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By Hugh Schofield
BBC News, Paris
French police are convinced there has been a murder. The problem is, not only is there no corpse. No-one has even been reported missing.
In a mystery straight out of an Inspector Maigret novel, Normandy's state prosecutor has appealed to the public to help discover who it is that might be dead.
"In my 23 years as a magistrate, I have never seen a situation like this," said Rémi Coutin.
"We don't have a body," he said.
"And we don't have an identity for the person who we believe to have been killed."
What police do have is a suspect. He is a 46-year-old Polish man who has lived in France for many years, working as a carpenter. The unnamed man is currently in custody.
Police think that on 9 March, the suspect, driving a black Audi under the influence of alcohol, knocked over a woman cyclist on a road near the town of Grand Bourgtheroulde, 140km (85 miles) north-west of Paris.
Thinking he had killed her, he went home to get a spade. But when he came back, he found she was still alive. So the police believe he "finished her off" with the spade, then buried her body. The bicycle he disposed of at a rubbish dump.
There is precious little material evidence that any of this actually took place. There is no body, no confession to police, no bicycle. Above all, there are no reports of a woman cyclist going missing.
The reason the police are sure that a murder did happen is that the man initially told people who were near to him that he had done it.
On 14 May - more than two months later - the suspect's former girlfriend went to the police station in Dieppe. She told a story about how her ex had phoned her in a drunken state on 9 March, to tell her he had killed a woman in an accident.
That was his initial version. The suspect then called her back to say it was OK: the woman was alive and had gone home. Concerned, she went to see the suspect the next day. He was not at home, but his car showed damage to the windscreen and a large red stain.
Then on 13 March, she confronted the suspect in person, and this time he told the full story, including the killing with the spade and burial. The female cyclist, he told her, had been a bit like a clocharde or tramp.
Three other witnesses heard him give different versions of the accident. One woman friend who went to his house later on the same day found him frantically cleaning his damaged car.
To her, he said that the woman cyclist had survived. Nonetheless, the friend took pictures of the car, which are now a key part of police evidence.
Another incriminating fact is that the suspect declared that the Audi, which was found burned out in mid-April, had been stolen. In fact he later admitted to police that he himself had set it on fire.
Arrested in June, the suspect initially said that the story of the accident was a "bad joke", invented in order to make his ex-girlfriend take pity on him. The damage to the car he claimed to have inflicted himself, using chicken blood for the stain.
According to police, he subsequently admitted there had indeed been an accident, but that the victim was unharmed, before once more repeating the "bad joke" explanation.
To take the case further, police need to find out urgently who the victim may have been.
"Plenty of people go missing without the police being told," said prosecutor Rémi Coutin.
Among the possibilities are that the woman was a tourist or a foreign second home-owner, or maybe someone on a long-distance cycling tour. She could also have been living rough, or in isolation, cut off from her family.
The appeal to witnesses issued by police in Rouen describes her as between 40 and 60 years of age, wearing a backpack, and on a bike possibly equipped with saddle-bags.