff71>\",\"WARC-Concurrent-To\":\"<urn:uuid:5f1ed12e-6e75-4886-be68-279d9bec7664>\",\"WARC-IP-Address\":\"85.25.210.18\",\"WARC-Target-URI\":\"https://everything.explained.today/Consani%E2%80%93Scholten_quintic/\",\"WARC-Payload-Digest\":\"sha1:JPI7XWHYUHRDQPJNIMAMW3EVRXACR5PN\",\"WARC-Block-Digest\":\"sha1:MUQYXK4SHWO7KNL3OIV6NNXHQ4ZPDPQG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224648858.14_warc_CC-MAIN-20230602204755-20230602234755-00327.warc.gz\"}"}